Refraction-static analysis in 3-D by using time fieldsAtul Jhajhria1,2 and Igor Morozov11 University of Saskatchewan, Saskatoon, Saskatchewan, Canada2 Now at Kurukshetra University, Kurukshetra, Haryana, Indiaa.jhajhria@kuk.ac.in

Introduction

In reflection seismic processing, the term statics denotes thehighly variable travel times of reflected waves accumulatedduring their propagation within the shallow subsurface(Telford et al., 1990). The near-surface layer (weathered zone) isloosely consolidated and significantly more heterogeneouscompared to the deeper layers. The uneven thickness of thenear-surface layers and low velocities lead to large (often up to~50 ms or more for P-P data), strongly variable time shifts of thereflected waves recorded from the deeper layers. Becausereflected rays propagate nearly vertically within the low-velocity weathered zone, such time shifts are practically inde-pendent of the depth of reflectors, and they are consequentlycalled statics.

All types of statics can be incorporated in the concept of “refrac-tion statics” (Yilmaz, 2001). Refraction statics represent a groupof methods based on constructing a realistic model of theshallow subsurface by inverting the refracted (first) arrivals.This model should incorporate the complete topography,depths of buried sources, as well as the variations in the struc-

ture of the weathered zone. This is the most complete andadvanced approach to developing statics solutions, and it isused in this paper.

Refraction-statics calculations are usually based on the use ofhead waves to model the first-arrival travel times. Severalrefraction-statics methods are in broad use, such as the Plus-Minus method (Hagedoorn, 1959), Generalized Reciprocalmethod (Palmer, 1981), and the Generalized Linear Inverse(Hampson and Russell, 1984). These methods take the first-arrival times as inputs and use different types of travel-timemodeling to derive estimates of the depths and/or subsurfacevelocities. Most of these travel-time models are based on thefollowing dependence of the head-wave travel time on thesource-receiver distance x in a horizontal one-layer case:

. (1)

Here, h1 is the thickness of the overburden layer, V1 is itsvelocity, V2 is the velocity of the refractor, q is the angle of theray direction from the vertical, and p = 1/V2 is the head-waveray parameter. This equation relates the observed property(time) to the physical properties (depths and velocities) of thelayers beneath the source receiver locations. By analysing thedependence of t on x, model parameters V1, and h1 in this equa-tion can be estimated. In practice, spatially-variable layer veloc-ities and thicknesses are used, and multiple layers may beneeded for accurate modeling of the subsurface structure.

The general objective of this study is to improve the existingapproaches to refraction statics in three-dimensional (3-D)seismic datasets by utilizing the concept of the first-arrival“travel-time field” instead of the conventional treatment oftravel-time data as individual “picks”, in three ways: 1) byusing pre-inversion data analysis and quality control (QC),which is not commonly performed in standard refraction inver-sion programs but could make great improvement in thequality of the inversion; 2) employing an improved model para-meterization and inversion technique, and 3) using an extendedQC of the inversion itself, including the resolution matrices andcheckerboard resolution tests. The time-field inversion yieldsgreat savings of processing times by allowing automaticconstruction of starting models and efficient iterations. Inparticular, the new procedure for constructing the startingmodel for inversion by using the Herglotz-Wiechert transformat every midpoint is likely to greatly improve the convenience,speed, and accuracy of the solution. In addition, resolution testsyield quantitative measures of the quality and attainable detailof the model, and they allow selection and analysis of the modeland algorithm parameters.

CJEG 12 June 2013

CANADIAN JOURNAL of EXPLORATION GEOPHYSlCSVOL. 38, NO. 1 (June 2013), P. 12-21

Abstract

Inversion for refraction statics is a key part of 3-D reflec-tion seismic data processing. This study has two primarygoals directed toward improvement of refraction staticsinversion. First, we implement a rigorous, model-inde-pendent data quality control by viewing the first-arrivaltravel times as “travel-time fields” (TTF’s). The TTFs aresurfaces in a five dimensional space of shot and receivercoordinates and time which allow utilization of thetravel-time reciprocity conditions for detecting errors ingeometry and in first-arrival picking. Second, the TTFconcept suggests a novel, efficient inversion approach forrefraction statics, which is particularly advantageous for3-D seismic datasets. Similarly to reflection records, TTFscan be viewed in the common-shot, common-receiver,common-midpoint (CMP), and common-offset forms.The CMP first-break TTF’s are decomposed by using at�-p parameterization, which allows an automatic deriva-tion of a high-quality initial subsurface model. Thismodel is further improved by using multi-layer raytracing and inversion to obtain an accurate subsurfacemodel. Finally, the surface-consistent statics are calcu-lated and applied to a part of a large real dataset fromsouthern Saskatchewan.

θ= +t x hV

px( ) 2 cos1

1

θ= +t x hV

px( ) 2 cos1

1

CJEG 13 June 2013

Seismic dataset

The method of this paper is illustrated on a 3-Dseismic dataset covering a ~400-km2 area in southernSaskatchewan (Figure 1a). For this study, only a partof this dataset is used, including 255 shots and 12lines of receivers near the western edge of the survey(red box in Figure 1a, and Figure 1b). The totalnumber of first-arrival travel-time picks within thissubset is 169,667. The smaller dataset allowed us toperform a significant number of tests while using arelatively slow implementation of the algorithms inMatlab. Nevertheless, the selected subset is largeenough to provide meaningful information aboutthe 3-D near-surface structure of the study area andto produce a sample stack. The iterative inverse usedin this study is readily scalable to datasets of any sizeand shapes of surface areas.

Travel-time field analysis

The key idea of our refraction-statics approach is inanalysing the travel times prior to their entering anymodel-based inversion. Inversion can only succeedwhen the input travel times are correct and consistentwith the general principles of seismic first-arrival propagation.Any travel-time errors caused, for example, by cycle skipping,errors in phase identification, or shot-by shot inconsistencyduring picking would be impossible to identify during the inver-sion and will adversely affect the solution. Errors in source-receiver geometry could also have severe impact on the quality ofthe refraction solution. Therefore, such errors need to be identifiedand removed prior to the inversion.

The travel-time analysis procedure is based on the concept ofthe travel-time field (TTF), in which the first-arrival time read-ings are viewed as representing a continuous TTF functiont(xS,xR) sampled in a five-dimensional space formed by the posi-tions of sources (xS) and receivers (xR) and time (t). For variouspurposes, common-shot, common-receiver, common-midpoint,or common offset-azimuth slices of this space can be created, ineach of which the resultant TTF represents continuous two-dimensional surfaces. Such continuity is a powerful criterionallowing establishing the internal consistency of the TTF andsometimes using interpolation to infer missing data.

The TTF possesses several important properties that can be usedto verify and establish its consistency regardless of the subsur-face model. These properties are: 1) travel-time reciprocity whenusing seismic sources and receivers located on a commonsurface, 2) similarity of the TTFs recorded from adjacent sourcesor receivers (i.e., TTF continuity); 3) great redundancy of travel-time sampling in 3-D recording, and 4) generally regular varia-tion of the refraction travel times with the source-receiverdistance and azimuth. This allows inverting the TTF for anempirical “travel-time model” before defining an inverseproblem that solves for a subsurface velocity structure. Indefining such a model, only general properties of the 3-D source-receiver coverage and the general character of the first-arrivaltravel-time inversion problem are utilized, as described below.

The TTF model is constructed by explicitly separating thecontributions from the receiver-, source, and offset/azimuth-related factors. For a source located beneath point xS and areceiver at point xR at the free surface in a 3-D space, theobserved travel time tSR(xR) can be expressed as:

, (2)

where the surface-consistent t(xS,xR) travel-time is:

. (3)

In these expressions, dSR=xR–xS is the source-receiver offsetvector, tR(x) is the receiver delay common to all sources tu+tsthe shot uphole time (shot time advance common to allreceivers), and dtSR(x) is the remaining travel-time delay of theparticular travel-time reading relative to a combination of theseterms. In eq. (2), an additional time shift tS is also added to themeasured shot uphole time tu to allow compensation for anyerrors in tu or for inclusion of additional shot-time corrections.

The different terms in the time-field decomposition (3) possessseveral properties making them useful for data quality control,travel-time inversion, and also in manual and automaticpicking. The distance-dependent term td(dSR) is a relativelysmooth function which is therefore close for the adjacent shots orreceivers. This whole dependence can therefore be interpolatedbetween the nearby shots to predict the travel-times in a hithertounpicked shot. The term tR(xR) is variable and comprises muchof the elevation- and “short-wavelength” receiver staticscommon to all shots (Hampson and Russell, 1984). By contrast,term dtSR(x) is highly variable, but it is also relatively limited inmagnitude and represents a continuous 2-D surface whenviewed as function of xR. Performing a Delaunay triangulationof this surface allows interpolation of this term and predicting itat any receiver location. Finally, the tu+ts term is common to theentire shot, and adjusting the ts parameter can be used toimprove the average travel-time reciprocity, as explained below.

Atul Jhajhria and Igor Morozov

xx xx xx= − +t t t t( ) ( , ) ( )SR R S R u S

xx xx xx= − +t t t t( ) ( , ) ( )SR R S R u S

xx xx dd xx xxδ= + +t t t t( , ) ( ) ( ) ( )S R d SR R R SR R

xx xx dd xx xxδ= + +t t t t( , ) ( ) ( ) ( )S R d SR R R SR R

Figure 1. Location maps: (a) Beaver Ranch 3-D seismic dataset. Receiver lines extend north-south, and source lines are oriented in NE-SW direction. Red box shows the data subsetchosen for this study. (b) Locations of 255 shots (blue dots) and 12 receiver lines (denselyspaced black dots) used in this study. Red line is the location of the stacked section shownin Figure 14.

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Using the 3-D TTF reciprocity for deriving shotand receiver statics

Regardless of the subsurface structure, the surface-consistentrefracted travel times (3) between any two points must satisfythe reciprocity relation:

. (4)

This relation can be tested and enforced prior to inversion. Inour approach, we calculate the reciprocal time misfits betweenall pairs of shot locations Si and Sj which contain reciprocal(reversed) recording:

, (5)

and therefore:

. (6)

The travel times dtSR(x) at the reciprocal-shot locations aredetermined by linear interpolation based on a Delaunay trian-gulation, as outlined above.

For a typical 3-D recording geometry, the system of linear equa-tion (6) is strongly over-determined and can be solved forparameters tS by using the Least Squares or SimultaneousIterative Reconstruction (SIRT) methods described later in thispaper. However, because of the simple form of the coefficientsin this linear system (only equal -1, 0, or 1), an approximate,SIRT-type solution can also be used:

, (7)

which is applied iteratively until all dtSi,Sj become sufficientlysmall. Here, NRi is the number of shots reciprocal to shotnumber i, and l ≤ 1 is a factor used for damping the iterationsin order to prevent oscillations when the number of reciprocalshots is small. This correction reduces the residual averagetravel-time misfit of shot #i with all of its reciprocal shots.

As a result of iteratively applying the shot-time corrections (7),the average travel-time discrepancies between the shots arereduced to zero, and consequently any systematic travel-timeerrors related to shot timing or depth uncertainties are removedprior to the inversion.

Similarly to the shots, receivers may also have systematictravel-time variations caused, for example, by small-scale near-receiver heterogeneity or arrival-time picking problems. Suchtravel-time variations are incorporated in our model as“receiver static” terms, which are somewhat similar to the“short-wave” statics in GLI3D program by CGG Hampson-Russell (now by GeoTomo). The receiver static terms tR(x) in eq.(3) are estimated by using an iterative procedure similar to theshot time correction (7):

. (8)

This summation, performed over all shots covering receiver #i,represents the average travel-time deviation associated withthis receiver location. Such a correction would typically alsoabsorb the receiver elevation static. By separating this term, thedistribution dtSR(x) values become centered and its variancereduced, and the offset-dependent trend td(dSR) can thereforebe determined more accurately.

Travel-time data quality control

Once all shot travel times are decomposed according to eq, (3),parameters dtSR(x) can be used for identifying erroneous travel-time picks. This is particularly important if automatic pickershave been used, especially those done in seismic processingsoftware (we used ProMAX by Halliburton Landmark) whichdoes not recognise the geometrical and reciprocal relationshipsbetween the travel times from different source-receiver pairs.

Because the range of dtSR(x) values in eq, (3), should bemoderate and centred at zero, its anomalous values can beeasily identified. For example, we can measure the standarddeviation or the first-arrival time errors:

. (9)

This quantity represents the average squared width (disper-sion) of the statistical distribution of dtSR(x). Values that are toolarge compared with this standard deviation, for example suchthat 3dtSR(x) > 3S[tSR], can be considered outliers and removedfrom further analysis and inversion. Seismic traces containingsuch errors can also be examined for geometry errors oranalysed interactively, as described below.

Similarly, the statistics of reciprocal travel-timemisfits can be measured, and their standard devia-tion determined. A histogram of reciprocal travel-time errors defined as D tR=t(xS,xR)–t(xR,xS) isshown in Figure 2. This histogram shows that therange of reciprocal travel-time mismatches is withinabout ±10 ms. Shots with significant reciprocal-timemismatches can be re-examined or excluded fromfurther travel-time analysis and reflection imaging.For example, in the Beaver Ranch data (Figure 1),such procedure helped to quickly identify shots witherrors in source-receiver geometry patterns. This isthe most common (and also relatively widespread)source of travel-time errors in the present dataset.

Refraction-static analysis in 3-D by using time fields

tN

t t t1Si

RiSi Sj ui uj

j

N

,1

Ri

∑λδ( )( )= −

+− +

=

tN

t t t1Si

RiSi Sj ui uj

j

N

,1

Ri

∑λ δ( )( )= −+

− +=

∑ ( ) ( ) ( )= − + − =

tN

t t t tx x d1 ,RiSi

S R u S d SRj

N

1

Si

∑ ( ) ( ) ( )= − + − =

tN

t t t tx x d1 ,RiSi

S R u S d SRj

N

1

Si

S tN

t11SR

RSR

j

N2

1

R

∑δ δ( ) =− =

S tN

t11SR

RSR

j

N2

1

R

∑δ δ( ) =− =

Figure 2. Histogram of residual reciprocal travel-time errors in the selected data subset.

t tx x x x, ,S R R S( ) ( )=

t tx x x x, ,S R R S( ) ( )=

δ ( ) ( )= − = − + −t t t t t t tx x x x, ,Si Sj Si Sj Sj Si ui uj Si Sj,

δ ( ) ( )= − = − + −t t t t t t tx x x x, ,Si Sj Si Sj Sj Si ui uj Si Sj,

δ− = − +t t t t tSi Sj Si Sj ui uj,

δ− = − +t t t t tSi Sj Si Sj ui uj,

CJEG 15 June 2013

The TTF’s for overlapping shots and the measured source- andreceiver time misties can be displayed in an interactive 3-D visu-alization program. Here, we use examples using the OpenGL 3-D visualization package in our IGeoS seismic processingpackage (Chubak et al., 2007) (Figure 3). This customizable inter-active tool can contain an image of the seismic section showinghow the TTF matches the actual picks made and the seismic data(Figure 3f). The TTF’s from adjacent and reciprocal shots can

also be used for effective guiding the automatic picking of firstarrivals in new shots entering the processing. An even betterapproach consists in “training” the program by interactive selec-tion of a waveform from one shot, which is further cross-corre-lated with the records in the vicinities of the first breaks. In othershots, this “seed” waveform is selected automatically fromreceivers located near shots that have already been picked. Thewaveforms collected from each shot record can be saved and

used later, for example, for deconvolution.

In cases with complex 3-D layouts, such as thedataset of this study (Figure 1), geometry errorsmay present great difficulties in analysing thetravel times. If present in the survey documen-tation, most geometry errors are not identifiedduring data loading and binning but appearupon examination of the first-break travel-timepatterns. However, this is a difficult andextremely tedious procedure requiring repeatedvisual inspections and periodic re-binning ofthe entire dataset, which was impractical in thisstudy containing about 15,000 shots. However,the statistics of the travel-time distribution [eq.(9)] can be used for detecting and correctingsuch pattern errors automatically (Morozov andJhajhria, 2008).

In an experimental approach to automatic geom-etry pattern analysis, the patterns were randomlyperturbed by shifting the receiver numbers up ordown for each line of shot-receiver pattern. Foreach random modification of the pattern, shottravel-time field was decomposed by using equa-tions (2) and (3), the standard deviations (9) weremeasured, and the number of travel-time outliersdetermined. By using the Genetic Algorithmsapproach, several of such modifications werecompared simultaneously, and those producingthe least outliers and the smallest S[tSR] wereintermixed and randomized again and kept inthe analysis. After many (~1000) trials, the algo-rithm found the patterns providing the best-quality TTF decomposition (2) and (3) andreported whether the originally specifiedpatterns appeared to be correct.

Inversion for depth model using t-p

The effectiveness and efficiency of iterative-inversion algorithms strongly depends on thequality of the data and also on the proximity ofthe initial models to the true solution. With thechosen type of model parameterization(constant-velocity, variable-thickness layers) anextremely efficient and fully automatic proce-dure can be formulated for deriving a high-quality starting model. This procedure is basedon the t-p parameterization of the midpoint TTFdescribed below.

The t-p formulation of the travel-time problemis convenient for inverting for the depths in a

Atul Jhajhria and Igor Morozov

Figure 3. Interactive travel-time analysis and surface-consistent travel-time picking: a) Base mapof one selected shot with reciprocal-time mismatch indicators in eq (5) (coloured rectangleslabeled “rt”). b) 3-D display of a shot (tan colour) and reciprocal times (red); c) Vertical travel-timeat a midpoint selected in base map b); d) Reciprocal-time shot mismatch diagram. Colours repre-sent the reciprocal-time misties in eq. (5). e) Map of the selected shot with reciprocal-time mistieindicators; f) Seismic section of one selected line for travel-time picking. Shots and lines can beselected from panels a) and b) and time reduction can be applied for convenient viewing of thewaveforms. In this section, reciprocal times extracted from the travel-time surfaces can be usedto guide picking. A 10-shot data subset is used for clarity of displays.

CJEG 16 June 2013

layered model. In a real dataset with dense first-break-recording, a nearly-continuous travel-time curve can be approx-imated by an infinite number of straight lines, which wouldcorrespond to a continuous (t-p) curve (dashed line in Figure4a,b). Such a continuous curve could result from a continuousdepth-velocity distribution, which can be obtained from theHerglotz-Wiechert transform (Aki and Richards, 2002):

. (10)

Here, z(V) is the depth as a function of wave velocity, p is the rayparameter (travel-time moveout), and X(p) is the source-receiver distance at which this ray parameter value is observed.

To construct a starting velocity-depth model for subsequenttomographic inversion, time-distance trends td(dSR) areextracted for each shot and “re-sorted” (interpolated within the2-D TTF surfaces in 5-D space) into common-midpoint (CMP)travel times. With moderate lateral variability, such CMP traveltimes are relatively closely related to the velocity distributionsbeneath the corresponding midpoints. These velocity-depthdistributions can then be obtained by a 1-D t-p inversionmethod based on eq. (10) (Bessonova et al., 1974). In numericalimplementations of this method, it is convenient to parame-terize the models as z(p), where z is the depth at which the ray-parameter value p is attained, and p is assumed to be decreasingwith depth. Therefore, to parameterize the t-p inversion, we uselayers with constant velocities Vl=1/pl and invert for thedepths zl to the bottom of lth layer. The values of pl are prede-fined (for example, in regular increments) selected byinspecting the general travel-time pattern of the TTF data. Inprinciple, there is no limit on the number of layers, some whichmay have zero thicknesses in certain areas. This parameteriza-tion is also convenient for ray tracing and 3-D travel-time inver-sion (below).

Because this 1-D inversion is performed at each (x, y) location (inour implementation, this is done at the receiver locations andfurther spatially interpolated), the resulting model becomesspatially-variant. In 3-D, a similar inversion was performed byMorozov et al. (2005) by using long-range seismic data.

Note that the starting-model inversion described here does notrequire interactive determination of any depths, velocities, orcontrol points. The interpolated CMP TTF (2) – (3) containssufficient information for determining the near-surface velocityand layer depths automatically and at all locations covered bythe source-receiver spreads. The resulting 3-D starting velocitymodel is derived at every point of the model grid and is quitedetailed (Figure 5). It usually reproduces the observed travel-times well and needs to be only moderately adjusted by the fullinversion.

3-D ray tracing

Once the starting depth model is constructed by “midpoint”t-p inversion, it is further refined by full 3-D modeling of headwaves. The same constant-velocity, variable-depth model para-meterization is used as the one constructed in the startingmodel. For each layer, the depth to its bottom is discretized ona Cartesian grid, with equal grid sizes in the X and Y directionstaken for simplicity. Denoting with the dimensions of the gridin the X and Y directions for layer i by Nxi × Nyi, the depth atpoint (x, y) can be obtained by using bilinear interpolation ofthe adjacent grid nodes, zi,j :

, (11)

where l and j are the grid numbers in X and Y directions. Toformulate the linear inverse problem, the values of zl,j from alllayers need to be combined in a single model vector. To achievethis, two subscripts i and j in eq. (11) are replaced with a singleone, denoted k and spanning all nodes within one layer. Thesetwo counters for the nodes within the lth layer are related as:

. (12)

To accurately reproduce the ray shapes and travel timesbetween each pair of sources and receivers, we use the ray-tracing method based on the same delay-time concept as in thet-p inverse from which the starting model was obtained. In thisapproximation, the rays are viewed as travelling within thevertical cross-section planes containing both the source andreceiver. Within this plane, the rays consist of head-wavesegments along the appropriate constant-p boundaries

Refraction-static analysis in 3-D by using time fields

z x y x y z x y z( , ) ( , ) ( , )l j l j k kkl j , ,, ∑∑ ϕ ϕ= ≡

z x y x y z x y z( , ) ( , ) ( , )l j l j k kkl j , ,, ∑∑ ϕ ϕ= ≡

k Nx Ny Ny l jm m im

i

1

1∑= + ⋅ +=

−

k Nx Ny Ny l jm m im

i

1

1∑= + ⋅ +=

−

z V X p

p Vdp( ) 1 ( )

v

v

2 20 1

1

∫π= −

− −−

−

z V X p

p Vdp( ) 1 ( )

v

v

2 201

1

∫π= −

− −−

−

Figure 4. Tau-p decomposition of travel times: a) Schematic refractiontravel times in time-offset (t-x) form. Picked first-arrival times areschematically shown by dots. Straight-line segments numbered 1, 2, and 3approximately fit these travel times. These segments can be thought of asdirect and head-wave travel times, with�t1 and t2 being the intercept times.b) The same travel times in the t-p form. Parameter p is the slowness meas-ured by the slopes of each of the three lines in plot a). The values of t arethe corresponding intercept times.

Figure 5. Depth to the refractors with velocities, 1.6, 2.0,and 3.0 km/sec(bottoms of layers 1, 2, and 3, respectively), obtained from midpoint t-pinversion.

CJEG 17 June 2013

combined with up- and downgoing segments connecting theboundary to the source and receiver. The result is the 2-D raypropagation model in which the delay-time terms for the corre-sponding ray parameter can be readily obtained.

For delay-time head-wave ray tracing, it is necessary to find theintersections of the ray with the refracting interface of varyingdepth (point T in Figure 6). To find these intersections, we usean iterative bisection technique. Starting from the surface anddenoting the incidence angle in ith layer by a, we can expresssin(a) in two ways: as the ratio of ray parameters pi–1 and pi(Snell’s law) and geometrically, as a function of the distance lmeasured from the position of the source-receiver midpoint:

, (13)

where z(l) is the depth at the intersection point. Once the inter-section point is found on boundary #i, its coordinates are usedfor finding the intersections on the deeper boundaries.

To test the ray-tracing approach, the travel times calculated bythe bisection method were compared with theoretical traveltimes for horizontal layers. A model with a single horizontallayer model at depth of 600 m with direct-wave velocity of 0.667km/sec and refractor velocity of 1.667 km/sec was used. Thegeometry of source receiver was a) receivers aligned along axisX, b) receivers aligned at 45° to the X-axis. In both cases, theerrors were of the order of numerical accuracy (~10-11).

Inversion

By analyzing the first-arrival travel-times in the Beaver Ranchdataset (Figure 1), we determined that a three-layer modelshould be sufficient to approximate all the travel-time curves inthe t-p form. The velocities of these three layers were selected as0.667 km/sec, 1.5 km/sec, and 2.0 km/sec, and the velocity ofthe medium below the model was set equal 3.0 km/sec. Thesevelocities were fixed, and the depths to three interfaces were setfrom the starting model and varied during the ray-tracingbased inversion. Note that because of the fundamental ambi-guity of the first-arrival inversion problem (Bessonova et al.,1974), different selections can be made. Selecting a large

number of layers (as in Morozov et al, 2005) would lead to anear-smooth (Herglotz-Wiechert, eq. (10)) solution which,however, would predict nearly the same travel times in thepresent dataset.

For inversion, we need to: 1) linearize the travel-time problem,and 2) solve the linearized problem iteratively. The first-arrivaltime TSR modelled from source S to receiver R depends non-linearly on model parameters (depths) zk, where k = 1…N:

. (14)

To linearize equation (14), consider a small perturbation of onedepth node zk within layer i. From the Taylor series expansionin terms of dz the resulting perturbation in TSR equals:

, (15)

where the summation is performed over all model nodes whichaffect the ray connecting points S and R. Ignoring the termsO(|dz|2), the first two terms in equation (15) give a linearapproximation for the relation between the perturbations oflayer depths and travel times. To obtain the partial derivativesin eq. (15), note that the delay-time term variation is related tothe variation of the depth, dz, at which the refraction, occurs:

, (16)

where pi is the slowness of the layer, and q is the angle of inci-dence with respect to the vertical direction. From (11), theperturbation of the depth at which the ray strikes the refractorequals:

, (17)

and consequently:

. (18)

Similar partial derivatives are accumulated for all nodes adja-cent to the points of ray incidence on the refracting boundaries.

The linearized travel-time problem becomes:

d=Lm , (19)

where d is the data (first-break travel times) vector of length Nd,m is the model vector of size Nm, and L is the forward kernelmatrix representing the travel-time derivatives (18). Usually,the system (19) is over-determined, and its inverse can beobtained by minimizing the error vector (misfit):

r=dobs –Lm. (20)

The norm of this error vector minimized in our inversionscheme is the second order, L2 norm given by

. (21)

In the present study, we tried two iterative schemes by usingthe Simultaneous Iterative Reconstruction Technique (SIRT)(van der Sluis and van der Vorst, 1987) and the Kaczmarz’s(1937) method (Jhajhria, 2009). Only SIRT-based inversion isdiscussed here. In this method, the model vector is initialized asm0 = 0, and in q-th iteration (q = 1,2,…) , it is updated by usingthe following equation:

Atul Jhajhria and Igor Morozov

t T T z z z zz( ) ( , , ,..., )SR SR SR N1 2 3= ≡

t T T z z z zz( ) ( , , ,..., )SR SR SR N1 2 3= ≡

T TTz

z Oz z z z( ) ( ) ( )SR SRSR

kkk

2∑δ δ δ+ = +∂∂

+

T TTz

z Oz z z z( ) ( ) ( )SR SRSR

kkk

2∑δ δ δ+ = +∂∂

+

TV

z p zcos cosSRi

iδθ

δ δ θ≈ ≡

TV

z p zcos cosSRi

iδ θ δ δ θ≈ ≡

z x y x y z( , ) ( , )k kϕ δ=

z x y x y z( , ) ( , )k kϕ δ=

Tz

x y p( , ) cosSR

kk iϕ θ

∂∂

=

Tz

x y p( , ) cosSR

kk iϕ θ

∂∂

=

idr r

L i

N 2

12∑=

=

idr r

L i

N 2

12∑=

=

lpp

L l

L l z lsin[ ( )]

( )

i

i 1 2 2α

( )= =

−

− +−

lpp

L l

L l z lsin[ ( )]

( )

i

i 1 2 2α

( )= =

−

− +−

Figure 6. Two-dimensional head-wave ray tracing. Label ℓ shows thedistance of the ray point from the midpoint, in terms of which the positionof the refraction point is parameterized. Labels pi-1, pi are the ray critical-parameters in the corresponding layers.

CJEG 18 June 2013

, (22)

where i is the ray number, j is the model node number,

, Sj is the normalizing factor equal the number ofrays contributing to model parameter #j. We also included afactor a < 1 used to control (slow down) the convergence andprevent oscillations. The optimal value of this constant was inthe range of 0.2–0.3 for the model considered.

During each step of iterations, ray tracing of all data isperformed, and vector d and model updates (22) are evaluated.The iterations are stopped when the increment in all mj becomesnegligibly small. At this point, theresidual error becomes related to thestatistical data error or insufficient modeldetail. Once the SIRT code establishes thebest model, further iterations may some-times cause fluctuation of the errorbefore converging back to minimum-error level. At this point, our code termi-nates the ray-tracing/inversion loop.

Model resolution analysis

Before applying the inversion to realdata, we examine the ability of the algo-rithm to invert for various levels ofdetail in the model. In particular, we usemodel resolution analysis to estimatethe optimal grid size and the effects ofray coverage in different areas of themodel. In addition to the three-layermodel above, we also tested simple one-layer inverse problems, and examinedtheir resolution.

In model resolution analysis, the prop-erties of data coverage are used to esti-mate the optimal inversion grid size.The actual source-receiver geometry forall of the recorded travel times is used,whereas the travel-time data arereplaced with synthetic patterns veri-fying certain aspects of the model.Resolution analysis provides under-standing of the areas and features of themodel which are better or poorerconstrained by the available travel-timedata. The model resolution matrix is:

Rm=Lg–1L, (23)

where L is the forward matrix in eq. (19)and Lg

–1 is the generalized inverse repre-sented by the selected iterative inversescheme (Menke, 1984). In our case,computation of the full matrices Lg

–1,and therefore of Rm, is impractical.However, perturbation of a selectedmodel node can still be performed in

model m0, and synthetic data generated and inverted as inequation (23) by using the SIRT method. Such a test provides agreat amount of detail showing interaction between differentparts of the model during forward modeling and inversion. Forthis test, a one-layer model was used first. The grid size for theinversion was chosen 335 m, and the starting model had auniform depth of 600 m, in which the depth at one node wasincreased by 10% (to 660 m). Ray tracing was further performedto predict the travel times for all observed source-receiver pairs.Starting from the initial uniform model, the iterative inversearrived at the model with the recovered perturbation close tothe desired 10% (Figure 7a). The recovered model reproducedthe position of the perturbation accurately, but the anomaly wasalso smeared, and a pattern of side lobes surrounding it could

Refraction-static analysis in 3-D by using time fields

Figure 7. Resolution-matrix test. a) Model perturbed at a single node, b) the model inverted from it byusing a 335-m inversion grid. Note the side-lobes in the recovered model. c-e) Perturbation test in a three-layer model with 200-m, 400-m and 600-m interface depths. The second interface is perturbed at a singlenode in the middle. f-h) Model recovered from this perturbation.

m mS

L rDj

qjq

j

ij iq

ii

N11

1d∑α

= +−−

=

m mS

L rDj

qjq

j

ij iq

ii

N11

1d∑α

= +−−

=

D Li ikk

N 21m∑≡=

D Li ikk

N 21m∑≡=

Figure 8. Results of single-layer checkerboard model tests. Grid sizes used in the algorithm were 67, 134,201, and 335 m (labels). A) Input and recovered models at depth 600 m. b) Zoom-in into the checkerboardpatterns in a). Note that the patterns are recovered well by using the 134-m and 201-m grids.

CJEG 19 June 2013

be seen. These side lobes correspond to the non-zero, non-diag-onal values in the resolution matrix. A similar resolution testwas performed in a three-layer model with interface depths of200, 400, and 600 m (Figure 7c-h). As seen in Figure 7d, a pertur-bation in layer number 2 also caused a slight perturbation in theother two layers (Figure 7f-h).

To assess the general quality of ray coverage, we also performedanother test commonly used in tomographic studies and calledthe “checkerboard test”. In this method, a regular alternatingspatial pattern is generated within the model, and the inversionis tested for its ability to recover this pattern. The tested inver-sion grid sizes are multiples of the nominal receiver spacing inthe Beaver Ranch seismic dataset, which was equal to 67 m.

Checkerboard tests were performed by constructing alternatingpositive and negative, 10% perturbations of the interfacedepths, followed by predicting the travel-times and invertingthem using the procedure (SIRT) intended for inverting theactual data (Figure 8a,b). The first checkerboard test wasperformed in a one-layer, 600-m depth earth model. The best-resolved models appeared to be somewhere between 134-m to201-m grid sizes. The 67-m grid was unable to recover thecheckerboard pattern, whereas the 335-m grid size recoveredthe checkerboard pattern well (Figure 8). A close-up view ofthe checkerboard test results is shown in Figure 8b.

Another checkerboard test was performed for the one-layerearth model by using grid size of 201 and 335 m but placed atshallower depth of 200 m (Figure 9). The model was generatedby using the sine function with a period of about 2 km. Notethat although the 201-m grid correctly predicted the checker-board pattern, it produced some spurious ”footprint” in therecovered model (Figure 9). Hence, although the resolution ofthe dataset could be better than 201 m, we did not use such gridsizes because of the footprint of the receiver spread. Grid size of335 m seems appropriate for recovering the subsurface featureswith the available travel-time data coverage at these depths.

For the three-layer earth model, the checkerboard test wasperformed by using only a single grid size of 335 m. All threelayers were perturbed by ±10% in depths and alternating in

space. The result of this test is shown in Figure 10. The 335-mgrid size was sufficient for recovering the checkerboard patternfor the three-layer case. Therefore, this grid size was used toperform the inversion of the real dataset of this study.

Finally, in the one-layer case, the convergence speed of the SIRTalgorithm was also measured in the first (one-layer) checker-board test with 335-m inversion grid. The error in each iterationwas calculated by using eq. (21). Error reduction was measuredduring inversion for the checkerboard model with grid size of335 m. The error decreased rapidly with the number of itera-tions, with some oscillations starting after about 60 iterations(Figure 11).

Resulting model and surface-consistent statics

Using the adjusted and reconciled travel-time field (TTF) data,selected velocity layering, grid sizes, and the starting modelfrom t-p inversion, the final depth model was obtained by SIRTiterations (Figure 12). In order to suppress the inversion edge-effect artefacts resulting from excessive perturbations along thenorthern, eastern, and southern edges of the models, the solu-tion was smoothened in these areas. All three layers in theresulting depth model show significant structures, with greateramount of detail recovered at the shallower depths (Figure 12).These variations should be generally related to the variations of

Atul Jhajhria and Igor Morozov

Figure 10. Results of three-layer checkerboard resolution test. a) Inputcheckerboard model; b) recovered model. Grid size of 335 m was used forthe inversion.

Figure 11. Error reduction as a function of the number of iterations duringa checkerboard test.

Figure 9. Checkerboard test using grid sizes of 201 m and 335 m. Linearfeatures (acquisition footprint caused by receiver lines) appear on themodel recovered by using grid size of 201 m.

CJEG 20 June 2013

the subsurface. As expected, the structure of the uppermostmodel layer (Figure 12, left) is closely related to the surfacetopography in the study area (Figure 13a).

When interpreting the results of inversion for the near-surfacestructure, it is important to realize that because of the funda-mental ambiguity of the first-arrival (including TTF) inversion,details of this structure in depth may not be accurately relatedto the physical parameters, such as velocities and depths of thediscontinuities. In particular, low-velocity and thin high-velocity layers can always be introduced without altering thefirst-arrival travel times (Telford et al., 1990). Similarly to mostother first-arrival inversion approaches, the result (Figure 13b)represents a “minimal” solution without low-velocity zones,such as given by the Herglotz-Wiechert transform (eq (10)).With increasing number of layers considered, this solution canalso be viewed as the smoothest vertically.

Surface-consistent statics were calculated in the resulting modelby evaluating the vertical travel-times through the model:

, (24)

where pi is the slowness of ith layer, Dzi is its thickness, Eb is theelevation of the bottom of the model, DS is the source depth, EDis the elevation of the datum, and VRepl is the replacementvelocity.

The resulting statics are significant, ranging from about –60 to110 ms, and their variations correlate with the surface topog-raphy and shallow subsurface structure (Figure 13b). Further,we compared the derived surface-consistent statics model tothe statics obtained from the popular GLI3D software (now byGeoTomo; Figure 13c). As expected, both solutions show similarvalues for the statics and similar structural patterns, particu-larly in the middle of the model where the ray coverage is thebest (Figure 13b, c). Both solutions also correlate with surfacetopography, because topography represents the strongestcontribution to the statics. Our solution appears to correlatewith the topography somewhat stronger, likely because of thethinner layer 1 and thicker layer 3 beneath the area of increasedelevation near the western edge of the area (Figure 13a).However, along the underconstrained edges of the model, thesolutions differ significantly, because of the differences in theTTF corrections and in the inversion schemes. Further compar-

ison of the two solutions is difficult, because of the differencesin the algorithms and particularly in regularizing the inversionnear the edges of data coverage.

Finally, the statics shown in Figure 13 (b) were applied to thereflection data. Figure 14 (d-f) shows an 8.41-km long reflectionline extracted in the middle of the 3-D spread (Figure 1b) inwhich the data coverage was good and the resulting staticssolution is of high quality. Application of static correctionsremoves irregularities from reflection events and makes themnearly hyperbolic (Figure 14(a-c). Smaller variations of traveltimes still remaining in the records can be further removed bynon-surface consistent and residual (waveform cross-correla-tion based) statics corrections which are not considered in thispaper. However, in some areas closer to the edges, the staticsolution was less successful. Broader data coverage and poten-tially a more sophisticated regularization algorithm is requiredin order to overcome these problems.

Stacked reflection records (Figure 14d-f) show that thecoherency and continuity of stacked events is improveddramatically by application of the surface-consistent refractionstatics correction (Figure 14b, right). Notably, the GLI3D staticsappear to be somewhat better in the shot gathers (Figure 14c,left). This could probably be explained by effectively finer grid-ding employed by the GLI3D model. The finer scale of thismodel’s gridding can also be seen in Figure 13c. It is known thatwhen larger number of parameters is used in the inversion, themodel may become less well constrained, the inversion canbecome less stable, and yet the data (travel times) fit wouldoften improve (Menke, 1984). However, one of our key objec-tives in this study was inversion for a best-resolved subsurfacevelocity structure, and the grid size was selected conservatively.Interestingly, in the application to the stack, our solutionappears somewhat better than that by GLI3D (Figure 14d-f).This observation could again suggest a relative roughness of theGLI3D model, which could therefore be slightly less coherentbetween different midpoints. From the continuity of reflectionsbetween ~1400 ms and 1700 ms (Figure 14d-f), our solutionappears to be better in long-range statics whereas GLI3D mayproduce more detailed short-wavelength statics. Thus, wesuggest that conservative construction of a well-constrained,albeit maybe not the most detailed model may be advantageousfor constructing the surface-consistent statics models.

Refraction-static analysis in 3-D by using time fields

Figure 12. Depth to the three model interfaces obtained by the inversion.

Figure 13. Final refraction model of the subsurface: a) Elevations of thesource-receiver locations. b) Predicted surface-consistent statics derivedfrom the depth model (Figure 12) and c) statics derived by using GLI3Dsoftware.

t p z p DE EVs i ii

N

Sb D

11 0Repl

l∑= ∆ − +−

−=

t p z p DE EVs i ii

N

Sb D

11 0Repl

l∑= ∆ − +−

−=

CJEG 21 June 2013

Because of the “prototype” implementation in Matlab, our algo-rithm is relatively slow and difficult to compare to productionpackages in terms of computational performance. In principle,with implementation in C++ and/or Fortran, computationalrequirements of this method should be only moderately higherthan those of GLI3D. With similar amounts of ray tracing anditerative matrix inversion, our algorithm only includes addi-tional data sorting, Delaunay tessellations and tau-p inversions.These operations represent only modest computational tasks.Using a detailed and relatively accurate starting model shouldactually reduce the amount of ray tracing and SIRT iterationswhen using comparable grids and datasets. Finally, resolutiontests certainly do represent significant additional analyst’s andcomputational efforts; however, these tests also yield additionalinformation not available from the traditional approaches.

Conclusions

The two principal contributions of this study are in two areas:1) using the concept of the first-arrival travel-time field (TTF)for analysing the refracted-wave travel times and efficientinverting for refraction statics, and 2) emphasizing the impor-tance and suggesting ways to perform travel-time data qualitycontrol (QC) prior to inversion. In the first of these areas, treat-ment of the first-arrival travel times as piecewise-contiguoussurfaces (TTF’s) in the five-dimensional space (shot X, shot Y,receiver X, receiver Y, and time) space allows obtainingnumerous constraints on the consistency of the travel-time datathat help in both manual and automatic picking. Travel-timereciprocity can be used to check for any errors in geometry andfirst-break picking. Checking for travel-time reciprocity is also apowerful way for identifying and correcting travel-time pickingand geometry pattern errors. Statistics of the reciprocal travel-time misfits were measured and used for estimating the averagedata error (~10 ms in this dataset) and identifying outliers.

A description of the TTFs by using a common-midpoint t-pparameterization leads to robust and efficient inversion for astarting depth model. This model is produced automatically, isrelatively detailed and accurate, and reproduces many featuresof the first-arrival TTFs. This model is further improved byhead-wave ray tracing providing an accurate refraction (andrefraction-static) model. Several types of model-resolutionanalysis help evaluating the optimal grid sizes for the inversion.In particular, checkerboard resolution tests are useful for assess-ment of data coverage and the ability of the travel-time data toresolve the structure of the subsurface.

By using the above techniques, surface-consistent statics werederived for a part of a large reflection dataset and compared tothe results from commercial GLI3D software. The resultssuggest comparable performance, with somewhat improvedquality of constraining the near-surface structure and substan-tial improvements in the TTF quality analysis and building theinitial model in the present approach.

Acknowledgments

This work benefited from many comments, critique and sugges-tions by Dr. Brian Russell. Funding for this study was providedby the Southern Saskatchewan University Grant and TrainingProgram and NSERC Discovery Grant RGPIN261610-03.

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Atul Jhajhria and Igor Morozov

Figure 14. Application of statics to seismic data. Left column: Shot gather:a) before application of statics, b) after application of the surface-consistentstatics of this paper, c) after application of GLI3D statics. Right column:8.41-km long segment of stacked section location (red line in Figure 2): (d)without statics; (e) with surface-consistent statics from this study; and (f)with GLI3D statics applied.