ROZPRAWA DOKTORSKA

IMAGING OF NEAR SURFACE VELOCITYHETEROGENEITIES BY MEANS OF REFRACTION

TOMOGRAPHY

mgr inŜ. Nasar El Zawam

Promotor: prof. dr hab. in Ŝ. Zbigniew Kasina

KRAKÓW 2007

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Each type of seismic dataRequired it’s own

Specialized processing strategy.

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Author’s Acknowledgments

I would like to thank Prof. Zbigniew Kasina for his insight and direction throughout

this scientific process. He has aided me in gaining a tremendous amount of knowledge, but

more importantly, he has shown me what it takes to be a true scientist. Thanks goes to the

many workers in the Department of Geophysics who have helped me get things done in all

my four years at AGH. Special thanks goes to workers of Zakład Surowców Energetycznych ,

who have provided me opportunity with countless hours of computer support to realize most

of the calculations using their facility ProMAX system. Very special and heart felt Thank

You goes to Mohamed and Aisha, my parents, for instilling the values of courage and trust,

and for making me believe that anything is possible if you just put your mind to it. Thank

You goes to my Brothers and Sisters, for without them nothing is possible. Last, but definitely

not least, I would like to thank my wife Madiha for Her kind words of encouragement and her

smiling face gave me a reason to fight on.

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CONTENTS

INTRODUCTION ................................................................................................................. 5I. DIFFERENT APPROACHES TO IMAGING THE NEAR SURFACE VELOCITY HETEROGENEITIES IN SEISMIC EXPLORATION ................................. 8II. THE DESCRIPTION OF THE APPLIED PROGRAMS .................................................. 16 II.1. Promax’s programs for velocity model editing, seismic modeling, turning ray tracing, turning ray tomography, interactive first break picking and database operations ............................................................................... 16 II.2. Fortran programs for direct and inverse problem of the head wave tomography ..... 38 II.2.1. The description of the program #RAYEDH for direct problem of the head wave tomography II.2.2. The description of the program # INVERSDH for inverse problem of the head wave tomography II.3. Auxiliary Fortran programs ....................................................................................... 64II.3.1. The program for conversion ASCII velocity files from ProMAX

to binary files and text SURFER files II.3.2. The program for conversion binary velocity files from # INVERSDH to text SURFER files II.3.3. The program for calculation RMSDV (Root Mean Square velocity error) II.3.4. The program for conversion ASCII files with picks from ProMAX to text FORTRAN format II.3.5. The program for 2D (spatial) smoothing of velocity fields II.3.6. The program for statics calculations II.3.7. The program for turning ray tracing II.4. Programs for graphical presentation ........................................................................ 84III. RESULTS OF THE SOLUTION OF DIRECT AND INVERSE PROBLEM OF THE REFRACTION TOMOGRAPHY .................................................................. 88III.1. The ray approach and the wave approach to seismic wave propagation and their role in tomographic inversion ................................................................ 88 III.2. Imaging of near surface velocity heterogeneities of the medium in wave pattern of acoustic modeling ................................................................................. 95 III.3. The solution of direct and inverse problem of the refraction tomography for selected models and statics estimation ................................................................. 102 III.3.1. The two layer model with gradient medium over refractor (depth 176 m) III.3.2. The two layer model with gradient medium over refractor (depth 120 m) III.3.3. The model of gradient half-space III.3.4. The two layer model with constant velocities III.3.5. The three layer model with constant velocities III.3.6. The four layer model with constant velocities III.4. The analysis of the results of direct and inverse problem solution for model data ................................................................................................... 156 III.5. The tomographic inversion of first breaks on field records ............... .............. 158CONCLUSIONS ............................................................................................................. 167REFERENCES ................................................................................................................. 168

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INTRODUCTION

The first layer below the ground surface, known as the weathering layer (or Low

Velocity Layer - LVL), is generally a thin layer made up of low velocity, unconsolidated

material. The investigation of LVL may be undertaken for civil and mining engineering

projects as well as in connection with structural investigations.

In the application to civil engineering projects, the most common aim is the

determination of the depth to the bedrock on dam sites and along high pressure tunnel routes.

In mining, determination of the LVL thickness may be required in order to plan safely the

underground operations or to determine the thickness of the overburden to be stripped if

opencast operations are planned. During seismic structural investigations lateral variations in

the thickness or velocity of the weathering layer can corrupt the continuity of events on the

stacked section and, perhaps more seriously, can introduce apparent structure into deep

reflectors. To avoid these effects we must define and introduce so called static corrections to

the seismic records. Before this process the model of the weathering layer must be estimated.

The recognition of the structure of near surface layer and its velocity distribution plays

the main role in estimation of field static corrections specially in vibroseis seismics. Field

statics and residual statics define total statics corrections. The accuracy of static corrections

estimation has essential effect on results of many processing procedures. The most important

of them are velocity analysis, stacking and migration. The errors in static corrections

estimation are the source of serious structural interpretation errors as well as the source of

decreasing of the effectiveness of many advanced seismic procedures: inversion, AVO which

play main role in direct hydrocarbon prospecting and in stratygraphic interpretation.

The main source of our knowledge about LVL are shot holes, uphole times in

dynamite seismics, velocity measurements in deep shot holes, well to well measurements,

interpretation of first breaks by means of refraction methods (GRM method, Delay Time

method and others) and by means of tomographic inversion (head wave tomography, turning

ray tomography, attenuation tomography). We can also use surface waves to LVL

recognition.

The refraction tomography - comprising head wave tomography and turning ray

tomography - may be treated as one of the most important inversion method of LVL

recognition in the case when we have no holes or wells to our disposal (Bohm 2006, Bridle

2006, De Amorim 1987, Ditmar 1999, Docherty 1992). Such a situation is commonly met

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when we have only to our disposal results of land vibrator seismics and we must obtain all the

information from first breaks on seismic records.

The effectiveness of tomographic inversion of first breaks depends on many

parameters of seismogeological models of LVL, on many parameters of seismic acquisition

and on many parameters of tomographic inversion. The main goal of the presented doctor

thesis is the estimation of the effect of these parameters on results of tomographic inversion of

head waves arrivals and on the accuracy of field statics estimation using model data

simulating mainly standard reflection land records. The application of the model data for

analysis of the tomographic inversion effectiveness is well known and commonly used

procedure (Bridle 2006, De Amorim 1987, Ditmar 1999, Docherty 1992, Rajasekaran 1996,

Scott 1991, Zhu 1992). Additionally the comparison will be carried out between results of

tomographic inversion of head waves and the results of turning wave tomography on selected

model data. Although the presented work has mainly the form of model study some results of

field data processing will be presented too.

The input model data were constructed using the advanced processes of seismic data

processing system ProMAX (Interactive Velocity Editor, Finite Difference Modeling,

Turning Ray Tracing) and using the original Fortran programs constructed in the Department

of Geophysics (University of Science and Technology AGH Faculty of Geology, Geophysics

& Environmental Protection). The tomographic inversion of turning wave arrivals was realized

using ProMAX tomographic processes (Create Turning Ray Velocity, Turning Ray Tracing,

Compute Residual Travel Times, Turning Ray Tomography). The tomographic inversion of

head wave arrivals was realized using original Fortran programs constructed in the

Department of Geophysics. All the original basic and auxiliary Fortran programs used in my

work have been created with my contribution in the phase of their numerical testing and

optimizing. It must be stressed that programs of head wave tomography are not actually

available in seismic data processing systems used in Poland in main geophysical companies

(system ProMAX and system OMEGA ). Only variants of turning wave tomography and

reflection tomography are available.

Access to the ProMAX system on Faculty of Geology, Geophysics & Environmental

Protection was possible within Landmark University Grant Program sponsored by Landmark

Graphics Corporation.

The following chapters of my work include: analysis of different approaches to

imaging the near surface velocity heterogeneities in seismic exploration, the theoretical

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principles of applied programs and description of input/output data, presentation of the

seismogeological models used in the analysis, results of the solution of direct problem

(defining of traveltimes and seismic ray trajectories) and inverse problem of refraction

tomography (determination of velocity fields from traveltimes), results of field statics

calculations.

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I. DIFFERENT APPROACHES TO IMAGING THE NEAR SURFACE VELOCITY HETEROGENEITIES IN SEISMIC EXPLORATION

Shallow refraction seismics has long been used for the determination of the near

surface layer structure. The most common goal of this investigation in seismic prospecting for

gas and oil was the definition of the static corrections. The model of near surface layer was

the result of refraction interpretation allowing to derive estimates of the thicknesses and

velocities of the near-surface layers by analyzing the first breaks of head waves on the field

records. Conventional analysis of first-break data makes use of intercept times and inverse

slopes of the refracted-arrival segments of traveltime-distance graphs to interpret the depth

and velocity structure of the shallow subsurface. Several methods have been proposed for the

interpretation of refraction data, such as the intercept–time method, the wavefront-

reconstruction method (Thornburgh 1930), the plus-minus method (Hagedoorn 1959), the

general reciprocal method (Palmer 1980, 1981), the delay time method (Barry 1967). All

these methods are very useful tool of seismic interpretation and are still used to define starting

model in more advanced modern interpretation techniques based on generalized linear

inversion (Hampson and Russell, 1984) or on tomographic inversion. However, all these

methods have certain drawbacks restricting their range of applications. First of all they were

designed only for interpretation of refraction data and it was not simple to include other types

of waves (for instance reflected waves). They cannot detect velocity inversions (a low-

velocity layer beneath a high-velocity) and cannot to resolve thin beds (known as the hidden

layer problem). The interpretation of velocity increases with depth within a layer can be

problematic with some implementations. Most of the refraction techniques were designed to

compute static corrections for a constant velocity weathering layer of slowly changing

thickness overlying a refractor of constant velocity. When these conditions do not exist, then

unacceptable errors arise in the computed statics.

Nowadays the CDP method (Common Depth Point method) is the dominating method

of surface acquisition. When the dynamite seismics is applied we measure so called uphole

time by using uphole geophone near each shot hole. This information may be used to estimate

static corrections. Additionally we can realize special velocity surveys in deeper shot holes

and use their results (traveltimes) to derive velocity model of LVL. Their is no such

a possibility in the case of Vibroseis method. In this case we must practically obtain all the

information about LVL only from reflection field records. This information is mainly inherent

in first breaks of refraction arrivals.

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The generalized linear inversion of first breaks (Hampson and Russell, 1984) is from

many years treated as one of the most effective tools of refraction statics estimation. In this

approach, an input model is designed using standard refraction interpretation and theoretical

first-break traveltimes are computed. The model is then perturbed iteratively until the

computed and observed traveltimes match according to some squared-error criterion.

Theoretical first break arrivals are calculated by ray tracing from each shot to each receiver.

Theoretical times are compared to pick first break times and the error is then minimized using

the GLI algorithm. As the error is minimized, the model is updated so that it matches the true

earth more correctly. The computer program for computation statics using GLI algorithm is

commonly known as GLI2D in two dimensional variant or GLI3D in three dimensional

variant.

There are still some limitations of the approach based on GLI algorithm. The

interpretation model used is the layered one limiting the range of velocity models taken into

consideration. The velocity in each layer cannot vary in vertical direction. The velocity in the

following layers must increase with depth. The number of the layers must be established

before interpretation. To overcome these limitations new solutions based on tomographic

inversion have been proposed for determination of the LVL structure and static corrections

(Bohm 2006, Bridle 2006, De Amorim 1987, Ditmar 1999, Docherty 1992). The static

corrections based on tomographic approach are named tomostatics (Zhu 1992).

Tomostatics has advantages over traditional refraction statics in regions where it is not

easy to identify refractors and where we can meet velocity inversion. The tomographic

method enables us to consider complex geological models with dipping or variously curved

layers and with strong lateral velocity variations and rough topography. Model

parametrization is much more flexible. The next advantage of this method is the possibility of

jointly inverting the different kinds of waves generated within a seismic experiment (turning

waves, head waves, reflected waves). Including different kind of waves increases the effective

aperture in the data providing much more reliable solutions. The velocity fields resulting from

tomographic inversion may be used for statics estimation as well as for determination of

migration velocity field in shallow part of the medium.

The tomographic inversions may be classified from different points of view. Using

different types of waves we can consider turning wave tomography, head wave tomography

and reflection tomography. Tomostatics is mainly based on turning wave tomography and

head wave tomography. Taking into account the theoretical principles of tomographic

inversion we can distinguish between ray tomography (traveltime tomography) and

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diffraction tomography (Lo, Inderwiesen 1994). The first one is based on ray approach to

wave propagation well known from theory of seismic modeling. Such a approach is

commonly applied in all production applications of seismic tomography in seismic

processing systems including system ProMAX and OMEGA used in Poland. The

diffraction tomography and wave-equation tomography based on the wave approach found

only limited applications in seismic prospecting (Nolet 1987, Pratt, Tura et al. 1994,

Woodward 1992; Harris, Wang 1996). Taking into account the main purposes of seismic

tomography we can distinguish between traveltime tomography and amplitude tomography

(attenuation tomography). The main goal of the traveltime tomography is determination of the

velocity distribution in the medium using propagation time of different waves (Scott, Fehler

1991, Michelena et al. 1993). The main goal of the amplitude tomography is to define the

attenuation distribution using amplitudes (Brzostowski, McMechan 1992, Liao, McMechan

1997, Neep et al. 1996, Leggett et al. 1993) or spectral characteristics of the waves (Quan,

Harris 1997, Chris et al. 2004).

Among different types of tomography only the traveltime tomography in the variants

of turning wave and head wave tomography have been widely applied to determination of the

near surface velocity characteristics and statics calculations. The typical tomographic

inversion of traveltimes based on ray approach consists of the following steps:

- determination of the traveltimes from field records using the picking procedure,

- construction of an initial velocity model using all available data,

- defining the theoretical traveltimes for assumed acquisition parameters and assumed initial

velocity field through the process of ray tracing (direct task of tomography),

- calculations the differences between theoretical and measured (picked) traveltimes,

- defining the corrections to an initial velocity field through the procedure of minimizing

the errors between the observed arrivals on field records and those computed by ray theory

through an initial model (inverse task of tomography),

- refining an initial velocity model using calculated corrections.

Effectiveness of the tomographic inversion of seismic data depends on many factors related to

geological structure, acquisition geometry as well as to interpretation methods. The most

important factors related to geological structure of medium are:

- shape and size of velocity heterogeneities,

- character of velocity variations (the occurrence of strong velocity gradients),

- position of velocity heterogeneities.

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The most important factors related to acquisition geometry are:

- the number of shots and shot interval,

- the number of receivers and receiver interval,

- position of shots in relation to position of receivers and velocity heterogeneities,

- dominant frequencies of seismic waves.

The most important factors related to tomographic inversion are:

- accuracy of traveltimes picking,

- the step of velocity discretization of the medium,

- the method of solving of the basic set of tomographic equations,

- constraints applied to velocity distributions and the way of taking into account

additional velocity information,

- the methods of smoothing of velocity fields (averaging, convolutional smoothing -

seismic quelling, regularization),

- the range of taking into consideration different components of wave field, mainly

different types of waves.

Different aspects of tomographic inversion applied to definition the near surface layer

characteristics and static corrections were analysed in many papers. The most important

conclusions will be presented below.

De Amorim et al. (1987) introduced the model of LVL consisting of an undulating

earth surface with a planar refracting horizon between two media divided into blocks of

constant velocity. Each block was of equal horizontal length and had an unknown constant

velocity. The traveltimes were computed for the waves that are refracted at the bottom of the

LVL between any source and receiver locations. These traveltimes were expressed in terms of

the velocities in the blocks. Testing various other models for the LVL with dipping and

curved refractor boundaries did not obtain improved field static corrections on the available

data. The L2 norm was used during inversion and the tomographic set of equation had typical

form:

∆tA∆V)IλAA( TT ˆˆˆˆ =+

where: Ir

- the unit matrix, $AT - transposition of coefficient matrix $A , λ - arbitrary

parameter, ∆V - column vector of unknowns (values of velocity corrections in the nodes of

computational grid), ∆t - column vector of traveltime differences (measured between recorded

and calculated - for assumed velocity distribution - traveltimes), $A - matrix of coefficients

defining relation between traveltimes and velocities.

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Olsen K.B. (1989) used the procedure aimed to model a laterally varying distribution

of velocities and the topography of n refracting interfaces. By the specification of n vertical

grid lines of equal spacing, each layer of the model is divided into a number of cells. The

upper limit of the model (the surface of the earth) is specified by known elevations. The cells

situated below the lower refracting interface are considered half-infinite. For each cell,

a constant velocity is determined in terms of the slowness. The depths to the base of the cells,

which define the refracting interfaces, are among the inversion parameters. The ray tracing

procedure is developed on the concept of minimal traveltime of the first arrivals. For each

station, the raypath representing the shortest traveltime of all the possible direct and refracted

waves from the shot is calculated. No refraction is assumed at the vertical cell boundaries.

Lawton (1989) used differences in first-arrival traveltimes between adjacent records in

multifold reflection surveys to compute the depth and velocity structure of near-surface

layers. The traveltime differences as a function of source-receiver offset provide a direct

indication of the number of refractors present, with each refractor being defined by an offset

range with a constant time difference. For each refractor, the time-difference value at

a common receiver from two shotpoints is used to partition the intercept time into the delay

time at each shotpoint. This procedure is repeated until the delay times at all shotpoints and

for all refractors have been computed. Refractor depths and velocities are evaluated from this

suite of delay times.

Zanzi and Carlini (1991) proposed a new method for refraction statics reducing the

computational time without reducing accuracy. The first arrivals, common-offset organized,

formed the data space. The method involves Fourier transforming any common-offset data

vector with respect to the common mid-point. As a result, the data are decomposed in

a number of subspaces, associated with the wave-numbers, which can be independently

inverted to obtain any wavelength of the near-surface model.

Docherty (1992) investigates the feasibility of computing the weathering model from

the traveltimes of refracted first arrivals. The problem is formulated in terms of the difference

in arrival time at adjacent receivers, resulting in a much sparser matrix for inversion. Lateral

variations in both the weathering thickness and velocity are sought. In most cases, it is

necessary to include a small number of constraints to obtain the true weathering model. Any

roughness in the solution that is not required to fit the data is most effectively removed using

a second difference smoothing technique. Two layers make up the model: a laterally

inhomogeneous weathering layer and a uniform, high speed refractor. The weathering layer is

divided into cells of constant velocity. Each cell is bounded above by the observation surface

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and below by the refractor. Boundaries between adjacent cells are vertical. The base of

weathering is described by a series of node points, joined by straight line segments. In this

study a constant refractor velocity is assumed.

Zhu et al. (1992) presented examples illustrated that turning ray tomography can

image near-surface velocities more accurately than refraction statics methods. The medium to

be imaged was discretized into a grid of small rectangular cells, each of which contains

a single velocity. Sources and receivers are both located on the surface. The updated

velocities were slightly smoothed (or damped) every few iterations. This was the Constrained

Damped Simultaneous Iterative Reconstruction Technique (CDSIRT). It was confirmed that

tomostatics is noticeably closer to the true statics where velocity inversions are significant.

Generally, long spatial wavelength statics appear to be estimated better using tomostatics,

although a tomostatics bias (bulk shift) exists with increasing depth. Due to damping and

smoothing in the tomography algorithm, the output image of a linear inversion was

remarkably robust to a wide range of reasonable initial models.

Stefani (1995) used turning-ray tomography for estimating near-surface velocity

structure in areas where conventional refraction statics techniques fail because of poor data or

lack of smooth refractor/velocity structure. The method comprises nonlinear iterations of

forward ray tracing through triangular cells linear in slowness squared, coupled with the

LSQR linear inversion algorithm.

Rajasekaran and McMechan (1996) performed the tomography on prestack time picks

using the simultaneous iterative reconstructive technique (SIRT) algorithm with modifications

to include reflected as well as turned rays. Traveltimes of head waves are well approximated

by rays turned in a small velocity gradient below a high contrast reflector, and so are included

automatically as a special case of turned rays. The reflections, which correspond to

predominantly near vertical propagation, define horizontal changes in the model, but not the

vertical changes. Conversely, the turned transmissions are better able to define the vertical

changes. Increasing the effective aperture by combining reflection and transmission data and

performing tomography on this composite data set produces a better image of the 2-D velocity

distribution.

Lanz et al. (1998) investigated the applicability of surface-based 2-D refraction

tomography (turning ray tomography) for delineating the geometry of a landfill. The depth of

the near surface model did not exceed 100 m. The velocity in this layer was defined by rapid

increases from about 1000 m/s to 1500 m/s. Geophone and source spacing were set to 2 and

8 m, respectively. Sampling interval 0,25 ms was used. The results have demonstrated that the

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tomographic refraction scheme may be an efficient means of studying the very shallow

subsurface but complementary geological and other geophysical data are required to make

interpretation reliable. Additonally the accurate first-break picking was critical because the

traveltime anomalies associated with waste disposal sites may be only a few milliseconds.

A two-stage picking strategy was proposed. In a first pass, the arrivals were determined with

a commercial phase picker based on neural networks. After visual inspection,

a crosscorrelation-based algorithm was used to refine these initial first-break picks. In the

process of ray tracing fast finite-difference eikonal solver has been used. The inversion cell

size was increasing from 1m to about 15 m.

Zhang and Toksöz (1998) presented a nonlinear refraction traveltime tomography

method that consists of a new version of the shortest path ray-tracing approach , a regularized

nonlinear inversion method that inverts “traveltime curves” rather than traveltimes alone, and

a Monte Carlo method for nonlinear uncertainty analysis of the final solution. Seismic

raypaths were defined by calculating the shortest traveltime paths through a network

consisting of nodes and representing the earth. They chose to solve an inverse problem that

explicitly minimizes data misfit as well as model roughness.

Ditmar et al (1999) developed a new algorithm for tomographic inversion of

traveltimes of reflected and refracted seismic waves. In the case of a very inexact initial

model, a ‘layer-by-layer’ inversion strategy was recommended as a first inversion step. It was

assumed that the model consists of several layers separated by interfaces represented by a set

of points connected by straight segments. Velocity distribution in each layer was described by

means of its own velocity grid, the layer being completely inside the grid. The velocity values

were specified at gridnodes; bilinear interpolation was used in between them.

Zhu (2002) summarized the results of tomostatics based on turning wave tomography.

The main conclusions are:

- if structures are relatively simple and the assumption for refraction statics is valid,

tomostatics will not be better than refraction statics;

- non-linear tomography is not sensitive to an initial velocity model but a good initial velocity

model improves the convergence rate;

- usually, the recording aperture (signed offsets) should be at least four times larger than the

desired imaging depth;

- if the first arrivals are contaminated by noise at near or far offsets, we should eliminate

those offsets;

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- it is recommended to remove any previously applied elevation and velocity statics before

tomostatics.

Bohm et al. (2006) used a joint inversion of both first and refracted arrivals in order to

obtain a well-resolved velocity field for the computation of static corrections. After the

analysis of the diving waves, they inverted the traveltimes associated with the refracted events

by using the velocity model obtained from the diving waves as the initial model. Also after

inverting the two refracted arrivals separately they used the resulting output velocity field as

a new initial model for jointly inverting again the direct arrivals and the traveltimes with the

first and second refracted waves, in order to obtain a more accurate velocity field in depth.

Bridle R., (2006) analysed the applications of refraction statics and tomostatics on test

lines. For longer deeper anomalies with irregular raypaths, refraction statics and tomostatics

were expected to provide major improvements; however, only marginal improvements were

observed. In the test line considered the refraction statics provided the best section visually in

terms of signal strength, sharpness and continuity, with a structure that seems geologically

reasonable. The image provided by tomostatics was similar in structure, but was much noisier.

However, only the tomostatics solution was able to focus some events in the most difficult

area.

Summaring the above presentation of tomographic publications we can state that the

application of tomographic inversion to imaging near surface heterogeneities and to

estimation static corrections is still the subject of many current investigations.

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II. THE DESCRIPTION OF THE APPLIED PROGRAMS

II.1. PROMAX’S PROGRAMS FOR VELOCITY MODEL EDITING, SEISMIC MODELING, TURNING RAY TRACING, TURNING RAY TOMO GRAPHY, INTERACTIVE FIRST BREAK PICKING AND DATABASE OPERATION S

During the presented analysis several programs of ProMAX system have been used

for the purposes of velocity model editing, seismic modeling, turning ray tracing and turning

ray tomography.

At the stage of constructing velocity fields the process named Interactive Velocity

Editor was used. This tool is designed for building and editing velocity fields and is useful for

creating a velocity and/or density field for finite difference modeling as well as for analyzing

and adjusting the results of a tomographic velocity inversion. The main approach to editing a

velocity field is to create a polygon and use the polygon to act on the velocity field. After

closing a particular polygon we can define a velocity field using option Input/Set Velocity and

then write the velocity field into the polygon using option Apply Velocity Edit. The example

of velocity model (gradient model) defined in the work window of Interactive Velocity Editor

is presented in fig. 2.1.

Fig. 2.1. The example of velocity model (gradient model) defined in the work window of Interactive Velocity Editor

The main parameters of the Interactive Velocity Editor process are:- the type of velocity functions to edit (interval velocity in depth was mainly used ),- CDP number on left and right edge of velocity field,- the spatial sampling of the output velocity field expressed in CDP’s,- the minimum and the maximum depth of the input velocity field (in meters),- depth sample interval of new velocity field (in meters).

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The presentation of work window of Interactive Velocity Editor is included in fig. 2.2.

Fig. 2.2. The presentation of interactive work window of Interactive Velocity Editor

The velocity field is the descrete one and is saved in the database as the velocity table

in which the velocity in the node of assumed grid is the function of CDP’s number and depth.

The constructed velocity field may be exported using special ProMAX ASCII format

including only nodes of the velocity grid which define velocity variations. Other nodes are

omitted.

The next applied ProMAX program was process named Finite Difference Modeling

(FDM). This process provides an accurate, but compute intensive method of forward

modeling the earth’s response. A highly complex velocity and density field may be easily

built using the Interactive Velocity Editor. FDM then divides the field into a very fine two

dimensional grid. Within each rectangular grid point, velocity and density values are

approximately constant, and a compressional wave can be accurately propagated through the

grid point. FDM offers two modes of modeling:

- Exploding Reflector being a poststack, zero-offset form of modeling,

- Point Source simulating the firing of a shot into an array of receivers with finite offsets.

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In the presented analysis the second mode of modeling was applied. FDM process is

the type of acoustic modeling (only P waves are used) resulting in the set of seismic traces.

The main parameters of the process are:

- the name of the input velocity/density field constructed in Interactive Velocity Editor,

- type of modeling mode (Exploding Reflector or Point Source),

- the maximum depth of field for modeling (in meters),

- length of time (in ms) to run modeling per shot,

- the dominant frequency that will be contained in the wavelet produced by this model,

- type of source wavelet (Zero/ Minimum Phase Ricker , Zero/Minimum Phase Ormsby,

Zero/Minimum Phase Klauder, Gaussian Wavelet),

- the grid adjustment factor changing the grid spacing and time steps,

- CDP spacing in meters,

- parameter of selection boundary condition at the surface (reflecting or absorbing multiples

boundary),

- CDP of left and right edge of modeling field,

- number of shots,

- CDP increment between shots,

- depth of the first shot (the same for all),

- number of receiver stations,

- CDP of the first receiver,

- distance (in CDPs) between shot and first group,

- receiver location increment (in CDPs).

The parametrization window of Finite Difference Modeling is presented in fig. 2.3. The

seismic record calculated with FDM using the velocity field from fig. 2.1 and parameters

from the fig. 2.3 are presented using ProMAX procedure Trace Display in fig. 2.4 .

The direct task of turning ray tomography in ProMAX system was solved using the

process named Turning Ray Tracing. This process produces the predicted travel times of

turning wave for a starting model which are compared with the picked travel times. It is also

used to produce the ray paths which are used in the tomographic inversion. In according with

the Promax manual process Turning Ray Tracing uses a Langan & Lerche style ray tracer,

which is a shooting method that traces rays through a gridded velocity field using triangles.

Since this ray tracer uses a shooting approach to find the ray from a source to a receiver, it

cannot trace rays into a shadow zone. The shooting approach works by first shooting a fan of

rays at all angles. Then the two rays that span each receiver are used to iterate to a receiver.

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Fig. 2.3. The parametrization window of Finite Difference Modeling

Fig. 2. 4. The seismic record calculated with FDM using the velocity field from fig.2.1 and parameters from the fig. 2.2 are presented using ProMAX procedure Trace Display

Ray tracers can fail to find a receiver for several reasons (to small width od the area with

relation to the position of first/last receiver, too small depth od the area, shadow zones). The

main parameters of the process are:

- the name of an interval vs. depth file from the database,

- option of creating a ray path file (needed in Turning Ray Tomography),

- the name of output ray path file to store in the RAYS database; although the ray paths are

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stored in the table database, they are not tables and cannot be viewed/exported/ or edited

using the menu features,

- the number of the first/last shot at which to start/end tracing rays,

- the shot increment for ray tracing,

- the nmber of arrivals for each receiver to store in the database, this value needs to be greater

than 1 to save the ray information when multiple arrivals reach a receiver,

- the order the arrivals are written to the database; the options are Minimum travel time,

Maximum travel time, Minimum amplitude, and Maximum amplitude; generally, a sort of

Minimum Travel Time First is used;

- CDP spacing of the velocity field in m, the x axis of the velocity field is defined by CDP

number,

- horizontal cell size in meters for ray tracing, values on the order of 75-150 m are

recommended,

- vertical cell size, it is recommended to make this the same as the horizontal cell size, or

a little smaller,

- maximum depth of velocity field to use for ray tracing, this depth should be large enough

to allow room for rays reaching the last receiver and beyond,

- buffer for ray tracing field beyond first and last station (enter the distance in CDPs to extend

the velocity field to the left and right of the first and last source or receiver positions, this

buffering is necessary for the fan shooting to send a ray,

- option of storing ray tracing information in the database under the default entry names,

- a robustness factor (a value greater than 1.0 will reduce program error, but it will increase

computation time, a reasonable range for this value is 0.3-4.0.

The parameterization window of Turning Ray Tracing is presented in fig. 2.5. The

results of Turning Ray Tracing process are saved in ProMAX Database and may by displayed

and edited using Database procedures. Example of such a display of calculated traveltimes are

presented in fig. 2.6 for the case of gradient model without anomaly (red graph) and with low

velocity anomaly (blue graph).

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Fig. 2.5. The parameterization window of Turning Ray Tracing process

Fig. 2.6. Example of a display of calculated traveltimes for the case of gradient model without anomaly (red graph) and with low velocity anomaly (blue graph).

If the depth of velocity model is too small the achieving receivers on far offsets may

be impossible. In this case traveltimes are not estimated and in Database on the positions of

distant receivers (stations) we have constant time extrapolated from last achieved receiver.

This case is illustrated in fig. 2.7 (horizontal part of the left branch of the traveltime graph)

and confirmed in fig. 2.8 by behaving of ray trajectories determied with the help of Fortran

programs (#RAYEDT and #RAYPLTT).

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Fig. 2.7. Example of a display of calculated traveltimes for the case of gradient model without anomaly when the model depth is too small for assumed velocity gradient; vertical axis – traveltimes in ms, horizontal axis – channel number

The presented problem of turning ray tracing by means of Turning Ray Tracing

procedure is very important in the case of low velocity layer which practically has small

thickness and the problem of achieving distant receivers arises.

Fig. 2.8. Example of a display of calculated ray trajectories for the case of gradient model without anomaly when the model depth is too small for assumed velocity gradient

The next two ProMAX programs were used before tomographic inversion. The first

named Create Turning Ray Velocity produces an approximate interval velocity model for the

ray tracing from the first break picks in depth using flat layer theory of refracted arrivals. This

approach is sometimes known as the slope/intercept method for refractor inversion. The travel

time versus offset points are converted to refractor velocity/depth under the assumption that

- -23

velocity is monotonically increasing in depth. Then the process Create Turning Ray Velocity

is used in the case of layered model when the velocity increases with depth and all picks of

head waves arrivals are available. In practice it is difficult to have all these conditions

satisfied.

The second ProMAX program named Compute Residual Travel Times subtracts the

predicted travel time database values from the picked travel time database values and stores

the result in a new database value. These subtracted values are the residual travel times used

for tomographic inversion. The picking of the first breaks is realized using interactive

procedure available from the work window of Trace Display procedure. The picker is

activated from the mouse button and is the type of the crosscorelation picker. The resulting

picks are saved to database file.

The last and the most important in my analysis ProMAX program is named Turning

Ray Tomography. This is the main program of the Turning Ray Tomography package that

performs the velocity inversion. It takes the travel time deviations between picked travel times

and those predicted for a starting model to adjust the velocity along the ray paths of the

starting model to reduce the travel time deviations. Following manual we can say that this

process is an inversion and should not be considered automatic. Inversions are very error

prone. Artifacts can cause problems except in the best of situations. Inversion should be

treated as a mathematical aid that assists the interpreter, but it is the interpreter making the

final decision. The main parameters of Turning Ray Tomography comprise:

- minimum eigenvalue to invert,

- horizontal smoothing (in meters),

- vertical smoothing (in meters),

- damping factor (in seismic ray number),

- maximum residual travel time to use (in ms),

- norm (L2 or L1) used in the process of minimization of the objective funtion,

- defining model weighting input to the inversion.

The parameterization window of the Turning Ray Tomography is presented in fig. 2.9. The

problem of selection the optimal values of the input parameters of Turning Ray Tomography

process is essentially important in the case of near surface thin layer where we have small

distances between receivers and medium heterogeneties. To select the proper values of

inversion parameters in the analyzed case the analysis has been accomplished

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Fig. 2.9. The parameterization window of the Turning Ray Tomography

(Kasina, Zawam 2006) on synthetic records generated with the finite difference method in the

acoustic variant.

The construction of input data to tomographic inversion

Several gradient models of near surface medium with velocity anomalies have been

used in the analysis of tomographic inversion effectiveness. The first model was the gradient

one with low velocity zone (Fig. 2.10a). The velocity was increasing from 800 m/s near the

surface to 2900 m/s for the maximum depth 300 m. The zone with low velocity (800 m/s)

was located in the range of depth 60-100 m and its width was 104 CDP (for CDP interval

equal 5 m). The velocity model used in the modeling process contained 1200 CDPs. In the

second model (fig. 2.10b) with high velocity anomaly the constant velocity layer was

embeded in the gradient model with velocity 2500 m/s on depth 160 m and the velocity in the

anomaly zone was changed from 800 m/s to 1500 m/s. For such a model we can observe the

turning wave in the first gradient layer and the head wave generated on the boundary between

first and second layer.

To model the full wave pattern of records with the finite difference method in acoustic

variant the procedure Finite Difference Modeling (FDM) was applied in the processing

system ProMAX. The surface point source was assumed with source signal in the version of

minimum-phase Ricker signal with pick frequency 40 Hz. In the modeling process the traces

of 10 split-spreads have been generated. Each spread contained 601 receivers with receiver

interval equal 10 m (2 CDPs). The shot interval was equal 100 m (20 CDPs). The modeling

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has been performed fo the following effective dimensions of the grid: ∆x = ∆z = 2.66 m and

for the time interval equal ∆t = 0.53 ms. These values of modeling parameters let us to avoid

numerical errors and to secure calculation stability. The selected synthetic record calculated

for the first velocity model (model with low velocity anomaly) is showed in Fig. 2.11. The

selected synthetic record calculated for the second velocity model (model with high velocity

anomaly) is showed in Fig. 2.12.

Fig. 2.10a. Velocity model of the gradient medium with low velocity anomaly (v = 800 m/s); horizontal axis – CDP’s number, vertical axis – depth in meters

Fig. 2.10b. Velocity model of the gradient medium with high velocity anomaly (v = 1500 m/s); horizontal axis – CDP’s number, vertical axis – depth in meters

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Fig. 2.11. Theoretical wave pattern of selected record obtained with FDM modeling for the first velocity model (from Fig. 1) with low velocity anomaly with results of first break

picking marked (red); horizontal axis – offset in meters

Fig. 2.12. Theoretical wave pattern of selected record obtained with FDM modeling for the second velocity model with high velocity anomaly with results of the first break picking marked (red); horizontal axis – offset in meters

The nest step of preparation input data to tomographic inversion was the defining the

traveltimes of turning ray wave for the starting velocity model. The starting models were

created from original velocity models (fig. 2.10a, 2.10b) by removing the anomalous zone.

Additionally, one starting gradient model in the case of second velocity model has been

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created using interpretation of first breaks on model records without velocity anomaly by

means of Create Turning Ray Velocity procedure available in ProMAX system.

Fig. 2.13. The traveltimes graphs of turning wave (defined with picking first breaks – blue) in the medium with high velocity anomaly and calculated traveltimes (red) in the medium without velocity anomaly; horizontal axis – trace number, vertical axis – traveltimes in ms

The traveltimes for starting velocity model have been defined using procedure Turning

Ray Tracing in ProMAX system or have been identified with first breaks traveltimes picked

on the records without velocity anomaly. The example of the graph containing first breaks

traveltimes from records with high velocity anomaly and the traveltimes calculated for the

second starting velocity model by means of Turning Ray Tracing is illustrated in Fig. 2.13.

The last step of preparation input data to tomographic inversion was to define time

differences between picked and calculated (for starting velocity models) traveltimes using

procedure Compute Residual Travel Times available in ProMAX system.

The tomographic inversion in discussed variant of turning waves is realized in

ProMAX system with Turning Ray Tomography procedure. According to the well known ray

tomography theory during the consecutive tomographic inversions, in which the resulted

velocity field of previous inversion is – after process of smoothing and adding constraints –

the input velocity field to the next inversion, we obtain – for correct selection inversion

parameters – the reducing of the differences between picked and calculated traveltimes. These

differences are named residual traveltimes and their reduction is the basic criterion of

evaluation the inversion effectiveness. The first step before inversion is to select the proper

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parameters of inversion. The basic principles of parameter values selection are presented

below in accordance with the instructions of ProMAX manual:

- the value of ray damping should be equal a fourth of the average ray density;

- if the residual travel times are not improved, then we use a smaller eigenvalue or less

smoothing,

- the L1 norm is useful when your travel time picks are noisy or have many bad picks,

- variable model weighting can be used to disallow velocity variations in certain parts of the

model and encourage them in others; we use the values of weights from the

- a smaller value of eigenvalue inverts more subtle features, even features with some imposed

smoothing, but increases noise effects; this value should be about 0,05 – 0,2,

- the scale of velocity field smoothing (parameter horizontal smoothing) should be decreased

in consecutive iterations.

The analysis of the tomographic inversion effectiveness

The main goal of realized calculations was the evaluation the effect of selection of

tomographic inversion parameters on the accuracy of reconstruction of near surface velocity

fields. The precision of such reconstruction has been evaluated by means of the analysis of

the resulted velocity fields, the difference between resulted and assumed velocity fields as

well as the differences between picked and theoretically calculated traveltimes. These time

differences are named residual traveltimes. Three consecutive iterations of tomographic

inversion have been accomplished for the selected parameters values and then the value of

selected parameter has been changed.

Three iterations of tomographic inversion have been accomplished for established

smoothing parameters of horizontal smoothing (600 m) and vertical smoothing (75 m) for

established minimum eigenvalue 0,2. The resulted velocity field is showed in Fig. 2.14. The

anomalous velocity zone which assumed position is marked by white rectangle is difficult to

interpret. The error of velocity estimation in this zone reaches values 100 - 230 m. The graph

of residual traveltimes is presented in Fig. 2.15. These times have been reduced from the

starting values (–4) ms – 24 ms to the values 0 ms – 13 ms after third inversion. The

differences between residual traveltimes after second and third inversion were very small.

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Fig. 2.14. The velocity field after 3 tomographic inversions without changing the parameter of horizontal (600 m) and vertical (75 m) smoothing; horizontal axis – CDP number, vertical axis – depth in meters

Introducing variation of horizontal smoothing parameter in consecutive inversions from

value 600 m, through value 300 m to the value 150 m after third inversion improved not so

much reconstruction of anomalous zone (Fig. 2.16) at the cost of some deformation of the

deepest part of velocity field below anomaly. We can observe essential decreasing the

residual traveltimes for the largest offsets to the value about 2 ms after third inversion

(Fig. 2.17).

In the next step the calculations have been accomplished for established horizontal

smoothing (600 m) and varying value of vertical smoothing through values 75, 50 and 25 in

consecutive inversions. We observe decreasing the residual traveltimes in the range of

intermediate offsets. The application of vertical smoothing 25 m and varying horizontal

smoothing parameter in consecutive inversions through values 600, 300 and 150 m brought

the best reconstruction of anomalous zone (Fig. 2.18) and caused the essential reduction of

residual traveltimes for small, intermediate and large offsets after third inversion (Fig. 2.19).

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Fig. 2.15. The graphs of residual times before first inversion (orange), after first inversion (green), after second inversion (red), after third inversion (blue); parameter of horizontal smoothing 600 m, parameter of vertical smoothing 75 m; vertical axis – residual time in ms

Fig. 2.16. The velocity field after 3 tomographic inversions for established value of vertical smoothing (75 m) and varying horizontal smoothing through the values 600, 300 and 150 m in consecutive inversions; horizontal axis – CDP number, vertical axis – depth in meters

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Fig. 2.17. The graphs of residual traveltimes calculated for established value of vertical smoothing (75 m) after first inversion (green), after second inversion (red), afer third inversion (blue) for varying horizontal smoothing through the values 600, 300 and 150 m in consecutive inversions; vertical axis - residual traveltimes in ms

Beside the smoothing parameters the effect of other parameters have been analyzed

too. Decreasing the minimum eigenvalue from value 0,2 to value 0,1 and established remaning

parameters (horizontal smoothing 600 m, vertical smoothing 75 m) caused some lowering

residual traveltimes for the largest offsets. The effect of changing damping parameter marked

only for intermediate and largest offsets. Among four values of this parameter (5, 10, 15, 20)

for the value 5,0 we can observe noticeable decreasing of residual traveltimes. Application the

norm L1 instead of the norm L2 didn’t introduce – as one could expect for data with good

quality of first break picks – essential changes.

In the above analysis it was assumed that the starting velocity model is the exact

gradient model of the medium without anomaly. Two starting gradient models have been

additionally considered to evaluate the effect of error of defining starting model: one with

vertical gradient increased by 10% (gradient value 7,7 m/s per meter) and the second one with

gradient decreased by 10% (gradient value 6,3 m/s per meter). Differences of residual

traveltimes for these two cases were essential only after first inversion. After second and

third inversion the residual traveltimes practically were the same. Lowering the value of

parameter named Maximum travel time residual to use from 10 to 5 after second and third

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inversion didn’t decreased the maximum values of residual traveltimes in the zone of

intermediate offsets.

Fig. 2.18. The velocity field after 3 tomographic inversions for established value of vertical smoothing (25 m) and varying horizontal smoothing through the values 600, 300 and 150 m in consecutive inversions; horizontal axis – CDP number, vertical axis – depth in meters

In the above analysis it was assumed that the starting velocity model is the exact

gradient model of the medium without anomaly. Two starting gradient models have been

additionally considered to evaluate the effect of error of defining starting model: one with

vertical gradient increased by 10% (gradient value 7,7 m/s per meter) and the second one with

gradient decreased by 10% (gradient value 6,3 m/s per meter). Differences of residual

traveltimes for these two cases were essential only after first inversion. After second and

third inversion the residual traveltimes practically were the same. Lowering the value of

parameter named Maximum travel time residual to use from 10 to 5 after second and third

inversion didn’t decreased the maximum values of residual traveltimes in the zone of

intermediate offsets.

Summaring the analysis results in the considered case of gradient model with low

velocity anomaly we can state that:

- tomographic inversion let to locate the low velocity anomaly; the accuracy of positioning is

greater in the horizontal direction than in the vertical direction;

- the velocities in anomalous zone are reconstructed for considered iteration number with the

error about 19-25% (the velocities are generally too high);

- the greatest lowering of the residual traveltimes ca be observed after first inversion;

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- when the starting model is taken with error the model with lowered gradient is much better;

- using the gradual lowering of horizontal smoothing after consecutive inversions with lower

value of vertical smoothing and lowered minimum eigenvalue made it possible to obtain

reduction of residual traveltimes; this reduction appeared to be offset dependent.

Fig. 2.19. The graphs of residual traveltimes calculated for established value of vertical smoothing (25 m) after first inversion (green), after second inversion (red), afer third inversion (blue) for varying horizontal smoothing through the values 600, 300 and 150 m in consecutive inversions; vertical axis – residual traveltimes in ms

Summaring the analysis results in the considered case of gradient model with low

velocity anomaly we can state that:

- tomographic inversion let to locate the low velocity anomaly; the accuracy of positioning is

greater in the horizontal direction than in the vertical direction;

- the velocities in anomalous zone are reconstructed for considered iteration number with the

error about 19-25% (the velocities are generally too high);

- the greatest lowering of the residual traveltimes ca be observed after first inversion;

- when the starting model is taken with error the model with lowered gradient is much better;

- using the gradual lowering of horizontal smoothing after consecutive inversions with lower

value of vertical smoothing and lowered minimum eigenvalue made it possible to obtain

reduction of residual traveltimes; this reduction appeared to be offset dependent.

The final calculations have been accomplished to evaluate the effect of the model

parameters on tomographic inversion effectiveness. The second model of the medium with

gradient layer over constant velocity basement (2500 m/s) has been considered. The high

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velocity anomaly (1500 m/s) was embeded in the gradient layer. In the starting model velocity

anomaly has been removed from the gradient layer. The calculations of the theoretical

traveltimes have been done using two ways. One way was to calculate traveltimes with the

procedure Turning Ray Tracing. The second way was to pick first breaks on the synthetic

records obtained with Finite Difference Modeling procedure to create gradient medium using

procedure Create Turning Ray Velocity and then to calculate traveltimes with the procedure

Turning Ray Tracing. The calculated traveltimes have been used to define residual

traveltimes. The inversion has been accomplished for the following parameters: minimum

eigenvalue 0.1, damping factor 5, vertical smoothing 25 m, horizontal smoothing changing

from 600 m through 300 m to 150 m in the consecutive inversions. The inversions have been

realized with and without velocity weights.

The results of calculations for the case of inversion without velocity weights are

showed in Figs. 2.20-2.22. The location of velocity anomaly after third inversion is very good

in horizontal and vertical direction (Fig. 2.20). The errors of inversion presented in Fig. 2.21

in the form of map of differences between assumed and resulted velocities have generally

small values with the exception of central part of anomaly (error values about –200 m/s) and

the deepest part of the map in its central part (error values about 300 m/s). The residual

traveltimes are significantly reduced after third inversion from starting values about –20 ms to

values about ± 4 ms (Fig. 2.22). If we look at the ray density map (Fig. 2.23) after first

inversion we can see that the significant velocity errors occur even in the area of higher ray

density. If we treat this area as well illuminated and if we select it as the area with great

velocity weights (that means as the area in which significant velocity changes are allowed) we

will obtain after third inversion the velocity field showed in Fig. 2.24. We can observe some

improvement of velocity field reconstruction appearing as better smoothing its values in

deeper part of the velocity field without significant changes of residual traveltimes (Fig. 2.22).

Applications of the picked times in computations of theoretical traveltimes practically didn’t

change the results of inversion.

Conclusions

The realized model computations and the analysis of tomographic inversion results

made it possible to formulate the following conclusions:

- the procedures of tomographic inversion realized in seismic data processing ProMAX in the

variant of turning ray tomography may be efficiently used to reconstruction of near surface

gradient velocity fields with anomalous zones basing on the picked first breaks of reflection

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seismics records,

- the effectiveness of tomographic inversion appeared to be much better in the case of high

velocity anomalous zones embeded in the gradient medium,

- the selection of the proper values of tomographic inversion parameters made it possible to

improve the quality of velocity field reconstructions and to minimize the values of residual

traveltimes,

- the horizontal and vertical smoothing parameters appeared to be the most meaning ones

among others,

- we can observe a little better velocity reconstruction in horizontal direction then in vertical

direction,

- the errors of velocity anomaly reconstruction can achieve values of about 25% in the case

of low velocity anomaly and about 15% in the case of high velocity anomaly,

- we can observe significant reduction of residual traveltimes after first inversion in all

considered cases.

Fig. 2.20. The velocity field after 3 tomographic inversions in the case of gradient model with high velocity anomaly for established value of vertical smoothing (25 m) and varying horizontal smoothing through the values 600, 300 and 150 m in consecutive inversions; horizontal axis – CDP number, vertical axis – depth in meters

- -36

Fig. 2.21. The map of differences between assumed and resulted velocities after 3 inversions in the case of of gradient model with high velocity anomaly; the graph on right side is defined on the vertical line crossing the central part of velocity anomaly

Fig. 2.22. The graphs of residual traveltimes in the case of of gradient model with high velocity anomaly calculated for established value of vertical smoothing (25 m) after first inversion (green), after second inversion (red), afer third inversion (blue) for varying horizontal smoothing through the values 600, 300 and 150 m in consecutive inversions; vertical axis – residual traveltimes in ms

- -37

Fig. 2.23. The map of ray density in the case of of gradient model with high velocity anomaly with marked area used for velocity weighting; horizontal axis – CDP’s number, vertical axis – depth in meters

Fig. 2.24. The velocity field after 3 tomographic inversions in the case of gradient model with high velocity anomaly for established value of vertical smoothing (25 m),varying horizontal smoothing through the values 600, 300 and 150 m in consecutive inversions and application of velocity weighting; horizontal axis – CDP number, vertical axis – depth in meters

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II.2. FORTRAN PROGRAMS FOR DIRECT AND INVERSE PROBL EM OF THE HEAD WAVE TOMOGRAPHY

Two original Fortran programs have been tested and used to solve direct and inverse

task of head wave tomography. The first one named #RAYEDH was appropriated for

defining head wave trajectories and traveltimes and the second one named #INVERSDH was

appropriated for reconstruction velocity fields in the process of head wave traveltimes

tomographic inversion. These two programs work for discrete velocity fields and can be use

to solve direct and inverse task of curved ray tomography.

II.2.1. The description of the program for direct problem of the head wave tomography

The algorithm of #RAYEDH is based on the following assumptions defining the

interpretation model:

- we consider 2D velocity model of the medium in the plane (x,z) consisting of the two

layers separated by linear refractor,

- the velocities of the first layer are defined in the nodes of rectangular regular grid,

- the velocity below the refractor is assumed to be constant,

- the position of the flat refractor is defined by coordinates (x,z) of two its points,

- the positions of shot points are defined by (x,z) coordinates,

- the positions of receiver points are defined by (x,z) coordinates where receiver depth

„z” is constant for all receivers,

- the curvilinear trajectory of seismic ray propagating from the source is defined by Fermat

principle.

The assumption of constant receiver depth is some simplification but my analysis is

mainly directed towards estimation of the effect of velocity heterogeneities in the first layer

not topography.

The scheme of the seismic ray trajectory against a background of assumed grid is

presented in fig. 2.25.

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Fig. 2.25. The scheme of the seismic ray trajectory against a background of assumed grid and positions of source (PW) and receiver (PO)

The consecutive points of seismic ray trajectory are determined through solution of the

following set of differential equations resulting from Fermat’s principle (Kasina, 1988):

dv

vx

vz

dsα ∂∂

α ∂∂

α= −

1sin cos dx ds= cosα

dz ds= sinα dtdsv

=

where: dα - angle between tangents to a ray trajectory defined in two points located at

distance ds; v - velocity function; α - angle between x-axis and ray trajectory. That set of

equations has been solved using numerical Runge-Kutta method of fourth order. Interpolation

of velocity function and its derivatives has been performed using linear or cubic interpolation

and velocity values defined in the nodes of rectangular grid. Before interpolation the

estimation of the first and second spatial derivatives of the velocities in the nodes of assumed

grid is realized. If we introduce the velocity grid consisting of I columns and J rows and the

dimensions of the cell ∆x and ∆z then the coordinates of the nodes are defined by the

relations:

x = i ∆x, i = 0, 1, 2, ..., I

z = j ∆z, j = 0, 1, 2, ..., J

If we introduce the following notation for velocities in the nodes:

v(x,z) = V(i ∆x, j ∆z) = Vi,j

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then we can define the spatial derivatives of velocities using central differences:

v x zV V

xxi j i j' , ,( , )=

−+ −1 1

2∆

v x zV V

zzi j i j' , ,( , )=

−+ −1 1

2∆

v x zV V V V

x zxzi j i j i j i j'' , , , ,( , )=

− − ++ + − + + − − −1 1 1 1 1 1 1 1

4∆ ∆

where:

x

vv

z

vv

zx

vv xzxz ∂

∂∂∂

∂∂ ==

∂= ''

2'' ,, .

The execution of the #RAYEDH program consists of several steps. First , the critical

angle is estimated for assumed velocities on both sides of refractor under the shot. Then for

assumed range of starting angles from the shot the proper starting angle is estimated for which

incidence angle of the ray on refractor is equal the critical angle with assumed accuracy. The

trajectory of the ray with critical incidence angle is defined and its intersection with refractor.

Next the coordinates of the following points of the ray trajectory along refractor are defined

for assumed propagation interval. For each point on refractor the emergence angle of head

wave is estimated. Then the trajectory of head wave ray is defined from refractor to level of

receivers with assumed step of propagation along ray. The coordinates of intersections of all

rays with receiver level is estimated. For the variant of established positions of receivers the

trajectories of rays hitting receivers with assumed accuracy are established. When the

positions of all rays is defined the coordinates of trajectory points are saved and the

traveltimes are calculated from the source to each receiver in accordance with assumed

velocity field.

The example of ray trajectories against a background of assumed seismogeological

model and positions of source/receivers is fig. 2.26.

Program #RAYEDH includes the master segment and the following subroutines:

DIFR, INTRP, TRAF, TRAS, INT1, INT2, GRAF. In the master program the input data are

red, the parameters used in the subroutines are defined, the results of calculations are printed

and saved. The subroutine DIFR is appropriated for the estimation of spatial derivatives of the

velocity field. The subroutines INTRP, INT1, INT2 are used to linear or cubic interpolation of

velocities and their spatial derivatives. The subroutine TRAS is used to define the seismic ray

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Fig. 2.26. The example of ray trajectories against a background of assumed seismogeological model and positions of source/receivers

trajectories through solution of the set of differential equations resulting from Fermat’s

principle. The subroutine TRAF is used to define the starting angles of the rays propagating

from the shot to assumed receivers. The subroutine GRAF is used for simplified presentation

of input and output small velocity fields by means of character map. The input data for the

case of two shot points are red in the following sequence:

READ(1,*) NAME2READ(1,*) IPRINT, NRMOD, PLOT, ITP, IPRNTREAD(1,*) ITAPEIF(ITAPE.NE.0) READ(1,*) NAME7IF(IPLOT.NE.0) READ(1,*) NAME3IF(ITAPE.NE.0) READ(7) LW, T, DX, DZ, LWT, (V(I), I= 1, LWT)IF(ITAPE.EQ.0) READ(1,*) LW, LT, DX, DZREAD(1,*) DS, DXMAX, DZMAX, DALFOP, DF1, INTLREAD(1,*) WXPS1, WZPS1, WXPS2, WZPS2READ(1,*) LPG, WZG, WXG(1), DXGREAD(1,*) X1RF, Z1RF, X2RF, Z2RF, V2, V1S1, V1S2, DSRF, SRFMAX, DIREAD(1,*) ALFOS1, ALFGS1, ALFOS2, ALFGS2IF(ITAPE.NE.0) READ(1,*) NRR, (NR1(I), NR2(I),VN(I), I=1, NRR)

where:

NAZWA2 – the name of output text file for printing the results of calculations

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NAZWA3 – the name of output binary file used for plotting rays

NAZWA7– the name of input binary file with velocities

IPRINT - indicator of printouts:

IPRINT=0 - variant without control printouts

IPRINT=1 - variant with control printouts

NRMOD - model number

IPLOT - indicator of saving data for plotting

IPLOT=0 variant without saving data for plotting

IPLOT=1 variant with saving data for plotting

ITP - indicator of calculation option

ITP=0 calculation for assumed receivers

ITP=1 calculation for assumed values of "s" parameter (DSRF) along refractor

IPRNT - indicator of additional control printouts (ray points)

ITP=1 and IPRNT=1 - all control printouts without ray points

ITP=1 and IPRNT=2 - all control printouts with ray points

ITP=0 and IPRNT=1 - limited control printouts without ray points

ITAPE - indicator of input velocity data

ITAPE=0 - reading from text file

ITAPE=1 - reading from binary file

LW - columns number of velocity matrix

LT - rows number of velocity matrix

DX - the dimension of the cell in „x” direction (in meters)

DZ - the dimension of the cell in „z” direction (in meters)

LWT – number of grid nodes

DS – ray step (in m)

DXMAX, DZMAX – assumed error of hitting receiver (in m)

DALFOP – increment of starting ray angle from the source (in deg)

DF1 - minimal increment of starting ray angle from the source (very small value)

INTL - indicator of interpolation type

INTL=0 - cubic velocity interpolation

INTL=1 - linear velocity interpolation

WXPS1,WZPS1,WXPS2,WZPS2 – coordinates of shot points 1 and 2 (in m)

LPG - number of receivers

WZG - coordinates „z” of receivers (in m)

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WXG - coordinates „x” of receivers (in m)

DXG – receiver interval (in m)

X1RF, Z1RF, X2RF, Z2RF – coordinates of two refractors points (in m)

V2 – head wave velocity (in km/s)

V1S1,V1S2 – overburden velocity near refractor under SP. 1 and SP. 2

DSRF - assumed propagation step along refractor (in m)

SRFMAX - assumed maximum propagation distance along refractor (in m)

DI - assumed accuracy of reaching critical angle (in deg) (very small value)

ALFOS1, ALFGS1, ALFOS2, ALFGS2 – minimum and maximum starting angle

from the SP.1 and SP. 2; for SP. 1 (source on the left of the spread)) angles are

decreasing from ALFOS1 to ALFGS1; for SP. 2 (source on the right) angles are

increasing from ALFOS2 to ALFGS2

VN(I) - input velocities red from text file in km/s and defined in grid nodes

NR1(I) , NR2(I) - pair of grid nodes between which velocity is equal VN(I)

NRR - number of pairs (NR1,NR2)

The main output data for the case of two shot points are written – excluding the

writing of input parameters - in the following sequence:

IF(IPLOT.NE.0) WRITE(3) NRM

IF(IPLOT.NE.0) WRITE(3) IPS, IG, (I, XA(I), ZA(I), VA(I), I=1, NRM)

WRITE(2,140) IPS, IG, (I, XA(I), ZA(I), VA(I), I=1, NRM)

WRITE(2,150) ((J, I, T(I,J), I=1, LPG), J=1, LPS)

IF(IPLOT.NE.0) WRITE(3)((J, I, T(I,J), I=1, LPG), J=1, LPS)

where:

NRM – the number of ray points for selected shot IPS and selected receiver IG

XA(I), ZA(I), VA(I) – the coordinates of I-th point of the ray and the value of

velocity in that point; these values are saved in binary file

(NAME3 on unit 3) and in text file (NAME2 on unit 2)

LPS – number of shot points

LPG – number of receivers

T(I, J) – the values of calculated traveltimes of head wave

- -44

The main difference in input data for the case of many shots and split-spreads refers

reading the parameters of aquisition:

READ(1,*) LPS, (WZPS(I), I=1,LPS), (WZPS(I), I= 1,LPS)

READ(1,*) LPG, WZG, WXG(1,1), WXG(LPG/2+1,1), DPS, DXG

where:

WXPS(I), WZPS(I) – coordinates (x,z) of I-th shot point (in m)

WXG(I,J) - coordinate (x) of I-th receiver for J-th shot (in m)

WXG(1,1) – coordinate (x) of the first receiver of first spread (first shot)

WXG(LPG/2+1,1) – coordinate (x) of the first receiver of right half of the first

split-spread

WZG – coordinate (z) of receivers (in m)

DPS – shot interval (in m)

DXG – receiver interval (in m)

The example of input and output data are presented below.

Example of the input data of #RAYEDH for the case of 2 shots placed outside of the spread

m200_2w_1200_1800z100_200a.out0 1 1 0 01m200_2w_1200_1800z100_200a.velm200_2w_1200_1800z100_200a.PLT

0.8 0.05 0.05 1. 1.E-20 1

4000. 0. 8050. 0.

700.4450. 50.

10. 200. 12000. 200. 2.5 1.8 1.8 5.0 4000. 1.E-1089. 10. 91. 160.

- -45

Example of the output data of #RAYEDH for the case of 2 shots placed outside of the spread

***HEAD WAVES -DIRECT TASK (KINEMATICS)***

INPUT DATA

MEDIUM PARAMETERS : COORDINATES OF TWO REFRACTOR POINTS (in m) - PAIRS(X,Z): X1RF= 10.00 Z1RF= 200.00 X2RF= 12000.00 Z2RF= 200.00 HEAD WAVE VELOCITY (in km/s)= 2.500 OVERBURDEN VELOCITY NEAR REFRACTOR UNDER SP.-1(in km/s)= 1.800 OVERBURDEN VELOCITY NEAR REFRACTOR UNDER SP.-2(in km/s)= 1.800 ASSUMED PROPAGATION STEP ALONG REFRACTOR (in m)= 5.00 ASSUMED MAX PROPAGATION DISTANCE ALONG REFRACTOR (in m)= 4000.00 ASSUMED ACCURANCY OF REACHING CRITICAL ANGLE (in deg)= 0.1E-09 FIELD PARAMETERS: COORDINATES(x,z) of SP (in m)- WXPS1= 4000.00 WZPS1= 0.00 WXPS2= 8050.00 WZPS2= 0.00 NUMBER OF RECEIVERS = 70 COORDINATES (z) OF RECEIVERS (in m)= 0.00 COORDINATES (x) OF RECEIVERS (in m))= 4450.00 4500.00 4550.00 4600.00 4650.00 4700.00 4750.00 4800.00 4850.00 4900.00 4950.00 5000.00 5050.00 5100.00 5150.00 5200.00 5250.00 5300.00 5350.00 5400.00 5450.00 5500.00 5550.00 5600.00 5650.00 5700.00 5750.00 5800.00 5850.00 5900.00 5950.00 6000.00 6050.00 6100.00 6150.00 6200.00 6250.00 6300.00 6350.00 6400.00 6450.00 6500.00 6550.00 6600.00 6650.00 6700.00 6750.00 6800.00 6850.00 6900.00 6950.00 7000.00 7050.00 7100.00 7150.00 7200.00 7250.00 7300.00 7350.00 7400.00 7450.00 7500.00 7550.00 7600.00 7650.00 7700.00 7750.00 7800.00 7850.00 7900.00 CALCULATION PARAMETERS: MODEL NR= 1 IPLOT= 1 ITP= 0 RAY STEP DS= 0.80(m) ERROR OF HITTING RECEIVER (in m) - DXMAX- 0.05 DZMAX= 0.05 LINEAR VELOCITY INTERPOLATION RANGES OF OUTPUT RAY ANGLES FROM THE SOURCE (in deg)- ALFOS1= 89.000 ALFGS1= 10.000 ALFOS2= 91.000 ALFGS2=160.000 INCREMENT OF SOURCE OUTPUT ANGLE(in deg) - DALFOP= 0.1E+01 DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 484 ROWS NUMBER- 151 INCREMENT OF X VARIABLE (in m)- 25.00 INCREMENT OF Z VARIABLE (in m) - 2.00 THE NUMBER OF UNKNOWS (GRID NODES)- 73084

- -46

Example of the output data of #RAYEDH for the case of 2 shots placed outside of the spread (c.d.)

CALCULATION RESULTS

REFRACTOR DIP = 0.000 CRITICAL ANGLE UNDER SP.-1 = 46.054 CRITICAL ANGLE UNDER SP.-2 = 46.054

XKR1= 4159.361 ZKR1= 200.000 AWYJ1= 61.529 TRAY1= 0.17434 CRITICAL EMERGING ANGLE OF RAY FROM REFRACTOR UNDER SP.1 - 316.054

NO SOLUTION FOR RECEIVER NO. 38

XKR2= 7890.639 ZKR2= 200.000 AWYJ2= 118.471 TRAY2= 0.17434 CRITICAL EMERGING ANGLE OF RAY FROM REFRACTOR UNDER SP.2 - 223.946

NO SOLUTION FOR RECEIVER NO. 67 NO SOLUTION FOR RECEIVER NO. 68 NO SOLUTION FOR RECEIVER NO. 69 NO SOLUTION FOR RECEIVER NO. 70

CALCULATED TRAVELTIMES

SHOT GEOPHONE TIME(S)

1 1 0.40119 1 2 0.42119 1 3 0.44119 1 4 0.46119 1 5 0.48119 1 6 0.50119 1 7 0.52119 1 8 0.54119 1 9 0.56119 1 10 0.58119 1 11 0.60119 1 12 0.62119 1 13 0.64119 .......................................... 1 69 1.76119 1 70 1.78119 2 1 1.66119 2 2 1.64119 .......................................... 2 65 0.38119 2 66 0.36119 2 67 -1.00000 2 68 -1.00000 2 69 -1.00000 2 70 -1.00000

- -47

II.2.2. The description of the program for inverse problem of the head wave tomography

The algorithm of the tomographic inversion of applied head wave tomography is based

(Kasina 2001) on the same interpretation model as used in the program # RAYEDH.

Let’s consider the seismic ray, propagating from the source to the assumed receiver.

The traveltime of the ray may be expressed using relation:

t sv Pr

kPk

= ∑∆ 1

( ) ... (2.1)

where v(Pk) is the propagation velocity along k-th segment of ray trajectory of length ∆s and

position of its center Pk. The velocity v(Pk) may be expressed as linear combination of

velocities in the neighbouring grid nodes. When m is the number of these nodes then:

v P c Vk m mm

( ) =∑ ... (2.2)

where Vm is the velocity in m-th grid node. If we use cubic interpolation m has the value 16.

Let’s define now the result of a subtraction:

∆t t tr r r= −' ... (2.3)

where t’ r is the recorded time and tr is the traveltime calculated during ray tracing for assumed

initial velocity field. If we treat this value as the error of defining traveltime (caused by the

errors of selection velocity in grid nodes) then we obtain:

∆ ∆∆

t sv P

v Pr

k

kPk

= − ∑( )

( )2 ... (2.4)

and

∆ ∆v P c Vk m mm

( ) =∑ ... (2.5)

where ∆v(Pk) is the error of velocity estimation in the point Pk, and ∆Vm is the velocity error

in m-th neighbouring node. Now the relation (2.4) has the form:

∆ ∆∆ ∆ ∆

t sv P

v P

v P

v P

v P

v Pr

k

k

= − + + + +

=

( )

( )

( )

( )...

( )

( )..1

21

22

22

= − + + + +

∑ ∑ ∑∆

∆ ∆ ∆s

c V

v P

c V

v P

c V

v P

m mm

n nn

r rr

k2

12

22( ) ( )

...( )

...

- -48

Finally, for the assumed r-th seismic ray we obtain the equation:

∆ ∆t a Vr r m mm

=∑ , ... (2.6)

where ar,m defines the sum of coefficients relating to the same velocity increment ∆Vm.

If we consider R seismic rays and by M we denote the number of used velocities in

grid nodes then we obtain the set of equation:

a V t r Rr m m rm

M

, , , ,... ,∆ ∆= ==∑ 1 2

1

... (2.7)

In the above set of equation we treat the increments ∆Vm as the unknowns which must be

added to the values of starting velocities to obtain the better estimation of the recorded

traveltimes using modeled traveltimes.

To avoid too big irregularities of the velocity in the solution we introduce the

constraints imposed on estimated velocity increments ∆Vm. Finally, we minimize – using

norm L2 - the quantity:

( )S a V t V Vr m m rm

M

m iir

R

= −

+ −

==∑ ∑∑ , ∆ ∆ ∆ ∆

1

22

1

λ ... (2.8)

where i denotes indexes of neighbouring nodes in relation to node with index m,

λ denotes the coefficient of constraints. If we introduce the matrix notation:

( , )

,( , ) ( , )

$ $ $, ,R M

r mM

mR

rA V ta V t= = =1 1

∆ ∆∆ ∆

then the equation (2.7) may be written in the form:

$ $ $A V t∆ ∆= ... (2.9)

The condition of minimizing the quantity S may written in the form:

∂

∂S

Vk M

k∆= =0 1 2, , ,..., ... (2.10)

As a result we obtain:

( )$ $ $ $ $A A V A tT T+ =λ Ω ∆ ∆ ... (2.11)

where $Ω is the matrix of constraints , $A T is the transposition of matrix $A .

The solution of the set of equation (2.11) gives us the column vector ∆ $V of the

velocity corrections. If we solve the set of equations using iterative method we obtain the

consecutive approximations of the solution. Adding defined corrections ∆ $V to the velocities

- -49

(assumed or resulting from previous iteration) in the grid nodes we obtain the consecutive

approximations of the desired velocity distribution.

The execution of the #INVERDH program consists of the following steps:

- for assumed starting velocity distribution - defining the initial model - we perform ray

tracing and traveltime calculations for established positions of sources and receivers

in accordance with the algorithm of direct task of tomography,

- in the process of ray tracing we generate the coefficient matrix A of equation set (2.9),

- the matrix A is modified using algorithm of convolutional quelling,

- generation of constraint matrix Ω,

- solving the set of equation (2.11) for λ = 0,

- we estimate (using the solution (2.11) for λ = 0) or we assume the value of observation

error ε of the traveltime,

- we solve the set of equation (2.11) for different, assumed values of constraint coefficient λk

and then we estimate the optimal value λopt,

- we solve the set of equation (2.11) for λopt,

- we modify resulting velocity matrix using once more operator of convolutional quelling,

- we add the obtained velocity corrections ∆ $V to the values of initial velocity field,

- we use (or not) 2D smoother (average type) to resulting velocity field,

- we treat modified velocities as input data to the next iteration step,

- we continue calculations until we exceed the assumed number of iterations or at the moment

when the maximum differences between velocities of consecutive iterations will be lower

then assumed value.

The program #INVERDH consists of master segment and the following subroutines:

TRAS, DIFR, INT1, INT2, INTRP, TRAF, GRAF, PLUS, OPTY, SORT, TRANS,

GRADSPR, COEF, OMEGA,SPLOT. Subroutines TRAS, DIFR, INT1, INT2, INTRP,

TRAF, GRAF are the same as in the program #RAYEDH.

Subroutine OMEGA is used to generate the matrix of constraints. Only nonzero

elements (INTEGER type) of the matrix are calculated and saved with their indexes. The

range of constraints between velocity value Vm (or the value of its increment ∆Vm ) in

selected grid node and values of neighbouring nodes is the matter of choice. If we limit this

range to 4 neighbouring nodes we minimize the quantity:

- -50

B V Vm mm m

m

k

k

= −=∑ ( )∆ ∆ 2

1

4

where:

Vm = Vi,j

m1 = (j - 2) W + i m2 = (j - 1) W + i - 1

m3 = (j - 1) W + i + 1 m4 = j W + i

Therefore:

∂

∂∆B

VV V

mm m

m m

m

k

k

= − ==∑2 0

1

4

( )∆ ∆

and we can write the relation defining the way of $Ω matrix generation (Kasina 2001):

λ λ$ $ ( )Ω ∆ ∆ ∆V V Vm im

m

m

M

m

M

= −===∑∑∑

111

4

where M is the number of grid nodes, $Ω is the matrix with dimensions M × M, ∆ $V is the

matrix with dimensions M × 1. In the considered case the values of $Ω matrix elements are

INTEGERS in the range (-1, 4).

Subroutine COEF is used to generate nonzero elements of the coefficient matrix A.

These coefficients are modified during introducing new seismic rays.

Subroutine TRANS defines the transposition of matrix A and realizes multiplication

matrix AT by A. Only nonzero elements of matrices are taken into account. Also the right side

of equation (2.11) is calculated.

Subroutine GRADSPR is used to solve the set of equation by means of conjugate

gradient method. During iterations the actual value of the sum of squared residuals GAMMA2

is defined. Application of iterative method to solve the set of tomographic equation is

necessary because the coefficient matrix is the singular one. Additionally, this matrix is sparse

one. So we can save the computational time using only nonzero elements of matrices in the

process of iterative solution. For solution our set of equation we can use also technique based

on Singular Value Decomposition (Michelena 1993), but for large set of equations met in

practice iterative methods (conjugate gradient method, Gauss-Seidel method) are much more

effective (Young 1971, Kasina 1994, Kasina 1999).

Subroutine PLUS i used to add matrix ATA and λ $Ω . Only nonzero elements are taken

into account. Subroutine SORT is used for sorting the elements of resulting matrix.

Subroutine OPTY is used to define optimal value of constraint coefficient λopt.

- -51

Subroutine SPLT is used to realize convolutional quelling. This is a technique

improving imaging of seismic tomography data. The results were found to be superior to a

simple, least-squares solution because convolutional quelling suppresses side bands in the

resolving function that lead to imaging artifacts (Meyerholtz et al. 1989, Kasina 2001).

The operator of convolutional quelling is a matrix with a special, banded structure.

After convolution with a smoother (in the case illustrated, the smoother uses the eight nearest

neighbors), the raypath is effectively broadened into a band with a width controlled by the

width of the smoothing function. Convolutional quelling with a smoother is equivalent to

specifying that the velocities in neighboring cells are to be highly correlated.

Fig. 2.27. Graphic illustration of the process of convolutional quelling. The upper corner shows a perspective picture of the smoother used in the convolutional quelling. The left diagonal line of boxes illustrates pixels crossed by a ray associated with a given datum (nonzero values of the matrix A). After quelling, the raypath is broadened by the width of the smoother. The result for the eight nearest-neighbor example is illustrated on the right. For this example, the boxes with the solid boundaries would have larger amplitudes than those with the dashed boundaries (after Meyerholtz et al. 1989).

In the presented program nine-points convolutional operator was used with the matrix

structure presented below

a a a

a b a

a a a

with symbolic notation q=b/a used in the description of input data.

- -52

If we look at the comparison of popular methods of tomographic inversion (Scott and

Fehler 1991) it will possible to classify the algorithm of inversion used in #INVERSDH. That

comparison have been carried out using synthetic crosswell data and was based on the misfit

with the true model, solution stability under different sets of noise of the same level, and

resolution-covariance relationships. In the mentioned below relations G is the coefficient

matrix of tomographic set of equations.

The basic solution named DLS (Damped Least Squares) is obtained by minimizing

a combination of data residual and solution length terms where the solution is really

a perturbation from some starting model:

TTg GIGGG 121 )( −− += ε

where I is the identity matrix. The damping parameter ε regulates the balance between data

residual and solution length terms. Damping reduces the total length of the solution

perturbation vector, but puts no constraint on the similarity of adjacent elements of the

solution. Thus, DLS images tend to be rough when data are noisy or inconsistent. Damped

least squares with a spatiul average smoother worked reasonably well in the smooth model

case, but performed poorly when applied to the high-gradient model data.

DLS plus averaging smoother means that we apply convolution with a spatial

averaging operator to smooth DLS images:

TTg GIGGSG 121 )( −− += ε

where S represents the smoothing operator. The smoother can have any shape. A given pixel

value is replaced with a weighted average of the surrounding pixels and itself. Smoothing is

applied without regard to how well constrained each model parameter may be.

Iteratively reweighted least squares (IRLS) is an iterative scheme based on least

squares that estimates the general Lp solution (Scales et al. 1988). The Lp solution is defined

as that which minimizes the sum of absolute values of the data residuals, each raised to the

power p. The L1 , L2 (least squares) norms are most commonly used in geophysical inversion.

The normal equations obtained by minimizing the Lp norm are:

dRGmGRG TT =

where m is model data, d is observed data, R is a diagonal matrix representing reweighting

with elements dti p-2

and dti are the data residuals. These equations must be solved

iteratively if p ≠ 2. For p < 2, a threshold must be set for very small dti.

- -53

The next solution named convolutional quelling (Meyerholtz et al. 1989) may be

described as a ray broadening. The basic equation has the form:

TTTg GWIWGGWWG )( 21 ε+=−

where the smoothing operation (operator W) distributes the path lengths to neighboring pixels

which broadens the raypath. Convolutional quelling proved the best of the damped least-

squares methods, but had resolution limited by the dimension and shape of the smoothing

function. Convolutional quelling performed the best of the damped least squares based

methods in the misfit and stability tests, and is a very low error technique. We can treat this

approach as some step towards solutions of wave approach type based on so called wavepaths

and Fresnel volumes (Vasco et al. 1995).

A slightly different class of methods falls under the category of “regularization”

(Scales et al., 1990). These techniques are widely used and involve the minimization of

a measure of model roughness which can be some combination of first and second spatial

derivatives in various directions. The solution has the form:

TTTTg GDDGGG 1)( −+= λ

where D represents a first or second difference operator, and λ regulates the balance between

the data residual and solution roughness. A first difference regularization method performed

the best in terms of rms misfit from the true model, and solution stability under different

realizations of noise.

From the analysis in the Scott and Fehler’s paper (1991) it can be concluded that

convolutional quelling is very effective technique, only a little weaker then the best one - first

difference regularization.

Taking into account the presented comparison of different tomographic inversions we

can classify the algorithm used in #INVERSDH (comprising matrix of constraints,

convolutional quelling and 2D smoother applied to resulting velocity fields) as very close to

the solution based on first difference regularization.

The input data to #INVERSDH for the case of two shot points are red in the followingsequence:

READ(1,*) NAZWA2READ(1,*) NAZWA3READ(1,*) NAZWA4READ(1,*) ICASE, IPRINT, LMAX, ITAPE, IPRNT, IPLOT, LOPT, LATAREAD(1,*) LBIEL,LSKAL,BL,SK,LOPT1,ATAMIN,LCON,LSPLTIF(IPLOT.NE.0) READ(1,*) NAZWA8IF(ITAPE.NE.0) READ(1,*) NAZWA7

- -54

IF(LCON.NE.0) READ(1,*) NAZW10READ(1,*) LW, LT, DX, DZ, ILWREAD(1,*) WXPS1, WZPS1, WXPS2, WZPS2READ(1,*) LPG, WZG, (WXG(I), I=1, LPG)READ(1,*) X1RF, Z1RF, X2RF, Z2RF, V2, V1S1, V1S2, DSRF, SRFMAX, DIREAD(1,*) DS, DXMAX, DZMAX, DALFOP, DF1, INTLREAD(1,*) LIT, ERESREAD(1,*) ALFOS1, ALFGS1, ALFOS2, ALFGS2READ(1,*) LEPS, DLMBD, DXMAXVIF(LEPS.NE.0) READ(1,*) EPSIF(LSPLT.EQ.1) READ(1,*) ((SS(I,J),J= 1,3), I=1,3)READ(1,*) ((J, I, TR(I,J),I=1, LPG), J=1, LPS)

where:

NAZWA2 - the name output data text file for printer

NAZWA3 - the name of binary file *.ATA (created as product of matrix A and AT)

NAZWA4 - the name of binary file *.ATB with matrix ATB (created as product of matrix AT

and B)

NAZWA7 - the name of input binary file with velocities

NAZWA8 - the name of output data binary file (with velocities) for plotter

NAZW10 - the name of input data file with invariable velocities

ICASE – indicator of calculation variant

ICASE.LE.0 – we start with velocities defined as constant (V1S1),

ICASE.GT.0 - we start with assumed velocity field from text file or binary file

(for data from ProMAX ICASE.EQ.1) or from previous

iteration (ICASE.EQ.2)

ITAPE - indicator of reading velocities V(I) for ICASE.GT.0 from text file (ITAPE=0)

or from binary file (ITAPE=1)

LMAX - maximum number of iteration

IPRINT – indicator of printouts

IPRINT = 1 - full control prints are available with (IPRNT=2) or without

(IPRNT.NE.2) ray points

IPRNT = 0 - only basic control prints are available (without ray points)

IPLOT - indicator of saving velocity field

IPLOT= 1 - saving resulting velocity field in binary file

IPLOT= 0 - without saving resulting velocity field in binary file

- -55

LOPT – indicator of constraints

LOPT= 0 variant without constraints

LOPT= 1 variant with constraints (LAMBDA>0.)

LATA – indicator of matrices (ATA, ATB) generation

LATA= 0 - first iteration, we save resulting matrices ATA, ATB on disk

LATA= 1 - we don’t generate but read resulting matrices from previous iteration

LCON – indicator of introducing selected non variable velocities

LCON=1 - we introduce from binary file non variable velocities

LCON=0 - we don’t introduce from binary file non variable velocities

ATAMIN - the value of element of matrix ATA below which we are zeroing value

LSPLT - indicator of convolutional quelling

LSPLT= 1 - we introduce convolutional quelling

LSPLT= 0 - we don’t introduce convolutional quelling

LBIEL – indicator of "whitening" the coefficient matrix

LBIEL = 1 – calculations with "whitening" the coefficient matrix by adding small

value BL to the diagonal elements

LBIEL = 0 – calculations without "whitening”

LSKAL – indicator of matric scaling

LSKAL= 1 - we introduce matrix scaling (multiplication matrix ATA and column

matrix DT by value SK)

LSKAL= 0 – variant without scaling

LOPT1 – indicator of optimization of coefficient LAMBDA

LOPT1 = 1 - we assume the value DLMBD of constant LAMBDA (for LOPT=1)

LOPT1 = 0 - we assume increment DLMBD of constant LAMBDA (for LOPT=1),

optimal LAMBDA is evaluated

LW - number of columns of velocity grid

LT - number of rows of velocity grid

DX, DZ - dimensions of grid cell in meters

ILW= LPS x LPG – number of observations

LPG - number of receivers

WZG - coordinates „z” of receivers (in m)

WXG(I) – coordinates „x” of receivers (in m)

X1RF, Z1RF, X2RF, Z2RF – coordinates of two refractors points (in m)

V2 – head wave velocity (in km/s)

- -56

V1S1,V1S2 – overburden velocity near refractor under SP. 1 and SP. 2

DSRF - assumed propagation step along refractor (in m)

SRFMAX - assumed maximum propagation distance along refractor (in m)

DI - assumed accuracy of reaching critical angle (in deg) (very small value)

DS – ray step (in m)

DXMAX, DZMAX – assumed error of hitting receiver (in m)

DALFOP – increment of starting ray angle from the source (in deg)

DF1 - minimal increment of starting ray angle from the source (very small value)

INTL - indicator of interpolation type

INTL=0 - cubic velocity interpolation

INTL=1 - linear velocity interpolation

LIT - maximum number of iterations of conjugate gradient method

ERES - value of sum of squared residuals below which we stop calculation

ALFOS1, ALFGS1, ALFOS2, ALFGS2 – minimum and maximum starting angle from the

SP.1 and SP. 2; for SP. 1 (source on the left of the spread)) angles are decreasing from

ALFOS1 to ALFGS1; for SP. 2 (source on the right) angles are increasing from

ALFOS2 to ALFGS2

LEPS – indicator of estimation the observation error

LEPS= 0 - we estimate the observation error

LEPS=1 - we read the known observation error

DLMBD - increment of constant LAMBDA (for LOPT1=0) or assumed value of constant

LAMBDA (for LOPT1=1)

DXMAXV - assumed value of maximum difference of consecutive velocity solutions below

which we stop calculations

EPS - assumed observation error

SS(I, J) - matrix of convolutional quelling

TR(I, J) - recorded traveltimes in seconds

Examples of input and output data of #INVERSDH are presented below.

- -57

Example of the input data of #INVERSDH for the case of 2 shots placed outside of the spread

INVm200_2w_1200_1800z100_200a.OUTINVKLINR.ATAINVKLINR.ATB0 0 1 0 0 1 1 00 0 0.E-0 0.E-0 1 1.E-60 0 1INVm200_2w_1200_1800z100_200a.vel484 151 25. 2. 144

4000. 0. 8050. 0.700.4450. 50.

10. 200. 12000. 200. 2.5 1.8 1.8 5.0 4000. 1.E-100.5 0.05 0.05 1. 1.E-20 130 1.E-789. 10. 91. 160.

0 10.E0 0.010.5 0.5 0.50.5 1.0 0.50.5 0.5 0.5 1 1 0.40119 1 2 0.42119 1 3 0.44119 1 4 0.46119 ....................................... 1 65 1.68119 1 66 1.70119 1 67 1.72119 1 68 1.74119 1 69 1.76119 1 70 1.78119 2 1 1.66119 2 2 1.64119 ....................................... 2 65 0.38119 2 66 0.36119 2 67 -1.00000 2 68 -1.00000 2 69 -1.00000 2 70 -1.00000

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Example of the output data of #INVERSDH for the case of 2 shots placed outside of the spread

***HEAD WAVE TOMOGRAPHY - INVERSE TASK***

INPUT DATA NUMBER OF ITERATIONS= 1

CALCULATION PARAMETERS- ICASE=0 IPRINT=0 ITAPE=0 IPRNT=0 IPLOT=1 LOPT=1 LBIEL=0 BL= 0.0000E+00 LSKAL=0 SK= 0.0000E+00 ATAMIN= 0.1E-59 PARAMETER OF INVARIABLE VELOCITIES=0 PARAMETER OF CONVOLUTIONAL SMOOTHING= 1

MEDIUM PARAMETERS : COORDINATES OF TWO REFRACTOR POINTS (in m) - PAIRS(X,Z): X1RF= 10.00 Z1RF= 200.00 X2RF= 12000.00 Z2RF= 200.00 HEAD WAVE VELOCITY (in km/s)= 2.500 OVERBURDEN VELOCITY NEAR REFRACTOR UNDER SP1(in km/s)= 1.800 OVERBURDEN VELOCITY NEAR REFRACTOR UNDER SP2(in km/s)= 1.800 ASSUMED PROPAGATION STEP ALONG REFRACTOR (in m)= 5.00 ASSUMED MAX PROPAGATION DISTANCE ALONG REFRACTOR (in m)= 4000.00 ASSUMED ACCURACY OF REACHING CRITICAL ANGLE (in deg)= 0.1E-09

FIELD PARAMETERS: COORDINATES(x,z) of SP (in m)- WXPS1= 4000.00 WZPS1= 0.00 WXPS2= 8050.00 WZPS2= 0.00 NUMBER OF RECEIVERS = 70 COORDINATE (z) OF RECEIVERS (in m)= 0.00 COORDINATES (x) OF RECEIVERS (in m)= 4450.00 4500.00 4550.00 4600.00 4650.00 4700.00 4750.00 4800.00 4850.00 4900.00 4950.00 5000.00 5050.00 5100.00 5150.00 5200.00 5250.00 5300.00 5350.00 5400.00 5450.00 5500.00 5550.00 5600.00 5650.00 5700.00 5750.00 5800.00 5850.00 5900.00 5950.00 6000.00 6050.00 6100.00 6150.00 6200.00 6250.00 6300.00 6350.00 6400.00 6450.00 6500.00 6550.00 6600.00 6650.00 6700.00 6750.00 6800.00 6850.00 6900.00 6950.00 7000.00 7050.00 7100.00 7150.00 7200.00 7250.00 7300.00 7350.00 7400.00 7450.00 7500.00 7550.00 7600.00 7650.00 7700.00 7750.00 7800.00 7850.00 7900.00

CALCULATION PARAMETERS: RAY STEP DS= 0.50(M) ERROR OF HITTING RECEIVER (in m) - DXMAX- 0.05 DZMAX= 0.05 LINEAR VELOCITY INTERPOLATION RANGES OF OUTPUT RAY ANGLES FROM THE SOURCE (in deg)- ALFOS1= 89.000 ALFGS1= 10.000 ALFOS2= 91.000 ALFGS2=160.000 INCREMENT OF SOURCE OUTPUT ANGLE(in deg) - DALFOP= 0.1E+01

- -59

Example of the output data of #INVERSDH for the case of 2 shots placed outside of the spread (c.d.)

DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 484 ROWS NUMBER- 151 INCREMENT OF X VARIABLE (in m)- 25.00 INCREMENT OF Z VARIABLE (in m) - 2.00

PARAMETERS OF CONJUGATE GRADIENT METHOD- MAXIMUM NUMBER OF ITERATIONS- 30 MINIMUM VALUE OF SUM OF SQUARED RESIDUALS- 0.1000E-06

OPTIMIZATION PARAMETERS- ASSUMED VALUE OF CONSTANT LAMBDA= 0.1000E+02

NUMBER OF UNKNOWS(GRID NODES)- 73084

MATRIX OF CONVOLUTIONAL SMOOTHING OPERATOR- 0.50 0.50 0.50 0.50 1.00 0.50 0.50 0.50 0.50

OBSERVED TIMES - TRIO (SP NUM,GEOPH,TIME(MS))

1 1 401.19 1 2 421.19 1 3 441.19 1 4 461.19 1 5 481.19 1 6 501.19 1 7 521.19 1 8 541.19 1 9 561.19-------------------------------------------------------------

2 57 541.19 2 58 521.19 2 59 501.19 2 60 481.19 2 61 461.19 2 62 441.19 2 63 421.19 2 64 401.19 2 65 381.19 2 66 361.19 2 67 -1000.00 2 68 -1000.00 2 69 -1000.00 2 70 -1000.00

CALCULATION RESULTS

REFRACTOR DIP = 0.000 CRITICAL ANGLE UNDER SP.-1 = 46.054 CRITICAL ANGLE UNDER SP.-2 = 46.054

XKR1= 4207.501 ZKR1= 200.000 AWYJ1= 43.946 TRAY1= 160.10845 XKR2= 7842.499 ZKR2= 200.000 AWYJ2= 136.054 TRAY2= 160.10845

CRITICAL EMERGING ANGLE OF RAY FROM REFRACTOR UNDER SP.1- 316.054 CRITICAL EMERGING ANGLE OF RAY FROM REFRACTOR UNDER SP.2- 223.946

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Example of the output data of #INVERSDH for the case of 2 shots placed outside of the spread (c.d.)

H1= 200.000 H2= 200.000 DEAD ZONE - X <4415.001 AND X >7634.999

GEOPH NO- 65 WITH COORD. 7650.000 IN DEAD ZONE OF SP NUM 2 WITH COORD.-7634.999 GEOPH NO- 66 WITH COORD. 7700.000 IN DEAD ZONE OF SP NUM 2 WITH COORD.-7634.999 XKR2= 7842.499 ZKR2= 200.000 AWYJ2= 136.054 TRAY2= 160.10845 CRITICAL EMERGING ANGLE OF RAY FROM REFRACTOR UNDER SP.2 - 223.946

RMSDT= 70.11

NUMBER OF ROWS OF SET EQUATION AX=B - 64 NUMBER OF ZERO VALUE COLUMNS OF COEFFICIENT MATRIX = 59660

NUMBER OF NONZERO ELEMENTS OF MATRIX OMEGA= 66636

GAMMA2= 0.3108E+09 GAMMA2= 0.2151E+06 GAMMA2= 0.2269E+05 GAMMA2= 0.9947E+04 GAMMA2= 0.1309E+05 GAMMA2= 0.1057E+05 GAMMA2= 0.2567E+04 GAMMA2= 0.5562E+03 GAMMA2= 0.3231E+03 GAMMA2= 0.6986E+03 GAMMA2= 0.1655E+04 GAMMA2= 0.8928E+03 GAMMA2= 0.2062E+03 GAMMA2= 0.8440E+02 GAMMA2= 0.1051E+03 GAMMA2= 0.1394E+03 GAMMA2= 0.6866E+02 GAMMA2= 0.4963E+02 GAMMA2= 0.7378E+02 GAMMA2= 0.1067E+03 GAMMA2= 0.4610E+02 GAMMA2= 0.2582E+02 GAMMA2= 0.4055E+02 GAMMA2= 0.6450E+02 GAMMA2= 0.4183E+02 GAMMA2= 0.2162E+02 GAMMA2= 0.3042E+02 GAMMA2= 0.4369E+02 GAMMA2= 0.1863E+02 GAMMA2= 0.9830E+01

NUMBER OF ITERATIONS IN CONIUGATE GRADIENT METHOD- 30 FINAL SUM OF SQUARED RESIDUALS - 0.9830E+01

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The effectiveness of the #INVERSDH program was tested on many models. The

results of selected test are presented below. The assumed velocity model is presented in fig.

2.28. To secure relatively good illumination of the object (velocity anomaly) the dimensions

of the anomaly are small. The ray trajectories (from # RAYEDH) on the background of the

seismogeological model and positions of shots/receivers are presented in fig. 2.29. The results

of inversion (from # INVERSDH) are presented in figs. 2.30 – 2.31.

Example of the output data of #INVERSDH for the case of 2 shots placed outside of the spread (c.d.)

ESTIMATED VELOCITIES FOR ITERATION NR - 1

DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 484 ROWS NUMBER- 151 INCREMENT OF X VARIABLE (in m)- 25.00 INCREMENT OF Z VARIABLE (in m) - 2.00

1 1.800 2 1.800 3 1.800 4 1.800 5 1.800 6 1.800 7 1.800 8 1.800 9 1.800 10 1.800 11 1.800 12 1.800 13 1.800 14 1.800 15 1.800 16 1.800 17 1.800 18 1.800 19 1.800 20 1.800 21 1.800 22 1.800 23 1.800 24 1.800 25 1.800 26 1.800 27 1.800 28 1.800 29 1.800 30 1.800.................................................................................................................

73056 1.800 73057 1.800 73058 1.800 73059 1.800 73060 1.800 73061 1.800 73062 1.800 73063 1.800 73064 1.800 73065 1.800 73066 1.800 73067 1.800 73068 1.800 73069 1.800 73070 1.800 73071 1.800 73072 1.800 73073 1.800 73074 1.800 73075 1.800 73076 1.800 73077 1.800 73078 1.800 73079 1.800 73080 1.800 73081 1.800 73082 1.800 73083 1.800 73084 1.800

END OF CALCULATIONS FOR ITERATION NR- 1

NUMBER OF ITERATIONS= 1

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Fig. 2.28. The assumed velocity model used for testing #INVERSDH; velocity in the first layer - 1250 m/s, velocity in the second layer - 2500 m/s, velocity of anomaly - 800 m/s

Fig. 2.29. The ray trajectories (from # RAYEDH) on the background of the seismogeological model and positions of 2 shots (black circles) and receivers (circles)

0 100 200 300 400 500 600 700 800 900 1000-100

-80

-60

-40

-20

0

800 1000 1200 1400 1600 1800 2000 2200 2400

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Fig. 2.30. The results of inversion (from # INVERSDH) presented as 2D and 3D velocity fields using #SURFER

Fig. 2.31. The result of inversion (from # INVERSDH) presented as 2D velocity field using #SURFER

0 100 200 300 400 500 600 700 800 900 1000-100

-80

-60

-40

-20

0

900 930 960 990 102010501080111011401170120012301260

900 930 960 990 102010501080111011401170120012301260

- -64

II.3. AUXILIARY FORTRAN PROGRAMS

II.3.1. The program for conversion ASCII velocity files from ProMAX to binary files and text SURFER files

The velocity fields used as input data to program #RAYEDH were edited in ProMAX

by means of the process named Interactive Velocity Editor and exported in the Promax ASCII

format. It was necessary to do conversion of these ASCII velocity fields to binary files used

as input data in #RAYEDH for the purpose of modeling. These binary files were also used as

input data (initial models) in #INVERSDH. Additionaly it was necessary to do conversion of

these ASCII files to text files used as input data (so called *.dat files) in SURFER program.

The original Fortran program was named PROMXOUTR.

The input data to # PROMXOUTR program are red in the following sequence:READ(1,*) NAZWA2READ(1,*) NAZWA3READ(1,*) NAZWA4READ(1,*) NAZWA7READ(1,*) ILCDP, DXCDP, CDPMAX, DH, HMAX, X0, Z0READ(1,*) IPRINTwhere:

NAZWA2 - the name of output ASCII file with input velocities from ASCII file of ProMAX

at all nodes and output ASCII file (x,z,V) for plotting in #SURFER

NAZWA3 - the name of output ASCII file from ProMAX process Interactive Vel Editor

NAZWA4 - the name of *.DAT file used as input data for SURFER

NAZWA7 - the name of output binary file with full model velocities at all nodes

DXCP - interval CDP in meters

DH - interval of depth in meters

HMAX - maximum depth in meters

ILCDP - number of CDP points (column number of velocity matrix)

X0, Z0 - coordinates of new coordinate origin

IPRINT - index of printing input velocities at all nodes

IPRINT.EQ.0 - variant without printing velocity fields

IPRINT.NE.0 - variant with printing velocity fields

The examples of input and output data of # PROMXOUTR program are presented below.

Example of input data of # PROMXOUTRAnom_Dip.outAnomaly_DipAnom_Dip.vel981 5. 1282. 1. 300. 10. 2.1

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Example of Promax ASCII velocity file named Anomaly_Dip

ProMAX Data Export CDP X Coor Y Coor DEPTH VEL_INT------------------------------------------------------ 302.0 302.0 0.0 0.0 2500.0 2.0 2500.0 3.0 2500.0 ....................... 72.0 2500.0 73.0 2500.0 158.0 2500.0 159.0 5000.0 ------------------- 215.0 5000.0 300.0 5000.0 309.0 309.0 0.0 2500.0 2.0 2500.0 ....................... 72.0 2500.0 73.0 2500.0 158.0 2500.0 159.0 5000.0 ......................... 214.0 5000.0 215.0 5000.0 300.0 5000.0-------------------------------------------------------- 1282.0 1282.0 0.0 2500.0 2.0 2500.0 --------------- 70.0 2500.0 198.0 2500.0 199.0 5000.0 202.0 5000.0 203.0 5000.0 204.0 5000.0 207.0 5000.0 208.0 5000.0 209.0 5000.0 224.0 5000.0 225.0 5000.0 226.0 5000.0 300.0 5000.0

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Example of output data of # PROMXOUTR

*** DANE WEJSCIOWE ***

CDPs number = 981 CDPs interval in meters = 5.000 Maximum CDP in ProMAX file = 1282.0 Depth interval in meters = 1.0 Maximum depth in meters = 300.0 Coordinates of new origin: X0= 10.0 Z0= 2.0 INPUT VELOCITIES-

DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 981 ROWS NUMBER- 301 INCREMENT OF X VARIABLE (in m)- 5.00 INCREMENT OF Z VARIABLE (in m) - 1.00

1 2500.000 2 2500.000 3 2500.000 4 2500.000 5 2500.000 6 2500.000 7 2500.000 8 2500.000 9 2500.000 10 2500.000 11 2500.000 12 2500.000 13 2500.000 14 2500.000 15 2500.000 16 2500.000 17 2500.000 18 2500.000 19 2500.000 20 2500.000 21 2500.000 22 2500.000 23 2500.000 24 2500.000 25 2500.000 26 2500.000 27 2500.000 28 2500.000 29 2500.000 30 2500.000...........................................................................................................................................

295266 5000.000 295267 5000.000 295268 5000.000 295269 5000.000 295270 5000.000 295271 5000.000 295272 5000.000 295273 5000.000 295274 5000.000 295275 5000.000 295276 5000.000 295277 5000.000 295278 5000.000 295279 5000.000 295280 5000.000 295281 5000.000

*** 2D VELOCITY DISTRIBUTION ***

COORDINATES OF NEW ORIGIN: X0= 10.0 Z0= 2.0 VELOCITIES IN ASCII FILE - TRIOS (X,Z,V):

0.0 0.0 2500.000 5.0 0.0 2500.000 10.0 0.0 2500.000 15.0 0.0 2500.000 20.0 0.0 2500.000 25.0 0.0 2500.000 .............................................. 4850.0 -296.0 5000.000 4855.0 -296.0 5000.000 4860.0 -296.0 5000.000 4865.0 -296.0 5000.000 4870.0 -296.0 5000.000 4875.0 -296.0 5000.000 4880.0 -296.0 5000.000

- -67

The examples of velocity field prepared in Interactive Velocity Editor is shown in fig. 2.32.

The same field after exporting in ASCII format and processing in # PROMXOUTR is

displayed in fig. 2.33 using SURFER program.

Fig. 2.32. The window of Interactive Velocity Editor, velocity field prepared to export in ASCII format

Fig. 2.33. The velocity field presented in SURFER , input data were prepared in # PROMXOUTR

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II.3.2. The program for conversion binary velocity files from # INVERSDH to text SURFER files

The ouput velocity fields from #INVERSDH have the form of binary files. The

Fortran program named #VEL2D has been prepared for conversion binary velocity files to

text SURFER files. The input data are red in the following sequence:

READ(1,*) NAZWA2READ(1,*) NAZWA3READ(1,*) NAZWA7READ(1,*) IPRINT, IPRNT

where:

NAZWA2 - name of output ASCII file (x,z,V) for plotting in #SURFER

NAZWA7 - name of input binary file with velocities (results from INVERSHE.EXE)

NAZWA3 - name of output ASCII file (x,z,V) for plotting in #SURFER: *.dat file

IPRINT - index of printing velocity or velocity corrections in file NAZWA2

IPRINT=1 IPRNT=0 - printing velocities,

IPRINT=1 IPRNT=1 - printing corrections,

IPRINT= 0 - we save only *.dat file

II.3.3. The program for calculation RMSDV

To create a possibility of fast evaluation differences beetwen assumed velocity model

and velocity field resulting from tomographic inversion the Fortran program was prepared.

The program named RMSDVEL calculates the RMS error in accordance of formula:

( )

RMSE

V V

R

i est ii

R

=−

=∑ , ,mod

/2

1

1 2

where:

V i,est – the value of velocity estimated in the i – th node,

Vi,mod – the known value of model velocity in the i – th node,

R – the number of nodes in velocity grid

- -69

The input data are red in the following sequence:

READ(1,*) NAZWA2READ(1,*) NAZWA7READ(1,*) NAZWA3READ(1,*) V2,V1S1,HMAX,LPROMXwhere:NAZWA2 – the name of output text file for printerNAZWA7 – the name of input binary file with model velocities (output from PROMAXOUT)NAZWA3 – the name of input binary file with tomoinvers velocities (output from INVERSDH)V2 - head wave velocity (in km/s)V1S1 - overburden velocity near refractor under shot point (in km/s)HMAX – max depth of calculations in mLPROMX - indicator of input data type LPROMX =1 for data from ProMAX programs LPROMX =0 for data from FORTRAN programs

Examples of input and output data from # RMSDVEL are presented below.

Examples of input data from # RMSDVEL

RSMDVEL.OUTPROMVEL-Velocity-every-one.OUtPROMVEL-Velocity-every-fifth.OUt6.0 1.4 120. 1

Examples of output data from # RMSDVEL

*** CALCULATION OF RMS VELOCITY ERROR ***

MODEL VELOCITY DATA FILE NAME - PROMVEL-Velocity-every-one.OUt TOMOINVERS VELOCITY FILE NAME - PROMVEL-Velocity-every-fifth.OUt

INPUT DATA

MEDIUM PARAMETERS :

HEAD WAVE VELOCITY (in km/s)= 6.000 OVERBURDEN VELOCITY NEAR REFRACTOR UNDER SP.-1(in km/s)= 1.400 MAX DEPTH (in m)= 120.000

DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 125 ROWS NUMBER- 26 INCREMENT OF X VARIABLE (in m)- 25.00 INCREMENT OF Z VARIABLE (in m)- 8.00

THE NUMBER OF GRID NODES- 3250 THE NUMBER OF GRID NODES USED- 2000

RMS VELOCITY ERROR (m/s) = 23.

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II.3.4. The program for conversion ASCII files with picks from ProMAX to FORTRAN text format

One of the essential problems of tomographic inversion is defining the traveltimes in

the process of picking. The comparison between picks obtained from model data generated

using Promax process named Finite Difference Modeling and traveltimes obtained using ray

approach (FORTRAN program #RAYEDH) needed exporting ProMAX picks and conversion

ProMAX ASCII files to FORTRAN text format. The FORTRAN program named

#PICKSOUT has been prepared. The input data of the # PICKSOUT program are red in the

following sequence:

READ(1,*) NAZWA2READ(1,*) NAZWA3READ(1,*) NAZWA4READ(1,*) LPS,LCHAN

where:

NAZWA2 – the name of output temporary ASCII file with times

NAZWA3 – the name of input ASCII file (result of picking) from ProMAX

NAZWA4 – the name of output ASCII file with times of picks

LPS – the number of shots

LCHAN – the number of channels

The example of input and output data of #PICKSOUT are presented below.

The example of input data of #PICKSOUT

PICKSOUT.OUTPicksPICKSOUT2.OUT1 102

- -71

The example of output ASCII data with picks from ProMAX

ASCII database file write for Area="kasina", Line="tomography2005"

Value list for Database Order TRC Test_05_100_1_sh1>TRCGEOMETRY06439168 TRC 06439168< 1|339999995214436424907732413799364296704.00 |< 2|339999995214436424907732413799364296704.00 |< 3|339999995214436424907732413799364296704.00 |< 4|339999995214436424907732413799364296704.00 |< 5|339999995214436424907732413799364296704.00 |< 6|339999995214436424907732413799364296704.00 |< 7|339999995214436424907732413799364296704.00 |< 8|339999995214436424907732413799364296704.00 |< 9|339999995214436424907732413799364296704.00 |< 10|339999995214436424907732413799364296704.00 |< 11|339999995214436424907732413799364296704.00 |< 12|339999995214436424907732413799364296704.00 |< 13|339999995214436424907732413799364296704.00 |< 14| 160.11 |< 15| 164.03 |< 16| 167.95 |< 17| 171.85 |< 18| 175.75 |< 19| 179.67 |------------------------------------------------------------------------------< 83| 427.98 |< 84| 431.86 |< 85|339999995214436424907732413799364296704.00 |< 86|339999995214436424907732413799364296704.00 |< 87|339999995214436424907732413799364296704.00 |< 88|339999995214436424907732413799364296704.00 |< 89|339999995214436424907732413799364296704.00 |< 90|339999995214436424907732413799364296704.00 |< 91|339999995214436424907732413799364296704.00 |< 92|339999995214436424907732413799364296704.00 |< 93|339999995214436424907732413799364296704.00 |< 94|339999995214436424907732413799364296704.00 |< 95|339999995214436424907732413799364296704.00 |< 96|339999995214436424907732413799364296704.00 |< 97|339999995214436424907732413799364296704.00 |< 98|339999995214436424907732413799364296704.00 |< 99|339999995214436424907732413799364296704.00 |< 100|339999995214436424907732413799364296704.00 |< 101|339999995214436424907732413799364296704.00 |< 102|339999995214436424907732413799364296704.00 |*END

- -72

The example of output data of # PICKSOUT

CALCULATED TRAVELTIMES

SHOT GEOPHONE TIME(S)

1 1 -1.00000 1 2 -1.00000 1 3 -1.00000 1 4 -1.00000 1 5 -1.00000 1 6 -1.00000 1 7 -1.00000 1 8 -1.00000 1 9 -1.00000 1 10 -1.00000 1 11 -1.00000 1 12 -1.00000 1 13 -1.00000 1 14 0.16011 1 15 0.16403 1 16 0.16795 1 17 0.17185-------------------------------------------------- 1 80 0.41626 1 81 0.42016 1 82 0.42407 1 83 0.42798 1 84 0.43186 1 85 -1.00000 1 86 -1.00000 1 87 -1.00000 1 88 -1.00000 1 89 -1.00000 1 90 -1.00000 1 91 -1.00000 1 92 -1.00000 1 93 -1.00000 1 94 -1.00000 1 95 -1.00000 1 96 -1.00000 1 97 -1.00000 1 98 -1.00000 1 99 -1.00000 1 100 -1.00000 1 101 -1.00000 1 102 -1.00000

- -73

II.3.5. Program for 2D (spatial) smoothing of velocity fields

The smoothing of velocity fields resulting from tomographic inversion is commonly

used procedure after each tomographic inversion and before application the velocity fields to

statics calculations. The FORTRAN program named #VELAVE has been prepared to smooth

velocity fields from binary files using 2D spatial rectangular operator. If we introduce the

following notation for velocities in the nodes:

v(x,z) = V(i ∆x, j ∆z) = Vi,j i = 0, 1, 2, ..., I; j = 0, 1, 2, ..., J

then for radius of velocity smoothing ( LAVEX, LAVEZ) equal 2 in both directions we usethe following formula to calculated the velocity Vs

i,j after smoothing:

9

2,1,2,1,,2,1,2,1,,

++−−++−− ++++++++= jijijijijijijijijis

ji

VVVVVVVVVV

The input data of #VELAVE are red in the following sequence:

READ(1,*) NAZWA2READ(1,*) NAZWA7READ(1,*) NAZWA3READ(1,*) IPRINT, LPROMAXREAD(1,*) LAVEX, LAVEZ

where:NAZWA2 – the name of output ASCII file (x,z,V) (with averaged velocities)NAZWA3 – the name of output binary file with averaged velocitiesNAZWA7 – the name of input binary file with velocitiesIPRINT - index of printing averaged velocity in file NAZWA2 IPRINT=1 - printing averaged velocities IPRINT=0 - no printing averaged velocitiesLPROMAX=0 velocities (in m/s) from PROMAX =1 velocities (in km/s) from FORTRANLAVEX - radius of velocity smoothing in X direction defined in DX (increment of X variable in m), LAVEX=2 means radius = 2*DXLAVEZ - radius of velocity smoothing in Z direction

The example of input and output data of #VELAVE are presented below.

The example of input data of #VELAVE

Velave.outINVanomaly.velINVanom_ave2_2.vel1 12 2

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II.3.6. Program for statics calculations

The estimation of static corrections based on the near surface velocity fields resulting

from tomographic inversion was one of the main goal of presented dissertation. The

The example of output data of #VELAVE

Radius of velocity smoothing in X direction= 2 Radius of velocity smoothing in Z direction= 2

INPUT VELOCITIES (in km/s)-

DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 213 ROWS NUMBER- 51 INCREMENT OF X VARIABLE (in m) - 5.00 INCREMENT OF Z VARIABLE (in m) - 2.00

1 1.200 2 1.200 3 1.200 4 1.200 5 1.200 6 1.200 7 1.200 8 1.200 9 1.200 10 1.200 11 1.200 12 1.200 13 1.200 14 1.200 15 1.200----------------------------------------------------------------------------- 10831 1.200 10832 1.200 10833 1.200 10834 1.200 10835 1.200 10836 1.200 10837 1.200 10838 1.200 10839 1.200 10840 1.200 10841 1.200 10842 1.200 10843 1.200 10844 1.200 10845 1.200 10846 1.200 10847 1.200 10848 1.200 10849 1.200 10850 1.200 10851 1.200 10852 1.200 10853 1.200 10854 1.200 10855 1.200 10856 1.200 10857 1.200 10858 1.200 10859 1.200 10860 1.200 10861 1.200 10862 1.200 10863 1.200

AVERAGED OUTPUT VELOCITY-

DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 213 ROWS NUMBER- 51 INCREMENT OF X VARIABLE (in m)- 5.00 INCREMENT OF Z VARIABLE (in m) - 2.00

1 1.200 2 1.200 3 1.200 4 1.200 5 1.200 6 1.200 7 1.200 8 1.200 9 1.200 10 1.200 11 1.200 12 1.200 13 1.200 14 1.200 15 1.200--------------------------------------------------------------------------- 10841 1.200 10842 1.200 10843 1.200 10844 1.200 10845 1.200 10846 1.200 10847 1.200 10848 1.200 10849 1.200 10850 1.200 10851 1.200 10852 1.200 10853 1.200 10854 1.200 10855 1.200 10856 1.200 10857 1.200 10858 1.200 10859 1.200 10860 1.200 10861 1.200 10862 1.200 10863 1.200

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FORTRAN program was prepared for statics calculations using as input data binary velocity

files obtained during inversion as well as the velocity fields of considered models. The statics

is calculated assuming vertical propagation of the rays from the shot/receiver to the datum

identified with refractor.

The input data of #STATICS are red - for the variant of two shots - in the following

sequence:

READ(1,*) NAZWA2READ(1,*) IPRINT,NRMOD,IPLOT,ITP,IPRNTREAD(1,*) ITAPEIF(ITAPE.NE.0) READ(1,*) NAME7READ(1,*) LMOD,LPROMAXIF(ITAPE.NE.0) READ(7) LW,LT,DX,DZ,LWT,(V(I),I=1,LWT)IF(ITAPE.EQ.0) READ(1,*) LW,LT,DX,DZREAD(1,*) DS,DXMAX,DZMAX,DALFOP,DF1,INTLREAD(1,*) WXPS1,WZPS1,WXPS2,WZPS2READ(1,*) LPG,WZG,WXG(1),DXGREAD(1,*) X1RF,Z1RF,X2RF,Z2RF,V2,V1S1,V1S2,DSRF,SRFMAX,DIREAD(1,*) ALFOS1,ALFGS1,ALFOS2,ALFGS2IF(ITAPE.NE.0)READ(1,*) NRR,(NR1(I),NR2(I),VN(I),I=1,NRR)

where the only new parameters with comparison with input data of #RAYEDH are:LMOD – indicator of the type of input velocity field LMOD=0 - we read output velocity field from #INVERSDH LMOD=1 - we read input velocity model from PromaxLPROMAX – indicator of the source of the velocity field from inversion LPROMAX = 0 - we read data from FORTRAN program LPROMAX = 1 - we read data from PROMAXALFOS1=ALFGS1=ALFOS2=ALFGS2=90.

The examples of input and outpu data of #STATICS are presented below.

Example of input data of #STATICS

statanom2_model.OUT0 1 0 0 01PROMVEL.OUT1 00.001 0.05 0.05 1. 1.E-20 1

55. 4. 1000. 4.

704.200. 10.

10. 58. 1060. 58. 2.5 1.2 1.2 1.0 945. 1.E-1090. 90. 90. 90.

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Example of output data from #STATICS

*** STATICS FOR THE ASSUMED VELOCITY MODEL ***

INPUT DATA MEDIUM PARAMETERS : COORDINATES OF TWO REFRACTOR POINTS (in m) - PAIRS(X,Z): X1RF= 10.00 Z1RF= 58.00 X2RF= 1060.00 Z2RF= 58.00 HEAD WAVE VELOCITY (in km/s)= 2.500 OVERBURDEN VELOCITY NEAR REFRACTOR UNDER SP.-1(in km/s)= 1.200 OVERBURDEN VELOCITY NEAR REFRACTOR UNDER SP.-2(in km/s)= 1.200 ASSUMED PROPAGATION STEP ALONG REFRACTOR (in m)= 1.00 ASSUMED MAX PROPAGATION DISTANCE ALONG REFRACTOR (in m)=945.00 ASSUMED ACCURANCY OF REACHING CRITICAL ANGLE (in deg)= 0.1E-09 FIELD PARAMETERS: COORDINATES(x,z) of SP (in m)- WXPS1= 55.00 WZPS1= 4.00 WXPS2= 1000.00 WZPS2= 4.00 NUMBER OF RECEIVERS = 70 COORDINATES (z) OF RECEIVERS (in m)= 4.00 COORDINATES (x) OF RECEIVERS (in m))= 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 380.00 390.00 400.00 410.00 420.00 430.00 440.00 450.00 460.00 470.00 480.00 490.00 500.00 510.00 520.00 530.00 540.00 550.00 560.00 570.00 580.00 590.00 600.00 610.00 620.00 630.00 640.00 650.00 660.00 670.00 680.00 690.00 700.00 710.00 720.00 730.00 740.00 750.00 760.00 770.00 780.00 790.00 800.00 810.00 820.00 830.00 840.00 850.00 860.00 870.00 880.00 890.00

CALCULATION PARAMETERS: MODEL NR= 1 IPLOT= 0 ITP= 0 RAY STEP DS= 0.00(m) ERROR OF HITTING RECEIVER (in m) - DXMAX- 0.05 DZMAX= 0.05 LINEAR VELOCITY INTERPOLATION

RANGES OF OUTPUT RAY ANGLES FROM THE SOURCE (in deg)- ALFOS1 and ALFGS1 for right part of the spread ALFOS2 and ALFGS2 for the left part of the spread ALFOS1= 90.000 ALFGS1= 90.000 ALFOS2= 90.000 ALFGS2= 90.000

DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 213 ROWS NUMBER- 51 INCREMENT OF X VARIABLE (in m)- 5.00 INCREMENT OF Z VARIABLE (in m) - 2.00

THE NUMBER OF UNKNOWS (GRID NODES)- 10863

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II.3.7. The program for Turning Ray Tracing

In the Promax system we have the possibility to calculate the traveltimes of turning

waves and ray trajectories using process Turning Ray Tracing. Unfortunately we have no any

Example of output data from #STATICS (c.d.)

CALCULATION RESULTS

REFRACTOR DIP = 0.000

CALCULATED STATICS FOR ASSUMED VELOCITY MODEL

SHOT GEOPHONE STATION(M) STAT_ALL(S)

1 1 200.00 0.09000 1 2 210.00 0.09000 1 3 220.00 0.09000----------------------------------------------------------- 1 22 410.00 0.09000 1 23 420.00 0.09000 1 24 430.00 0.09405 1 25 440.00 0.09655 1 26 450.00 0.09905 1 27 460.00 0.09572----------------------------------------------------------- 2 69 880.00 0.09000 2 70 890.00 0.09000

CALCULATED RECEIVER STATICS FOR ASSUMED VELOCITY MODEL

SHOT GEOPHONE STATION(M) STAT_RCV(S)

1 1 200.00 0.04500 1 2 210.00 0.04500 1 3 220.00 0.04500-----------------------------------------------------------

1 68 870.00 0.04500 1 69 880.00 0.04500 1 70 890.00 0.04500

CALCULATED SHOT STATICS FOR ASSUMED VELOCITY MODEL

SHOT TIME(S)

1 0.04500 2 0.04500

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tools for displaying or exporting ray trajectories. Therefore the new original FORTRAN

program has been constructed for turning ray tracing (#RAYEDT) and another for ray

trajectories displaying (#RAYPLTT). The ray tracing in gradient medium is realized using the

same algorithm as in RAYEDH program. Two variants of calculations are possible. In the

first case we can calculate the ray trajectories for assumed range of starting angles from the

shot (fig. 2.34). In the second case we calculate the ray trajectories from the source to the

selected receivers (fig. 2.35). The velocity field is prepared in the ProMAX system using

process Interactive Velocity Editor.

Fig.2.34. The ray trajectories for assumed range of starting angles from the shot

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Fig. 2.31. The ray trajectories of turning wave for assumed positions of receivers

The input data of #RAYEDT are red in the following sequence:

READ(1,*) NAZWA2READ(1,*) IPRINT, NRMOD, IPLOT, ITP, IPRNTREAD(1,*) ITAPEIF(ITAPE.NE.0) READ(1,*) NAME7IF(IPLOT.NE.0) READ(1,*) NAZWA3IF(ITAPE.EQ.0) READ(1,*) LW, LT, DX, DZREAD(1,*) DS, DXMAX, DZMAX, DALFO, DF1, INTLREAD(1,*) LPS, (WXPS(I),I=1, LPS), (WZPS(I), I=1, LPS)READ(1,*) LPG, WZG,WXG(1,1), DPS, DGEOFREAD(1,*) ALFO, ALFGIF(ITAPE.eq.0) READ(1,*) NRR, (NR1(I), NR2(I),VN(I), I=1, NRR)

where:

NAZWA2 – the name of output text file for printing the results of calculations

NAZWA3 – the name of output binary file used for plotting rays

NAZWA7– the name of input binary file with velocities

IPRINT - indicator of printouts:

IPRINT=0 - variant without control printouts

IPRINT=1 - variant with control printouts

NRMOD - model number

IPLOT - indicator of saving data for plotting

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IPLOT=0 variant without saving data for plotting

IPLOT=1 variant with saving data for plotting

ITP - indicator of calculation option

ITP=0 calculation for assumed receivers

ITP=1 calculation for assumed values of "s" parameter (DSRF) along refractor

IPRNT - indicator of additional control printouts (ray points)

ITP=1 and IPRNT=1 - all control printouts without ray points

ITP=1 and IPRNT=2 - all control printouts with ray points

ITP=0 and IPRNT=1 - limited control printouts without ray points

ITAPE - indicator of input velocity data

ITAPE=0 - reading from text file

ITAPE=1 - reading from binary file

LW - columns number of velocity matrix

LT - rows number of velocity matrix

DX - the dimension of the cell in „x” direction (in meters)

DZ - the dimension of the cell in „z” direction (in meters)

LWT – number of grid nodes

DS – ray step (in m)

DXMAX, DZMAX – assumed error of hitting receiver (in m)

DALFOP – increment of starting ray angle from the source (in deg)

DF1 - minimal increment of starting ray angle from the source (very small value)

INTL - indicator of interpolation type

INTL=0 - cubic velocity interpolation

INTL=1 - linear velocity interpolation

LPS – the number of shots

WXPS(I), WZPS(I) – coordinates (x,z) of I-h shot

LPG - number of receivers

WZG - coordinates „z” of receivers (in m)

WXG(I,J) - coordinates „x” of receivers (in m)

DPS – shot interval (in m)

DGEOF – receiver interval (in m)

ALFO, ALFG - maximum and minimum starting angle from the shot

(ALFO > ALFG, ALFO is drceasing, 180>ALGO,ALFG>0)

NRR - number of pairs (NR1(I),NR2(I))

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VN(I) - input velocities red from text file in km/s and defined in grid nodes

NR1(I) , NR2(I) - pair of grid nodes between which velocity is equal VN(I)

Examples of input and output data of #RAYEDT program are presented below.

Example of input data of #RAYEDT

Anomaly_gradr_ps3.OUT1 1 1 1 0127-11-200m-grad.velAnomaly_gradr_ps3.PLT

0.5 0.05 0.05 0.5 1.E-20 1

11500. 0.

240.1050. 1000. 50.

179. 1.

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Example of output data of #RAYEDT

***TURNING WAVES -DIRECT TASK (KINEMATICS)***

INPUT DATA

SP NUMBER COOR. XPS

1 1500.00

SP NUMBER COOR. ZPS

1 0.00

COORDINATES (z) OF RECEIVERS (in m)= 0.00 NUMBER OF RECEIVERS = 24 COORDINATE WXG(1,1) = 1050.00 SHOT INTERVAL = 1000.00 RECEIVER INTERVAL = 50.00

SHOT NUMBER= 1 COORDINATES (x) OF RECEIVERS (in m))=

1050.00 1100.00 1150.00 1200.00 1250.00 1300.00 1350.00 1400.00 1450.00 1500.00 1550.00 1600.00 1650.00 1700.00 1750.00 1800.00 1850.00 1900.00 1950.00 2000.00 2050.00 2100.00 2150.00 2200.00

MODEL NR= 1 IPLOT= 1 ITP= 0

RAY STEP DS= 0.50(m) ERROR OF HITTING RECEIVER (in m) - DXMAX- 0.05 DZMAX= 0.05 LINEAR VELOCITY INTERPOLATION ALFO= 179.000 ALFG= 1.000

INCREMENT OF SOURCE OUTPUT ANGLE(in deg) - DALFOP= 0.5E+00

DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 125 ROWS NUMBER- 26 INCREMENT OF X VARIABLE (in m)- 25.00 INCREMENT OF Z VARIABLE (in m) - 8.00

THE NUMBER OF UNKNOWS (GRID NODES)- 3250

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Example of output data of #RAYEDT (c.d.)

NO SOLUTION FOR RECEIVER NO. 10

CALCULATED TRAVELTIMES

SHOT GEOPHONE TIME(S)

1 1 0.41823 1 2 0.37923 1 3 0.33819 1 4 0.29507 1 5 0.24988 1 6 0.20273 1 7 0.15381 1 8 0.10343 1 9 0.05199 1 10 -1.00000 1 11 0.05199 1 12 0.10343 1 13 0.15381 1 14 0.20273 1 15 0.24988 1 16 0.29507 1 17 0.33819 1 18 0.37923 1 19 0.41823 1 20 0.45525 1 21 0.49040 1 22 0.52378 1 23 0.55552 1 24 0.58570

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II.4. PROGRAMS FOR GRAPHICAL PRESENTATION

Several programs were used for graphical presentation the input data (seismogeolo-

gical models) and output data (velocity fields from inversion, ray trajectories, hodographs,

statics).

In the ProMAX system the following tools were applied for graphical presentation:

- Interactive Velocity Editor for creating velocity models and displaying the resulting velocity

fields from Turning Ray Tomography,

- Velocity Viewer/Point Editor for displaying the resulting velocity fields from Turning Ray

Tomography and differences between velocity fields,

- DATABASE for displaying results of picking, traveltimes from Turning Ray Tracing.

The velocity fields resulting from inversion realized by means of #INVERSDH were

presented using SURFER program. The hodographs and statics distributions were presented

using GRAPHER.

The ray trajectories of head waves resulting from # RAYEDH were presented using

original FORTRAN program named #RAYPLTH. That program was constructed using

Lahey/Fujitsu Fortran 95 (LF95) with application the tool Winteracter Starter Kit (WiSK)

and procedures of High Resolution Graphics.

The input data to the #RAYPLTH were red using the following sequence:

READ(2,*) NAZWA3READ(2,*) TEXTREAD(2,*) LW,LT,DX,DZ,ITPREAD(2,*) ILMODDO 99 I=1,ILMOD

READ(2,*) LMOD(I),(BX(K,I),BZ(K,I),K=1,LMOD(I))99 CONTINUEREAD(2,*) DS,DXMAX,DZMAX,DSRF,DF1,INTLREAD(2,*) LPS,(WXPS(I),I=1,LPS),(WZPS(I),I=1,LPS)READ(2,*) LPG, WZGW,WXG(1),DGEOFREAD(2,*) SRFMAX,SRF1,SRF2,NRDELREAD(2,*) ILNAP,ITPNAP,LTEXT,LSTEPDO 1 I=1,ILNAP READ(2,*) XNAP(I),ZNAP(I) 1 READ(2,*) NAPIS(I)READ(2,*) ILPS,(NRPS(I),I=1,ILPS)READ(2,*)LPAUSEREAD(2,*) XMAX,ZMAX,XPOCZ,ZPOCZ,DELX,DELZREAD(2,*) RADIUSREAD(2,*) XSIZE,ZSIZE

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READ(2,*)xleft,zlower,xright,zupperREAD(2,*) xl,zlow,xr,zup

where:

NAZWA3 – the name of binary file with resulted ray trajectories from direct task (RAYEDH)

TEXT - the name of model

LW - column number of velocity field

LX - row number of velocity field

DX - interval of variable "x"

DZ - interval of variable "z",

ITP - indicator of calculation option:

ITP=0 - drawing rays for assumed receivers

ITP=1 - drawing rays for assumed values of "s" parameter along refractor

ILMOD - the number of subareas of the model (max 10)

LMOD(I) - number of points defining I-th area

BX(K,I), BZ(K,I) - coordinates of points defining the subarea I-th

DS - interval of ray tracing (in meters) above refractor

DXMAX, DZMAX - precision of hitting the receiver (in meters)

DSRF - interval of ray along refractor

DF1 - minimal increment of source output angle during defining critical angle

INTL - parameter of interpolation option:

INTL=0 - cubic interpolation,

INTL=1 - linear interpolation

LPS - shot points number

(WXPS(I),WZPS(I)) - coordinates of shot points

LPG - receiver points number,

WZGW - "z" coordinate of receivers (one value),

WXG(I) - "x" coordinates of receivers

DGEOF - receiver interval

SRF1,SRF2 - the range of values (min and max) of parameter "s"(actual length of trajectory

along the refractor for which we are drawing trajectories of rays)

SRFMAX – maximum value of „s” parameter

ILNAP - number of descriptions

ITPNAP - parameter with value 0

LSTEP - interval of plotting ray points (=1 - each point)

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(XNAP(I),ZNAP(I)) - coordinates of I-th description

NAPIS(I) - the text of I-th description

ILPS - number of shots for graphical presentation

NRPS(I) - identification number of I-th shot (I=1,ILPS)

LPAUSE = 1 - drawing ray by ray with pause (= 0 without pause)

XMAX, ZMAX - maximum coordinates of drawing (in meters);

XPOCZ,ZPOCZ - initial coordinates (in meters)

DELX, DELZ - increment of "x" and "z" (in meters)

RADIUS - radius of circle used for graphical presentation of shots

XSIZE,ZSIZE - parameters defining window dimensions (in the range 0.0 - 1.0)

for values 1.0 we have the largest available window

xleft,zlower,xright,zupper - range of values (number of pixels) in the window

xl,zlow,xr,zup – window parameters: graphics on full window: xl=0.0, zlow=0.0, xr=1.0, zup=1.0 graphics on one quarter (top right)of window:xl=0.5, zlow=0.5, xr=1.0, up=1.0 , for (0.0,0.0,1.0,1.0) we have full graphics window

Example of input data of # RAYPLTHKLIN101r.PLTKLIN101r43 25 5. 5. 125 70. 25. 110. 25. 95. 40. 85. 40. 70. 25.2 0. 10. 190. 10.1. 0.05 0.05 3 1.E-20 12 11. 180. 10. 10.2410.50. 3.90. 0. 90. 24 0 1 2200. 45.V=1.2 km/s122. 33.V=0.8 km/s60. 70.V=2.5 km/s180. 65.FI=0 deg.2 1 20190. 90. 0. 0. 5. 5.2.00.75 0.750.0 0.0 639.0 349.00.0 0.0 1.0 1.0

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The example of ray trajectories against a background of assumedseismogeological model and positions of sources/receivers is presented in fig. 2.36.

Fig. 2.36. The example of ray trajectories against a background of assumed seismogeological model and positions of sources/receivers

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III. RESULTS OF THE SOLUTION OF DIRECT PROBLEM OF THE REFRACTION TOMOGRAPHY

Most of the tomographic inversions are based on the ray approach to wave

propagation. But the wave pattern on the records is the realization of wave process. So it is

very important to ascertain that for considered models of medium and considered parameters

of acquisition we can use ray approach to predict the proper behaving of the waves and to

estimate the correct traveltimes. In the case of low velocity layer and small distances between

source, receiver and heterogeneities we can observe - using ray approach - strong deviation of

the rays, complicated ray trajectories, the shadow zones. All these features may not occur in

real wave process of propagation. One of the most commonly used tool of investigation wave

process of propagation is finite difference modeling. Using this tool we shall ascertain that for

discussed models we can recognize the effects of near surface heterogeneities on first break

traveltimes and that these effects may be properly estimated using ray approach.

III.1. THE RAY APPROACH AND THE WAVE APPROACH TO SE ISMIC WAVE PROPAGATION AND THEIR ROLE IN TOMOGRAPHIC INVERSION

In the traveltime tomography we reconstruct the velocity fields using the differences

beetwen real traveltimes picked from the records and modeled traveltimes estimated usually

using ray approach. If the traveltimes are picked with errors or the modeled traveltimes are

not proper we obtain serious errors of tomographic inversion.

One of the most important question is what is the smallest feature that can be

reconstructed by traveltime tomography. This smallest feature is dependent on traveltime

uncertainties (frequency bandwidth of the signal) and medium velocity. Williamson (1991)

stated that the size of the smallest feature which can be accurately reconstructed using

traveltime tomography has a dimension that is about equal to the first Fresnel zone radius. He

also suggested that detail finer than this minimum size will not necessarily disappear but may

be significantly defocused.

The first Fresnel zone radius is given by (L where L is the propagation distance

and is the wavelength. Using 850 Hz as the high frequency limit and 4 km/s as the medium

velocity, this Fresnel zone criterion suggests that the smallest resolvable feature at the center

of the tomograms (L = 100 m) in the reservoir interval is approximately 20 m. Choosing a

grid size less than 20 m may cause the inverse problem to be an underdetermined system.

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Krajewski et al. (1989) suggested that a low velocity anomaly can be recognizable in

a tomogram if it causes a traveltime delay along an affected raypath that is larger than the

traveltime uncertainty. If we take 0.5 ms for the picking error, 4 km/s for the medium

velocity, and assume a 15% velocity decrease as a result of the temperature change, then the

minimum resolvable body size would be 11 m. So we can use a 9 x 9 m grid for the

tomographic inversion.

Additional condition was proposed by Lee et al. (1995): in the final velocity images ,

the velocity at cells not covered by more than five rays should be reset to the background

velocity (Lee et al. 1995).

Therefore in the process of constructing models we can’t introduce too small features

of the velocity fields. In the process of inversion we can’t use too small dimensions of the

cells to avoid underdetermination of the set of tomographic equation.

If we want establish relation between ray approach and wave approach for considered

seismogeological models we must take into account seismic signal. Using ray approach we

assume so called high frequency approximation of wave propagation. It means that instead of

signal with limited frequency band we assume propagation of spike. In real data we have

signals with limited frequency band and dominating pick frequency. The problem arises what

kind of the signal we should to use and which phase of this signal we must pick on field

records (simulated in our analysis using finite difference modeling) to obtain good relation

between model (theoretical) traveltimes and observed traveltimes. To answer this question the

analysis has been undertaken with application of signals of different phase characteristics.

Using these signals as input to Finite Difference Modeling (FDM) procedure the records were

calculated for the considered models and the first break arrivals were picked. The resulting

traveltimes of head waves and turning waves were compared with theoretically calculated

traveltimes using #RAYEDH and Turning Ray Tracing procedure.

To estimate the parameters of the first break signals before modeling the land seismic

records were introduced as input data to Interactive Spectral Analysis procedure of ProMAX

system. Two type of land seismic records were used: dynamite records and vibroseis records.

In fig. 3.1 typical dynamite record (after AGC application) is presented and in fig. 3.2 typical

vibroseis record from the same area (after application AGC and frequency filter 4-8-60-80) is

presented.

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Fig. 3.1. Typical dynamite record (after AGC application)

Fig. 3.2. Typical vibroseis record (after application AGC and frequency filter 4-8-60-80)

The results of interactive spectral analysis of first breaks of the records from fig. 3.1

and 3.2 are presented in fig. 3.3 and 3.4.

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Fig. 3.3. The results of interactive spectral analysis of first breaks of the records from fig. 3.1 (the blue window defines the area of spectral analysis, the amplitude spectrum is displayed on the right)

Fig. 3.4. The results of interactive spectral analysis of first breaks of the records from fig. 3.2 (the blue window defines the area of spectral analysis, the amplitude spectrum is displayed on the right)

On the base of the results of interactive spectral analysis of many dynamite and

vibroseis records it was established the representative signal of first breaks as the minimum-

phase and zero-phase Ricker signal with dominating frequency 40 Hz. To model theoretical

records the model of near surface layer with low velocity anomaly presented in fig. 3.5 was

chosen. The parametrization window of FDM procedure with used parameters is presented in

fig 3.6. The selected modeled records are presented in figs. 3.7 – 3.9

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Fig. 3.5. The model of near surface layer with low velocity anomaly applied in FDM

Fig. 3.6. The parametrization window of FDM procedure with applied parameters

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Fig. 3.7. The selected modeled record for the case of minimum phase signal

Fig. 3.8. The enlargement of selected modeled record for the case of minimum phase signal with result of picking phase maksimum (snap peak), for offsets 750 – 1050 m the effect of low velocity anomaly is visible (longer times of first breaks of head wave)

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Fig. 3.9. The enlargement of selected modeled record for the case of minimum phase signal with result of picking phase zero (snap zero + to −)

In fig. 3.10 the comparison is presented of the picked traveltimes for the case of zero-phase signal (snap peak) and minimum phase signal (pick zero) with predicted traveltimesusing ray approach (#RAYEDH).

Fig. 3.10. Comparison of the picked traveltimes for the case of zero-phase signal (snap peak) and minimum phase signal (pick zero) with predicted traveltimes using ray approach (#RAYEDH).

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The analysis of the results confirmed that the traveltimes predicted using ray approach

are close to the traveltimes obtained in the process of first break picking of records with

minimum phase signal and zero phase selected for picking. It was confirmed too that the

behaving of the zero-phase records are very similar to minimum-phase records in the zone of

first breaks.

III.2. IMAGING OF NEAR SURFACE VELOCITY HETEROGENEI TIES OF THE MEDIUM IN WAVE PATTERN OF ACOUSTIC MODELING

For the purpose of the analysis of first and later breaks of the head waves on reflection

records the 2-D finite difference acoustic modeling has been applied (Kasina, Nasar 2005).

The main goal of this modeling was to define the effect of heterogeneities dimensions,

velocity distributions in the near-surface layer as well as the seismic signal parameters on the

wave pattern of head waves breaks and on head waves traveltimes. The analysis has been

undertaken to assess the possibility of applications these breaks for recovering near-surface

velocity distributions by means of head wave tomography.

VELOCITY MODELS AND MODELING PARAMETERS

The velocity model of near-surface medium was composed of three horizontal layers

with parameters described in fig. 3.11. In the second layer the low velocity body was inserted

of effective width 20 m. The velocity of this body named in the analysis „velocity anomaly”

was changing during consecutive computation from 600 m/s through 800 m/s to final value of

1000 m/s. The thickness of the body was changing from 50 m through 20 m to 10 m. The

velocity model (fig. 3.11) has been defined for 4001 CDP points with CDP interval equal 1 m.

The off-end spread has been used with 400 receiver stations (channels) and receiver

interval 10 m. The shot has been located at the left edge of the velocity model on the surface.

The zero-phase Ricker signal has been used as point source wavelet with dominant frequency

50 Hz, 70 and 100 Hz. The input computation grid had dimensions 1 m x 1 m, and has been

automatically modified during computation to effective grid dimensions protecting against

numerical errors.

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Fig. 3.11. The velocity model of near-surface layer used in synthetic records calculation

THE WAVE PATTERN OF SYNTHETIC RECORDS

The case of the near-surface medium without velocity anomaly

The synthetic record calculated for the layered model from fig. 3.11 but without velocity

anomaly using zero-phase Ricker source signal with dominant frequency 50 Hz is presented

fig. 3.12 for the case of split spread. Additionally the results of approximate interactive

apparent velocity analysis have been presented in this figure for the purpose of wave

identification. Analysis of the wave pattern of these figures lets us to draw the following

conclusions:

- the waves which can be identified without any difficulties in first breaks comprise the direct

wave and the head wave connected with the deepest refractor with velocity 2500m/s,

- the breaks of head waves are well distinguished in first and later breaks,

- the head wave connected with the deepest refractor with velocity 2500m/s is dominated in

first breaks and can be identified in broad range of offsets,

- it is possible to realize the picking of head waves not only in first breaks but in later breaks

too,

- the differences in apparent velocity are as usual very useful in head waves identification.

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The case of the near-surface medium with velocity anomaly

In the next step the calculation of the records have been accomplished for the case of

near-surface medium with velocity anomaly in the second layer. The results of the

calculations for the selected thickness of anomaly (50 m) and for the selected velocity of

anomaly (600 m/s) are presented in fig. 3.13, 3.14, 3.15. It can be seen that the traveltime

delay related to low velocity anomaly is clearly marked in the first breaks of the head wave

propagating from the deepest refractor and in the breaks of head wave from the second

boundary.

Fig. 3.12. The synthetic record calculated for the case of the split spread and the velocity model without anomaly with results of approximate interactive apparent velocity analysis

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Fig. 3.13. The synthetic record calculated for the layered model with velocity anomaly (thickness 20 m, velocity 800 m/s) using zero-phase Ricker source signal with dominant frequency 50 Hz; the zone of breaks in the area of imaging traveltime delays related to velocity anomaly is located in the range of offsets 1800 – 2510 m

Fig. 3.14. Enlargement of first breaks zone of the synthetic record from fig. 4 in the area of imaging traveltime delays related to velocity anomaly (thickness 20 m)

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Fig. 3.15. Enlargement of later breaks zone of the synthetic record from Figure 3.14 in the area of imaging traveltime delays related to velocity anomaly (thickness 20 m)

The relation of the traveltimes after changing velocity of anomaly to 800 m/s and 1000

m/s can be defined from the differences of first breaks traveltimes presented in fig. 3.16.

These differences achieve the values from 6 ms to 16 ms ms in the range of anomaly offsets

and their graph is nearly the same for pick frequency 50 and 70 Hz.

Fig. 3.16. Traveltimes differences between first breaks of head wave for the model without and with anomaly for different values of anomaly velocity: a) V = 600 m/s, b) V = 800 m/s, c) V = 1000 m/s; dominant frequency 50 Hz, anomaly thickness 20 m

The next model parameter taken into consideration was the thickness of velocity

anomaly. It was changed from 50 m through 20 m to 10 m for the velocity 600 m/s. The

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resulting difference curves of first breaks traveltimes have been presented in fig. 3.17. These

differences achieve the values from 5 ms to about 18 ms in the range of anomaly offsets.

Fig. 3.17. Traveltimes differences between first breaks of head wave for the model without and with anomaly for different values of anomaly thickness: a) ∆h = 50 m, b) ∆h = 20 m, c) ∆h = 10 m; pick frequency 50 Hz, anomaly velocity 800 m/s

In the last step the effect of pick frequency of source signal on the first breaks

traveltimes has been analysed. Three pick frequency have been used during modeling: 50, 70

and 100 Hz. The difference curves of first breaks traveltimes are presented in fig. 3.18. We

can observe some dependence of difference curve values on the offset. The average value of

difference curve achieves the greatest value about 7 ms for the case of pick frequencies 50

and 100 Hz. For the frequency pairs (50 Hz, 70 Hz) and (70 Hz, 100 Hz) the average value is

about 3,5 ms.

Fig. 3.18. Traveltimes differences between first breaks of head wave for the model with anomaly for different values of pick frequency of source signal: a) ∆t = t(50 Hz) – t(100 Hz), b) ∆t = t(50 Hz) – t(70 Hz), c) ∆t = t(70 Hz) – t(100 Hz), anomaly velocity 800 m/s,anomaly thickness 20 m

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The results of undertaken model calculations let to draw the conclusions important

from the point of view of application seismic tomography for imaging near-surface velocity

heterogeneities:

- the breaks of head waves are well distinguished for the discussed near-surface layered

medium in first and later breaks,

- the main factors affecting the first breaks traveltimes in the model with velocity anomaly are

the anomaly thickness and anomaly velocity; the change of traveltimes achieves values from

about 6 to 18 ms for the discussed models,

- analysis of first breaks traveltimes obtained for different pick frequencies of source signal

confirms that the effect of variation of pick frequencies from 50 to 90 Hz may achieve

several milliseconds.

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III.3. THE SOLUTION OF DIRECT AND INVERSE TASK OF T HE REFRACTION TOMOGRAPHY FOR SELECTED MODELS AND STATICS ESTIMATIO N

In this chapter several models of near surface layer have been used to solve the direct

task of head wave tomography: the model of gradient layer with velocity anomaly over half-

space with constant velocity, the model of gradient half-space without any refractor, the

model of constant velocity layer with anomaly over half-space with constant velocity, the

model of two constant velocity layers (with velocity anomaly in the first or in the second

layer) over constant velocity half-space, the model consisting of three constant velocity layers

(with velocity anomaly in the second layer) over constant velocity half-space.

For each model with a refractor the traveltimes and ray trajectories of head waves

were defined using Fortran programs (#RAYEDH, #RAYEDT, #RAYPLT). For most of the

models the velocity grid consisted of the cells with dimensions: 25 m in „x” direction and 8 m

in „z” direction (125 columns and 26 rows). For multi-layer models the cells had dimension

1 m in vertical dimension. The points of rays were defined with the interval 0.5 m. The

distance beetwen receivers had the value 50 m typical for land seismics. The calculations

were realized for two shots placed outside of the spread (one shot on the left and one on the

right) or for many shots for the variant of seismic profiling with split-spreads (48 receivers,

shot interval 100 m, 10 shots). For the model of gradient half-space the traveltimes were

calculated using Turning Ray Tracing process in ProMAX system. Additionally the wave

pattern of the records was estimated using Finite Difference Modeling process in ProMAX

system.

The traveltimes of head waves estimated for the models with the velocity anomaly

were used as input data to tomographic inversion realized with the help of #INVERSDH. The

starting models were created from the models used in #RAYEDH through the elimination of

the velocity anomalies. Generally it means that the starting models have been estimated using

additional information from shot holes or we have realized standard refraction interpretation

of first breaks (e.g. using commonly available Hampson-Russel program named GLI2D or

GLI3D). It must be stressed that the main goal of presented analysis is the estimation of

tomographic effectiveness in reconstruction of the velocity anomalies included beetwen

refractors and the application of reconstructed velocity fields to static corrections estimation.

Additionally, the Turning Ray Tomography process in ProMAX system was used for the case

of the model of gradient half-space.

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In the process of inversion by means of #INVERSDH the following values of main

parameters were used selected during many tests:

- the velocity grid consisted of the cells with dimenions: 25 m in „x” direction and 8 m in „z”

direction (125 columns and 26 rows),

- the number of iterations in gradient conjugate method: 30,

- operator of convolutional quelling q=1.0/0.5,

- the value of constraint coefficient λ=10,

- the dimensions of spatial smoothing operator of resulting velocity fields: 2×2,

- the number of inversion iterations: 3.

For the purpose of quantitative evaluation of the inversion effectiveness the following

errors were calculated:

( )RMSE

V V

R

i est ii

R

=−

=∑ , ,mod

/2

1

1 2

( )

RMSDT

t t

N

i est ii

N

=−

=∑ , ,mod

/2

1

1 2

where:

V i,est – the value of velocity estimated in the i – th node,

Vi,mod – the known value of model velocity in the i – th node,

R – the number of nodes in the velocity grid,

ti,est – the traveltime for i – th ray calulated for the velocity field estimated in the

inversion process,

ti,mod – the observed (known from seismic modeling) traveltime for i-th ray,

N – the number of considered seismic rays

The first error RMSE was estimated using #RMSDV Fortran program, the second

error RMSDT was estimated in the #INVERDH Fortran program. Additionally the bitmaps of

resulting velocity fields will be presented (using SURFER program) as well as the bitmaps of

differences between assumed model velocity fields and resulting from inversion.

Before tomographic inversion the refractor in starting models is removed and the layer

below is replaced by the medium with velocity equal to the velocity a little above the

refractor. This was done to avoid the effect of layer below refractor on behaving rays a little

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above the refractor. As a result all the velocity fields resulting from inversion do not include

refractor and do not include the layer below the refractor.

At the end for each velocity field resulting from tomographic inversion the static

corrections will be estimated using the #STATICS Fortran program and diplayed using

GRAPHER program. The statics corrections (statics) are calculated assuming datum at the

position of the refractor.

III.3.1. The two layer model with gradient medium over refractor

The first analysed model consists of gradient medium over half-space with constant

velocity (fig. 3.19). The depth of the refractor is 176 m, its velocity – 3500 m/s. In the first

layer the velocity anomaly is placed with the following parameters: depth 48 m, width 12

CDP’s (300 m), velocity 800 m/s. The velocity in the first layer is incresing from 960 m/s

near surface to 1670 m/s near refractor. This starting model was modified through changing

anomaly velocity from 800 m/s to 600 m/s and 400 m/s for the case of low velocity anomaly.

Higher anomaly velocities (1500 m/s and 1800 m/s) were considered too. Additionaly the

anomaly depth was varied from 48 m to 80 m. The dimensions of the velocity cell were 1

CDP (25 m) in the horizontal direction and 8 m in the vertical direction.

Fig. 3.19. Two layer velocity model with velocity anomaly and gradient medium over constant velocity half-space, refractor depth 176 m

The parameters used in #RAYEDH for calulations of traveltimes and ray trajectories

had the following values: positions of shots 1000 and 2000 m, receiver interval 50m, spread

of 24 geophones. The example of ray trajectories for velocity anomaly 600 m/s are presented

in fig. 3.20.

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Fig. 3.20. The ray trajectories for the model with velocity anomaly 600 m/s, 2 shot points, refractor depth 176 m; horizontal axis – distance in m, vertical axis – depth in m

Behaving of the traveltime graphs for the considered 3 variants of velocity anomaly

(400, 600 and 800 m/s) is presented in fig. 3.21.

Fig. 3.21. The traveltime graphs for the considered 3 variants of velocity anomaly (400, 600 and 800 m/s), refractor depth 176 m

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In the next figures 3.22 – 3.27 the results of tomographic inversions are presented for

anomaly velocity 400 m/s, refractor depth 176 m and for different iteration number. The

position of assumed anomaly is marked as white rectangle. For the starting velocity model

the traveltime error RMSDT was equal 33.28 ms.

Fig. 3.22. The result of inversion in 1-st iteration (without smoothing) in the case of anomaly velocity 400 m/s (RMSDV= 122 m/s , RMSDT= 17.90 ms)

Fig. 3.23. The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) in the case of anomaly velocity 400 m/s (RMSDV= 122 m/s , RMSDT= 17.90 ms)

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Fig. 3.24. The result of inversion in 1-st iteration after smoothing in the case of anomaly velocity 400 m/s (RMSDV=124 m/s)

Fig. 3.25. The result of inversion in 2-nd iteration (without smoothing) in the case of anomaly velocity 400 m/s (RMSDV=124 m/s, RMSDT=1.19 ms)

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Fig. 3.26. The result of inversion in 2-nd iteration (with smoothing after 1-st iteration) in the case of anomaly velocity 400 m/s (RMSDV= 119 m/s)

Fig. 3.27. The result of inversion in 3-nd iteration (without smoothing after 2-st) in the case of anomaly velocity 400 m/s (RMSDV=117 m/s, RMSDT=0.33 ms)

The results of statics calculations for different iterations in the case of anomaly

velocity 400 m/s are presented below.

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Fig. 3.28. The results of statics calculations for different iterations in the case of anomaly velocity 400 m/s, refractor depth 176 m

In the next figures 3.29 – 3.34 the results of tomographic inversions are presented for

anomaly velocity 600 m/s, refractor depth 176 m and for different iteration number.

Fig. 3.29. The result of inversion in 1-st iteration (without smoothing) in the case of anomaly velocity 600 m/s (RMSDV= 80 m/s , RMSDT= 2.85 ms)

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Fig. 3.30. The differences between the true model and the reconstructed velocity Field in 1-st iteration (without smoothing) in the case of anomaly velocity 600 m/s (RMSDV= 80 m/s , RMSDT= 2.85 ms)

Fig. 3.31. The result of inversion in 2-nd iteration (without smoothing) in the case of anomaly velocity 600 m/s (RMSDV=79 m/s, RMSDT=0.33 ms)

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Fig. 3.32. The result of inversion in 2-nd iteration (with smoothing after 1-st) in the case of anomaly velocity 600 m/s (RMSDV=79 m/s)

Fig. 3.33. The result of inversion in 3-nd iteration (without smoothing after 2-nd iteration) in the case of anomaly velocity 600 m/s (RMSDV=79 m/s, RMSDT=0.03 ms)

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Fig. 3.34. The results of statics calculations for different iterations in the case of anomaly velocity 600 m/s, refractor depth 176 m

In the next figures 3.35 – 3.39 the results of tomographic inversions are presented for

anomaly velocity 800 m/s, refractor depth 176 m and for different iteration number.

Fig. 3.35. The result of inversion in 1-st iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 53 m/s , RMSDT= 0.88 ms)

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Fig. 3.36. The differences between the true model and the reconstructed velocity field in the 1-st iteration (without smoothing) inr the case of anomaly velocity 800 m/s (RMSDV= 53 m/s , RMSDT= 0.88ms)

Fig. 3.37. The result of inversion in 2-nd iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV=53 m/s, RMSDT=0.08 ms)

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Fig. 3.38. The result of inversion in 2-nd iteration (with smoothing after 1-st) in the case of anomaly velocity 800 m/s (RMSDV=52 m/s)

Fig. 3.39. The result of inversion in 3-nd iteration (without smoothing after 2-ndt) in the case of anomaly velocity 800 m/s (RMSDV= 53 m/s, RMSDT= 0.01 ms)

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Fig. 3.40. The results of statics calculations for different iterations in the case of anomaly velocity 800 m/s, refractor depth 176 m

In the next figures 3.41 – 3.45 the results of tomographic inversions are presented for

anomaly velocity 1500 m/s, refractor depth 176 m and for different iteration number.

Fig. 3.41. The result of inversion in 1-st iteration (without smoothing) in the case of anomaly velocity 1500 m/s (RMSDV= 34 m/s , RMSDT= 0.53 ms)

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Fig. 3.42. The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) in the case of anomaly velocity 1500 m/s (RMSDV= 34 m/s , RMSDT= 0.53 ms)

Fig. 3.43. The result of inversion in 2-nd iteration (without smoothing) in the case of anomaly velocity 1500 m/s (RMSDV=36 m/s, RMSDT=0.06 ms)

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Fig. 3.44. The result of inversion in 2-nd iteration (with smoothing after 1-st) in the case of anomaly velocity 1500 m/s (RMSDV=36 m/s)

Fig. 3.45. The result of inversion in 3-nd iteration (without smoothing after 2-st) for the case of anomaly velocity 1500 m/s (RMSDV= 36 m/s, RMSDT= 0.01 ms)

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Fig. 3.46. The results of statics calculations for different iterations in the case of anomaly velocity 1500 m/s, refractor depth 176 m

In the next figures 3.47 – 3.51 the results of tomographic inversions are presented for

anomaly velocity 1800 m/s, refractor depth 176 m and for different iteration number.

Fig. 3.47. The result of inversion in 1-st iteration (without smoothing) in the case of anomaly velocity 1800 m/s (RMSDV= 76 m/s , RMSDT= 1.13 ms)

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Fig. 3.48. The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) in the case of anomaly velocity 1800 m/s (RMSDV= 76 m/s , RMSDT= 1.13 ms)

Fig. 3.49. The result of inversion in 2-nd iteration (without smoothing) in the case of anomaly velocity 1800 m/s (RMSDV= 74 m/s, RMSDT= 0.13 ms)

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Fig. 3.50. The result of inversion in 2-nd iteration (with smoothing after 1-st) in the case of anomaly velocity 1800 m/s (RMSDV= 75 m/s)

Fig. 3.51. The result of inversion in 3-nd iteration (without smoothing after 2-st) in the case of anomaly velocity 1800 m/s (RMSDV= 74 m/s, RMSDT= 0.01 ms)

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Fig. 3.52. The results of statics calculations for different iterations in the case of anomaly velocity 1800 m/s, refractor depth 176 m

Table 1. The values of RMSDV ( m/s) for different values of anomaly velocity and for different options of smoothing application

Anomalyvelocity(m/s)

Starting Values

First Iteration

Second iteration Withoutsmoothing

Second iteration With

smoothing

Thirditeration

without smoothing 400 129 126 124 119 117

600 81 80 79 79 79

800 55 53 53 52 53

1500 38 34 36 36 36

1800 78 76 74 75 74

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Table 2. The values of RMDT (ms) for different values of anomaly velocity and for different number of iterations

III.3.2. The two layer model with gradient medium over refractor (depth 120 m)

The next analysed model consists of gradient medium over half-space with constant

velocity (fig. 3.53). The depth of the refractor is 120 m, its velocity – 2500 m/s. In the first

layer the velocity anomaly is placed with the following parameters: depth 48 m, width 12

CDP’s (300 m), velocity 800 m/s. The velocity in the first layer is incresing from 960 m/s

near surface to 1440 m/s near refractor. The dimensions of the velocity cell were 1 CDP (25

m) in the horizontal direction and 8 m in the vertical direction.

Fig. 3.53. Two layer velocity model with velocity anomaly 800 m/s and gradient medium over constant velocity half-space, refractor depth 120 m

Anomalyvelocity(m/s)

Starting Values

First iterationwithout

smoothing

Seconditeration without

smoothing

Thirditerationwithout

smoothing 400 33.88 17.90 1.19 0.33

600 16.88 2.85 0.31 0.03

800 10.06 0.88 0.08 0.01

1500 4.07 0.53 0.06 0.01

1800 7.37 1.13 0.13 0.01

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Fig. 3.54. The ray trajectories for the model with velocity anomaly 800 m/s, 2 shot points, refractor depth 120 m; horizontal axis – distance in m, vertical axis – depth in m

Fig. 3.55. The result of inversion in 1-st iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 59 m/s , RMSDT= 0.71 ms)

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Fig. 3.56. The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 59 m/s , RMSDT= 0.71 ms)

Fig. 3.57. The result of inversion in 2-nd iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 59 m/s, RMSDT= 0.06 ms)

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Fig. 3.58. The result of inversion in 2-nd iteration (with smoothing after 1-st) in the case of anomaly velocity 800 m/s (RMSDV= 59 m/s)

Fig. 3.59. The result of inversion in 3-nd iteration (without smoothing after 2-nd) for the case of anomaly velocity 800 m/s (RMSDV= 58 m/s, RMSDT= 0.01 ms)

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Fig. 3.60. The results of statics calculations for different iterations in the case of anomaly velocity 800 m/s, refractor depth 120 m

III.3.3. The model of gradient half-space

In the third analysed model of gradient half-space the velocity increased with depth from

960 m/s near surface to 1670 m/s near the bottom of the model at depth 200 m. In the gradient

medium the anomaly is placed with low velocity 800 m/s.. The anomaly is located at depth

range 48 to 80 m with effective width 12 CDP’s (from 80 to 92 CDP, 300 m), The

dimensions of the velocity cell were 1 CDP (25 m) in the horizontal direction and 8 m in the

vertical direction. The discussed velocity model is presented in fig. 3.61, the example of ray

trajectories are displayed in fig. 3.62. In figs. 3.63 and 3.64 the records resulting from Finite

Difference Modeling are presented.

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Fig. 3.61. The model of gradient half-space with velocity anomaly 800 m/s.

Fig. 3.62. The example of ray trajectories for the model of gradient half-space with velocity anomaly 800 m/s. horizontal axis – distance in, vertical axis – depth in m

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Fig. 3.63. The example of ray trajectories for the model of gradient half-space with velocity anomaly 800 m/s. horizontal axis – distance in, vertical axis – depth in m

Fig. 3.64. The example of modeled record obtained for the model of gradient half- space with velocity anomaly 800 m/s by means of Finite Difference Modeling; the first breaks picks are shown using red colour

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Fig. 3.65. The enlargement of record from fig. 3.62 with first breaks influenced by low velocity anomaly (increased traveltimes)

For the analysed model the direct task of turning ray tomography was solved using

ProMAX process named Turning Ray Tracing for model with anomaly. The calculated

traveltimes were then treated as input data to tomographic inversion carried out with the help

of ProMAX process named Turning Ray Tomography. The gradient model without anomaly

has been applied as the starting velocity model. To define the effect of the possible errors of

the starting model estimation the assumed velocity gradient was decreased or increased by

10% and 20%. The calculations were realized for the case of many shots (20shots were used)

taking into account every one shot, every third, every fourth and every fifth. The split-spread

was use with 49 receivers (49 channels), receiver interval 50 m, shot interval 100 m, shot on

position of 25-th receiver. The dimensions of velocity grid: 25 m in horizontal direction and

8m in vertical direction.

The results of tomographic inversion for discussed model are presented in

figs. 3.66 – 3.71. The estimated statics is showned in fig. 3.72.

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Fig. 3.66. The result of inversion obtained in ProMAX system by means of the process Turning Ray Tomography with application of many shots (every third shot, shot interval 300 m), RMSDV= 60 m/s

Fig. 3.67 The differences between the true model and velocity field reconstructed by means of the Turning Ray Tomography process in the case of anomaly velocity 800 m/s, RMSDV= 61 m/s

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Fig. 3.68. The result of inversion obtained in ProMAX system by means of the process Turning Ray Tomography with application of many shots (every third shot, shot interval 300 m); gradient of starting velocity model decreased by 10%, RMSDV= 61 m/s

Fig. 3.69. The differences between the true model and velocity field reconstructed by means of the Turning Ray Tomography process in the case of anomaly velocity 800 m/s and gradient gradient of starting velocity model decreased by 10%

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Fig. 3.70. The result of inversion obtained in ProMAX system by means of the process Turning Ray Tomography with application of many shots (every third shot, shot interval 300 m); gradient of starting velocity model increased by 10%, RMSDV= 60 m/s

Fig. 3.71. The differences between the true model and velocity field constructed by means of the Turning Ray Tomography process in the case of anomaly velocity 800 m/s and gradient gradient of starting velocity model increased by 10%, RMSDV= 60 m/s

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Fig. 3.72. The results of statics calculations using velocity fields resulting in ProMAX system from the process Turning Ray Tomography with application of many shots (every third shot, shot interval 300 m) and different errors of of starting velocity model gradient

III.3.4. The two layer model with constant velocities

The next analysed model consists of constant velocity (1200 m/s) layer over half-space

with constant velocity 2500 m/s (fig. 3.73). The depth of the refractor is 120 m. In the first

layer the velocity anomaly is placed with the following parameters: depth 48 m, width 12

CDP’s (300 m), velocity 800 m/s. The dimensions of the velocity cell were 1 CDP (25 m) in

the horizontal direction and 8 m in the vertical direction.

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Fig. 3.73. The two layer model with constant velocities, refractor depth 120 m

Fig. 3.74. The example of ray trajectories for the velocity anomaly 800 m/s in first layer , horizontal axis – distance in, vertical axis – depth in m

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Fig. 3.75. The enlargement of record with picked first breaks influenced by low velocity anomaly (increased traveltimes)

Fig. 3.76. The result of inversion in 1-st iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 58 m/s , RMSDT= 1.10 ms)

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Fig. 3.77. The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 58 m/s , RMSDT= 1.10 ms)

Fig. 3.78. The result of inversion in 2-nd iteration (without smoothing) for the case of anomaly velocity 800 m/s (RMSDV= 55 m/s, RMSDT= 0.17 ms)

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Fig. 3.79. The result of inversion in 2-nd iteration (with smoothing) for the case of anomaly velocity 800 m/s (RMSDV= 55 m/)

Fig. 3.80. The result of inversion in 3-rd iteration (without smoothing-after 2nd) in the of anomaly velocity 800 m/s (RMSDV= 55 m/s, RMSDT= 0.02 ms)

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Fig. 3.81. The results of statics calculations for different iterations in the case of anomaly velocity 800 m/s, refractor depth 120 m

III.3.5. The three layer model with constant velocities

The fifth analysed model consists of two layered over half-space with constant

velocity (fig. 3.82). The depth of the refractor is 176 m, its velocity – 2500 m/s. In the first

layer the low velocity anomaly is placed with the following parameters: depth 48 m, width 12

CDP’s (300 m), velocity 800 m/s. The velocity in the first layer is 1200 m/s, in the second -

1500 m/s . The dimensions of the velocity cell were 1 CDP (25 m) in the horizontal direction

and 1 m in the vertical direction. The same model but without anomaly was used as starting

velocity model in first iteration. Additionaly the values of first and second layer velocities

have been change (in the range of ± 5% and ± 10%) to simulate the errors of the starting

model. The position of low velocity anomaly (1100 m/s) in the second layer was also taken

into consideration.

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Fig. 3.82. The two layer model with constant velocities over half-space with constant velocity, velocity anomaly 800 m/s

Fig. 3.83. The example of ray trajectories for the velocity anomaly 800 m/s in first layer , horizontal axis – distance in, vertical axis – depth in m

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Fig. 3.84. The result of inversion in 1-st iteration (without smoothing) for the sase of anomaly velocity 800 m/s (RMSDV= 53 m/s , RMSDT= 1.39 ms)

Fig. 3.85. The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 53 m/s , RMSDT= 1.39 ms)

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Fig. 3.86. The result of inversion in 2-nd iteration (without smoothing) for the case of anomaly velocity 800 m/s (RMSDV= 52 m/s , RMSDT=0.07)

Fig. 3.87. The result of inversion in 2-nd iteration (with smoothing) for the case of anomaly velocity 800 m/s (RMSDV= 52 m/s )

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Fig. 3.88. The result of inversion in 3-rd iteration (without smoothing) for the case of anomaly velocity 800 m/s (RMSDV= 53 m/s , RMSDT= 0.01 ms)

Fig. 3.89. The results of statics calculations for different iterations in the case of anomaly velocity 800 m/s in first layer, refractor depth 176 m

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Fig. 3.90. The two layer model with constant velocities over half-space with constant velocity, velocity anomaly 1100 m/s

Fig. 3.91. The example of modeled record of the two layer model with constant Velocities over half-space with constant velocity obtained by means of Finite Difference Modeling, velocity anomaly 1100 m/s in first layer; the first breaks picks are shown using red colour

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Fig. 3.92. The enlargement of record with picked first breaks influenced by low velocity anomaly (increased traveltimes)

Fig. 3.93. The example of ray trajectories for the velocity anomaly 1100 m/s in second layer , horizontal axis – distance in, vertical axis – depth in m

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Fig. 3.94. The result of inversion for first iteration without smoothing in the case of anomaly velocity 1100 m/s (RMSDV= 56 m/s , RMSDT= 0.41 ms)

Fig. 3. 95. The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) for the case of anomaly velocity 1100 m/s (RMSDV= 56 m/s , RMSDT= 0.41 ms)

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Fig. 3.96. The result of inversion for second iteration without smoothing for the case of anomaly velocity 1100 m/s (RMSDV= 56 m/s , RMSDT= 0.01 ms)

Fig. 3. 97. The result of inversion for second iteration with smoothing in the case of anomaly velocity 1100 m/s (RMSDV= 56 m/s )

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Fig. 3. 98. The result of inversion for third iteration without smoothing in the case of anomaly velocity 1100 m/s (RMSDV= 56 m/s , RMSDT= 0.00 ms)

Fig. 3.99. The results of statics calculations for different iterations in the case of anomaly velocity 1100 m/s in second layer, refractor depth 176 m

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Fig.3.100. The results of inversion for starting model with first and second layer velocity decreased by 10%. Third iteration without smoothing in the case of anomaly velocity 800 m/s (RMSDV= 169 m/s ,RMSDT= 0.06 ms)

Fig. 3.101. Statics results for starting model with first and second layer velocity decreased by 10% in the case of low velocity anomaly 800 m/s

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Fig. 3.102. Statics results for starting model with first and second layer velocity increased by 10% in the case of low velocity anomaly 800 m/s

Fig. 3.103. Statics results for starting model with first and second layer velocity decreased by 5% in the case of low velocity anomaly 800 m/s

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Fig. 3.104. Statics results for starting model with first and second layer velocity increased by 5% in the case of low velocity anomaly 800 m/s

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III.3.6. The four layer model with constant velocities

The last analysed model consists of three layers with constant velocities over half-

space with constant velocity (fig. 3.105). The depth of the refractor is 199 m, its velocity –

2500 m/s. The velocity in the first layer is 800 m/s. Second layer with velocity 1200 m/s, third

layer with velocity 1500 m/s . The low velocity anomaly 800 m/s was placed in second

layer. Additionaly the anomaly depth was varied from 74 m to 123 m. The dimensions of the

velocity cell were 1 CDP (25 m) in the horizontal direction and 1 m in the vertical direction.

The results of calculations for this model are presented in figs. 3.106-3.113.

Fig. 3.105. The model of fourth layers with constant velocities.

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Fig. 3.106. The example of ray trajectories for the velocity anomaly 800 m/s in second layer , horizontal axis – distance in, vertical axis – depth in m

Fig. 3.107. The example of modeled record of the fourth layer model with constant velocities over half-space with constant velocity obtained by means of Finite Difference Modeling, velocity anomaly 800 m/s in second layer; the first breaks picks are shown using red colour

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Fig. 3.108. The result of the inversion in first iteration without smoothing in the case of anomaly velocity 800 m/s (RMSDV= 60 m/s , RMSDT= 1.45 ms)

Fig. 3.109. The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 60 m/s , RMSDT= 1.45 ms)

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Fig. 3.110. The result of the inversion in second iteration without smoothing in the case of anomaly velocity 800 m/s (RMSDV= 59 m/s , RMSDT= 0.06 ms)

Fig. 3.111. The result of the inversion in second iteration with smoothing in the case of anomaly velocity 800 m/s (RMSDV= 58 m/s )

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Fig. 3.112. The result of the inversion in third iteration without smoothing in the case of anomaly velocity 800 m/s (RMSDV= 59 m/s , RMSDT= 0.00 ms)

Fig. 3. 113. The results of statics calculations for different iterations in the case of anomaly velocity 800 m/s in second layer, refractor depth 199 m

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III.4. THE ANALYSIS OF THE RESULTS OF DIRECT AND IN VERSE PROBLEM SOLUTION FOR MODEL DATA

The analysed results of direct and inverse problem solution comprised the following

data: the graphs of traveltimes, the bitmaps of velocity fields, the bitmaps of differences

between assumed model velocity fields and fields resulting from tomographic inversion, the

RMSDV and RMSDT errors defined in introduction to chapter III.3, the theoretical records

calculated using Finite Difference Modeling process, the graphs of ray trajectories and the

graphs of static corrections. The most important conclusions from the analyses relating to

tomographic inversion effectiveness and to static corrections estimation are presented below.

The analyses of the ray trajectories graphs of turning wave in the case of gradient

medium with velocity anomalies confirmed that the behaving of rays in discrete velocity

fields defined with application of finite difference methods is very complicated. The strong

rays deviations and oscillations may be observed (e.g. fig.3.62a) causing essential and

artificial increasing of wave traveltimes not observed in the theoretical records (e.g. 3.62b)

obtained using wave approach (finite difference modeling). It means that applications of these

traveltimes in the tomographic inversion for estimation the differences between observed

(resulting from picking) and modeled traveltimes may be the source if serious errors and may

decrease in many cases the effectivenes of turning wave tomography (figs 3.63 – 3.69). It

seems that application of wave approach (finite difference modeling) to solution direct

problem of tomography let us to avoid some of these errors. Otherwise we must very carefully

analyse the calculated traveltimes and remove some of the erroneous traveltimes before

inversion. Additionally in the case of turning wave tomography we can observe anomalous

extending of the resulting velocity anomalies in the main, horizontal direction of turning

wave propagation.

In the case of head wave tomography behaving of head wave rays is generally much

more regular (figs 3.20, 3.54, 3.83, 3.85, 3.93). Such a behaving of rays in the case of thin

low velocity layer does not create optimal conditions for suitable illumination of the velocity

anomalies. To obtain good results of the tomographic inversion we need the rays going

through the anomaly as well as the rays which omitt the anomaly zone. This condition may be

satisfied only in the case of small velocity anomalies (figs 2.28 - 2.31). In other case the

resolution of tomographic solutions in the vertical direction is very weak and we can’t define

the proper position of anomaly in depth.

Analysing the velocity fields resulting from head wave tomography in the case of

different models of low velocity layer (chapter III.3) we can notice that the vertical resolution

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of velocity fields is very weak but horizontal resolution is very good. Then although we

cannot define the proper depth of anomaly we can estimate quite well its horizontal position.

Additionally we can observe very interesting compensating effect in vertical direction

(figures 3.23, 3.30, 3.36, 3.42, 3.56, 3.76, 3.85, 3.89, 3.96): decreasing of velocity in one zone

(connected with low velocity anomaly) is compensated by increasing of velocity outsite this

zone (below and above) . Thanks to this compensation effect the static corrections defined as

always for vertical propagation of rays are in all cases estimated with good (fig. 3.28) or very

good accuracy (figures: 3.34, 3.40, 3.46, 3.52, 3.60, 3.80, 3.89, 3.91, 3.100) if several

iterations of tomographic inversion is applied and proper spatial smoothing of velocity fields

is done. Even if we introduce errors in the velocities of starting models the resulting the

graphs of static corrections are moved by constant value in vertical direction retaining the

same shape (figures: 3.95, 3.100, 3.105, 3.110).

If we analyse the behaving of RMSDV and RMSDT errors of resulting velocity fields

and obtained differences between calculated and assumed traveltimes we can observe that in

consecutive iterations RMSDV errors do not change seriously while RMSDT errors are

decreasing very fast. Decreasing of RMSDT errors correlates with better estimation of static

corrections.

The proper estimation of starting model – which should be very close to the model

without velocity anomaly - plays very important role in the inversion of first breaks by means

of head wave tomography. But this is the well known problem of tomographic inversion. We

cannot reconstruct the boundaries when the propagating rays are perpendicular to these

boundaries. To define position of these boundaries we must use commonly used, very

effective tool known as GLI inversion based on standard refraction interpretation of

traveltimes. In the case of medium consisting of several gradient layers we can use also

turning ray tomography which is very effective tool for defining horizontal layering.

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III.5. THE TOMOGRAPHIC INVERSION OF FIRST BREAKS ON FIELD RECORDS

The evaluation of the tomographic inversion effectiveness is very complicated task

because the facters affecting this effectiveness are connected not only with many parameters

of inversion but in great part with the quality of input data and with the type of

seismogeological model of investigated medium. This is the main reason why all the serious

analyses of this type which appeared in the geophysical papers were based only on model data

(e.g. Dyer and Worthington 1988, Ivansson 1985, Krajewski et al.1989, Peterson et al.1985,

Philips and Fehler 1991). In practice we are first of all interested in the evaluation of the

usefulness of inversion results (e. g. velocity fields) in other essential processing procedures

(statics estimation and migration). If we want to realize such a evaluation in broad range

much volume of field data from areas of different seismogeological models (in our case

models of near surface layer) must be included. Taking into account that presented

dissertation has mainly the form of model study it was very difficult to select such a big

volume of field data and include these data in the presented analysis. However some attempt

was undertaken to show the main aspects connected with application of refraction

tomography to inversion of first breaks on field records.

The selected vibroseis field records from the area of investigations are presented in

fig. 3.114 and 3.115. These are raw field records after application only AGC and bandpass

filter (4-8-40-60). They were recorded with split-spread geometry using 120 channels,

receiver interval 50 m, shot interval 100 m.. The parts of records with visible influence of

statics are marked using red rectangles.

Fig. 3.114. The vibroseis record from shot 184, parts of record with visible influence of statics is marked using red rectangle

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Fig. 3.115. The vibroseis record from shot 201, part of record with visible influence of statics is marked using red rectangle

Fig. 3.116. The vibroseis record from shot 207, part of record with visible influence of statics is marked using red rectangle

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Fig. 3.117. The vibroseis record from shot 215, part of record with visible influence of statics is marked using red rectangle

From analysed records two of them have been selected with numbers 184 and 215.The positions of channels 61-120 of first record (no. 184) and 1-60 of second record (no. 215)were the same. The result of first break picking on these two records are displayed infig. 3.118 –3.119.

Fig. 3.118. The vibroseis record from shot 184 with marked results of first break picking

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Fig. 3.119. The vibroseis record from shot 215 with marked results of first break picking

The picks from right half of the first record (184) and the left part of the second one (215)

were selected as input data to tomographic inversion. These parts of records are presented in

fig. 3.120.

Fig. 3.120. The parts of two records selected as input data to tomographic inversion

In the area of investigation besides vibroseis lines also dynamite lines were recorded.

The graphs of uphole times, elevation and shot depths for dynamite line placed near analysed

vibroseis line are displayed from the line database in fig. 3.121 - 3.123. The data from these

graphs were used to construct the starting model for tomographic inversion.

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Fig. 3.121. The graphs of uphole times, elevation and shot depths for dynamite line

Fig. 3.122. The graph of elevation for dynamite line

Fig. 3.123. The graph of uphole times for dynamite line

From the analysis of the above graphs the parameters of starting model have been evaluated:

- the depth of the bottom of LVL was equal to the constant depth of shot holes (18 m),

- the velocity in the first layer was calculated from uphole times (800 m/s),

- the velocity in the second layer was estimated from apparent velocity of first

breaks (1800 m/s).

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The graphs of traveltimes resulting from first break picking are presented in fig. 3.124.

These times were the input data to tomographic inversion. The #INVERSDH program was

used to realize inversion with the folowing parameters:

- the velocity grid size: 397 x 20,

- the cell dimensions: 25 m x 1 m,

- number of iteration in conjugate gradient method: 300,

- constraint coefficient λ = 10,

- parameter of convolutional quelling: q = 1.0/0.5,

- spatial smoothing operator: 50 m x 2 m,

- number of inversion iterations: 1.

Fig. 3.124. The graphs of traveltimes resulting from picking vibroseis records 184 and 215

The velocity fields resulting from inversion before and after smoothing are displayed

in fig. 3.125 – 3.126. The estimated statics calculated for these two velocity fields using the

bottom of starting model as datum (depth 18 m) are presented in fig. 3.127.

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Fig. 3.125. The velocity field resulting from tomographic inversion

Fig. 3.126. The velocity field resulting from tomographic inversion after spatial smoothing

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Fig. 3.127. The estimated total statics calculated (before and after smoothing) using the bottom of starting model as datum (depth 18 m), horizontal axis – positions of receivers (common for two shots), shot statics was the same

The estimated total statics was applied to traces of two analysed records using Hand Staticsprocedure. The part of records before and after statics are displayed in fig. 3.128 – 3.129.

Fig. 3.128. The part of record 184 before statics (on the left) and after statics (on the right)

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Fig. 3.129. The part of record 184 before statics (on the left) and after statics (on the right)

Analysing the results we can state that the velocity field resulting from tomography

includes some low and high velocity anomaly. Generally the values of statics are small for so

thin low velocity layer. On the records we can see some small improvement in the case of

record 215 in the red rectangle. But such a small range of analysis cannot be the base of

evaluation the effectiveness of tomographic inversion.

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CONCLUSIONS

The main conclusion resulting from the analysis defines the effectiveness of statics

estimation by means of tomographic inversion of first breaks. It was confirmed that for the

discussed models of low velocity layer the head wave tomography based on first breaks

inversion of typical land records may be treated as a effective tool of field statics estimation

although the vertical resolution of resulting velocity fields is relatively very weak.

Analysis the velocity fields result from head wave tomography in the case of

different models of low velocity layer we can notice that the vertical resolution of velocity

fields is very weak but horizontal resolution is very good. Then although we cannot define the

proper depth of anomaly we can estimate quit well its horizontal position. Additionally we

can observe very interesting compensation effect in vertical direction: decreasing of velocity

in one zone (connected with low velocity anomaly) is compensated by increasing of velocity

outside this zone (below and above). Thanks to this compensation effect the static corrections

– defined as always for vertical propagation of rays – are in all cases estimated with good

accuracy if several iterations of tomographic inversion is applied and proper spatial

smoothing of velocity field is done.

The analysed approach to statics estimation together with typical refraction

interpretation (eg. GLI inversion) creates possibility to take into account velocity

heterogeneities occurring between the refractors of near surface layer. Additionally the

analysed approach may be applied not only to the gradient models of low velocity layer like

in the case of turning ray tomography.

Generally it was confirmed that in the discussed case of low velocity layer the vertical

resolution of head wave tomography was weak while the horizontal resolution was very good.

It was quit enough to estimate statics but not enough to identify the velocity anomalies. The

solution to this problem may be to use simultaneously head wave tomography and turning ray

tomography. Of course such a solution may be applied only in case of gradient layers.

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