tesi di nome cognome - d1rkab7tlqy5f1.cloudfront.net · master of science in applied ysics geoph...

Delft University of Te hnology

Fa ulty of Civil Engineering and Geos ien e

Joint Master of S ien e in Applied Geophysi s

The e�e t of a water layer and sea�oor

topography on S holte-wave inversion

Maria Kotsi

August 2014

mi

Idea League Joint Master's in

Applied Geophysi s

Delft University of Te hnology, The Netherlands

ETH Züri h, Switzerland

RWTH Aa hen, Germany

Dated: August 8, 2014

Supervisors:

1. Dr. Pauline Kruiver (Deltares, Utre ht, The Nether-

lands)

2. Dr. Guy Drijkoningen (Delft University of Te hnology,

The Netherlands)

Committee members:

1. Dr. Guy Drijkoningen (Delft University of Te hnology,

The Netherlands)

2. Dr. Marina Hruska (RWTH Aa hen, Germany)

3. Dr. Pauline Kruiver (Deltares, Utre ht, The Nether-

lands)

Contents

1 Introdu tion 1

2 Theoreti al ba kground of S holte waves 3

2.1 MASW te hnique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Literature on S holte waves . . . . . . . . . . . . . . . . . . . . . . . 5

3 Methodology 6

3.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 A quisition geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3 Modeling: fdelmod . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.4 Frequen y �ltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.5 Dispersion urves: Geopsy . . . . . . . . . . . . . . . . . . . . . . . . 10

3.6 Inversion: Dinver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.7 Full Velo ity Spe trum inversion: winMASW . . . . . . . . . . . . . 11

4 Variation of re eiver height 13

4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.4 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Variation of verti al heterogeneity 19

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.2 Input models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.2.1 Three-layer model . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.2.2 Five-layer model . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2.3 Ten-layer model . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.4 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

i

CONTENTS ii

6 Combination of water layer and layered subsurfa e 25

6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.4 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7 Variation of sea�oor topography 29

7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7.2 Regular sandwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7.2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7.2.3 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7.3 Mega ripples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7.3.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.3.3 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8 Full-Velo ity-Spe trum inversion: the winMASW software with ex-

amples and omparisons 50

8.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

8.2.1 Ten-layer model . . . . . . . . . . . . . . . . . . . . . . . . . . 51

8.2.2 Regular sandwaves . . . . . . . . . . . . . . . . . . . . . . . . 52

8.2.3 Mega ripples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

8.3 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

9 Dis ussions 57

10 Con lusions 59

Abstra t

Depth varying sti�ness measurements are an important hara teristi of sub-bottom

materials for geote hni al and environmental appli ations in marine environments.

Seismi ally, these measurements an be made through the dispersion analysis of

Rayleigh-type surfa e waves, i.e. S holte waves for a water layer on top of an elasti

layer. Sediment layering, as well as �nite water depth disperses the S holte wave.

Phase and group velo ities of S holte waves between 1 and 20 Hz are primarily

sensitive to S-wave velo ity of the uppermost few tens of meters of the sediment

�oor. Multi hannel Analysis of Surfa e Waves (MASW) provides an e� ient way in

estimating the V S pro�le of the sea�oor sediments: surfa e waves are ex ited by an

air-gun, travel through the sea �oor and then re orded by a hydrophone streamer.

In this resear h, a �nite di�eren e s heme has been used to model S holte waves

propagation for 1D models by varying the re eiver height, the verti al heterogen-

eity and the sea�oor topography. Based on the inversion results, the in�uen e of

water-layer thi kness has been illustrated for both a simple and a multi-layered

model with redu tion of frequen y ontents and thereby loss of resolution at shallow

depths. However, when the multi-layer model was tested, di�eren es also o urred in

the range of 15 to 20 Hz where a relative in rement in phase velo ities was observed

ompared to the simple 1D model. Con erning the experiments in verti al hetero-

geneity, a general trend of in reased velo ities with depth was aptured, even though

the inversion performed better when smoother models were used. Moreover, two dif-

ferent topographies were implemented, verifying the big in�uen e on S holte waves.

When ripples were present signi� ant e�e ts on inversion results were observed,

whereas the experiments on sand waves advo ated that data should be a quired in

two dire tions in order to evaluate the e�e ts of sea-�oor topography. Additionally,

a new inversion approa h based on the full velo ity spe trum has been used, showing

more reliable and a urate results, ompared to velo ity-dispersion urve, espe ially

when omplex stru tures are studied.

iii

A knowledgments

I would like to express my gratitude to my supervisors Pauline Kruiver, Guy Drijkonin-

gen and Giovanni Diaferia for the useful omments, remarks and engagement through

the learning pro ess of this master thesis. In addition, I would like to thank Del-

tares for giving me the opportunity to work on this thesis and all the people that

work there. With their patien e, support and friendliness they reated an enjoyable

working environment for me.

Furthermore, I want to thank Jan Thorbe ke, the developer of the FD modeling

programme that was used in this resear h. He has been always available providing

his help to all of my questions.

Last but not least, I would like to thank my loved ones who have supported me

throughout the entire pro ess. I will be grateful forever for your love.

iv

Chapter 1

Introdu tion

Shear wave velo ity (VS) provides important information in the hara terization of

sediments be ause V S is dire tly related to the sti�ness of the material. Moreover,

V S has a higher sensitivity to lithology variation and less to �uid ontent than

ompressional wave velo ity (VP ). On land, the MASW (Multi hannel Analysis of

Surfa e Waves) method was developed for the determination of shear-wave velo ity

by re ording and analyzing the dispersion of Rayleigh-type surfa e waves using an

impulsive seismi sour e and re eivers. Similarly, in marine environments MASW

te hnique an be applied by using an airgun and a towed hydrophone streamer.

The method an be used to evaluate sea�oor stability and shear strength for var-

ious appli ations su h as installation of submarine pipelines, oil-rigs, wind-farms

and dredging. Commonly signi� ant hanges in shear-wave velo ities are observed

espe ially in the �rst 10-15 m beneath sea�oor. The appli ation of surfa e waves

provides many advantages over body shear waves (Klein et al. 2005, Kugler et al.

2005). Surfa e waves are generated and propagate whenever there is an interfa e

that separates media with di�erent elasti properties. However, in marine environ-

ments the behavior of surfa e waves slightly hanges due to the intera tion with

the water layer. Hen e, for the water over solid material ase they are a modi�ed

version of Rayleigh waves, alled S holte waves (S holte, 1947). Re orded data are

�rst analyzed to derive the dispersion hara teristi s and afterwards the shear-wave

velo ity an be estimated using inversion pro ess.

Deltares (Utre ht, The Netherlands), the resear h institute where this work was

performed, is interested in the possibility of using S holte wave to hara terize the

near surfa e in marine environments. For this purpose, two similar studies were

performed in the past. The �rst one, intended to investigate whether the MASW

te hnique that was developed and operational on land (using Rayleigh waves) ould

be extended to appli ations under water; S holte waves were observed only in ertain

1

CHAPTER 1. INTRODUCTION 2

lo ations where the water layer was maximum 4 m during re�e tion seismi survey

at Wadden Sea, The Netherlands (Kruiver et al., 2010; Deák, 2009). This suggested

that either the sour e was not adequate for the generation of S holte waves or the

a quisition set-up was not suitable. Based on that, a se ond study was performed to

derive the optimal a quisition parameters to dete t S holte waves in shallow marine

environments. (MS thesis Diaferia, 2012). The aim of this thesis was to test the

in�uen e of the bathymetry on the generation and propagation of S holte waves

as well as the derived V S pro�le. To start with, the water layer e�e t has been

tested for both a simple and a multi-layered model. In addition, several numeri al

experiments on verti al heterogeneity were performed. Furthermore, two di�erent

topographies were implemented, showing the in�uen e on S holte waves. Finally, a

new inversion approa h taking into a ount the entire velo ity spe trum has been

used.

Chapter 2

Theoreti al ba kground of S holte

waves

2.1 MASW te hnique

Multi hannel Analysis of Surfa e Waves (MASW) is a non-invasive method for the

estimation of seismi shear-wave velo ity. The method has been adapted from the

Spe tral Analysis of Surfa e Waves (SASW) in whi h the dispersive nature of surfa e

waves in analyzed to provide a verti al V S pro�le by using an impulsive sour e and a

single pair of re eivers (Stokoe et al. 1994). MASW improves the SASW method by

in reasing the signal-to-noise ratio, providing better on�den e in the identi� ation

of surfa e waves and improving �eld produ tion rates through the use of multiple

re ording hannels (Xia et al. 1999, Park et al. 1999). Both methods are arried out

on land with the use of low frequen y geophones that are either planted on the soil or

pulled along the surfa e with a land streamer. Luke and Stokoe (1998) showed that

SASW method an be su essfully applied in underwater environments. Park et al.

(2000) have shown that onventional hydrophones laid at or near the sea bottom

an be as e�e tive as underwater geophones for dete ting surfa e waves. Therefore,

MASW has been su essfully applied to analyze surfa e waves a quired either by an

o ean-bottom hydrophone able (OBC) (Klein et al., 2000) or by towed hydrophone

streamer lose to the sea�oor (Klein, 2003).

When elasti properties of the near surfa e materials hange with depth, surfa e

waves be ome dispersive. This means that ea h frequen y omponent of the surfa e

wave travels with a di�erent propagation velo ity alled phase velo ity (cf ), resulting

in a di�erent wavelength (λf) for ea h frequen y propagated. Lower frequen ies

travel faster be ause they have bigger penetration. In ontrast, higher frequen ies

have limited penetration and therefore travel with usually lower velo ity to the

3

CHAPTER 2. THEORETICAL BACKGROUND OF SCHOLTE WAVES 4

shallow parts of the subsurfa e. Additionally, the parti le motion of S holte waves

is the same as for Rayleigh waves, meaning a retrograde ellipse, ounter lo kwise

against the dire tion of propagation, an be seen in �gure 2.1 .

Figure 2.1: Propagation of S holte waves through the sea�oor. S holte waves are

ex ited by an impulsive sour e and re orded by a towed hydrophone streamer (left).

The parti le motion of S holte wave depends on the geome hani al properties of the

di�erent strata through whi h the wave travels (right).

Analysis of this dispersion behavior of surfa e waves is done via the frequen y-

wavenumber domain (f, k). Sin e data are re orded in the o�set-time domain (x, t)

two onse utive Fourier transform are applied to onvert the data to the (f, k)

domain as shown in Bra ewell (1999).

As S holte waves are a modi�ed version of Rayleigh waves in a marine environ-

ment, the same pro essing work�ow of MASW te hnique an be adapted here as well.

In other words, analyzing interfa e waves the following work-�ow should be pursued:

1. Data a quisition

2. Data Pro essing:

� Frequen y �ltering

� 2D Fourier transformation for generation of dispersion plots

� Depending on the inversion s heme: A urate pi king of fundamental mode

� Inversion of dispersion urves to solve for the shear-wave velo ities of the

subsurfa e

CHAPTER 2. THEORETICAL BACKGROUND OF SCHOLTE WAVES 5

2.2 Literature on S holte waves

In order to obtain reliable results, S holte waves need to be adequately a quired.

Thus, all the parameters de�ning sour e and re eivers should be a urately hosen

(Diaferia, 2012).

In the �rst pla e, the wave�eld need to be adequately re onstru ted both in

time and spa e; it means that Nyquist riterion should be ful�lled (see Appendix

A). If temporal sampling is not orre t, higher frequen ies are aliased, introdu ing

� titious low frequen ies in the a quired data. On the other hand, orre t sampling

in spa e is needed in order to avoid spatial aliasing.

Moreover, it is also of great importan e to use a powerful low-frequen y sour e

in ombination with re eivers that an provide good responses to low frequen ies.

An airgun is suggested for the e� ient generation of S holte waves due to its power

and its apa ity on reating su� ient low frequen y energy. A ording to Bohlen

et al. (2004) and Kugler et al. (2007) a 0.6 l airgun demonstrated to be su essful

in water depths less than 50m. To a quire the signal either a towed streamer or

an o ean bottom able an be used. The �rst is time and ost e�e tive with small

position errors; nonetheless, the noise level in the data an be quite high due to the

movement of the ship, while the a quisition parameters an be modi�ed only to a

ertain degree. On the other hand, using an o ean bottom able (OBC) leads to

better S/N although the positioning is un ertain and time onsuming.

Chapter 3

Methodology

3.1 Introdu tion

In this hapter the a quisition parameters as well as the modeling s heme and the

basi model of this thesis will be presented. Additionally the pro essing work�ow

and the used methods will be explained.

3.2 A quisition geometry

It has been de ided for investigational purposes to use the results of the previous

work on this topi (Diaferia, 2012). Diaferia (2012) demonstrated the optimal a -

quisition set-ups for a quiring S holte waves in order to assure a good resolution of

the fundamental mode and thus reliable inversion results. Consequently the follow-

ing input parameters have been applied:

� Nearest o�set (x1) = 10 m

� Re eiver spa ing (∆x) = 1 m

� Number of hannels = 48 (Re eiver spread length (x) = 47 m)

3.3 Modeling: fdelmod

The �nite-di�eren e method has been used in order to solve the 2D wave equa-

tion and ompute the syntheti seismograms. The program �fdelmod � has been

developed by Jan Thorbe ke (Thorbe ke and Draganov, 2011), fellow resear her at

TU Delft, is open sour e and makes use of Seismi Unix (SU) parameter interfa e

and output �les. The program omputes a solution of the 2D wave equation by

6

CHAPTER 3. METHODOLOGY 7

approximating the derivatives in the wave equation by �nite di�eren es. The wave

equation is de�ned through a �rst-order linearized system of Newton's and Hooke's

law.

For the aim of this work the elasti s heme has been used, and therefore the

linearized equation of motion (Newton's se ond law) and equation of deformation

(Hooke's law) apply:

∂Vx

∂t= −

1

ρ

(

∂σxx

∂x+

∂σxz

∂z

)

(3.1)

∂Vz

∂t= −

1

ρ

(

∂σxz

∂x+

∂σzz

∂z

)

(3.2)

∂σxx

∂t= −

1

κ

∂Vx

∂x+ λ

∂Vz

∂z(3.3)

∂σzz

∂t= −

1

κ

∂Vz

∂z+ λ

∂Vx

∂x(3.4)

∂σxz

∂t= −µ

(

∂Vx

∂z+

∂Vz

∂x

)

(3.5)

where σij , Vi, κ, λ, µ are the symmetri stress tensor, parti le velo ity, ompres-

sion modulus, Lame's �rst parameter and shear modulus, respe tively.

The ode tests for stability and numeri al dispersion riteria. In order to persuade

stability, time steps and spatial dis retization steps must satisfy the Courant ri-

terion (For the 4th order spatial derivatives the Courant number is 0.606). Thus,

this approximation requires:

△t <0.606△h

cmax

(3.6)

where △t, △h and cmax are time step, grid size and maximum propagation velo ity,

respe tively.


In order to avoid numeri al dispersion, as a rule of thumb 5 sampling points per

minimum wavelength must be assured. For this assumption, the following approx-

imation is required:

△h <cmin

5fmax

(3.7)

∆h <λmin

5(3.8)

where cmin , fmax and λmin are the minimum propagation velo ity, the maximum

input frequen y and the minimum wavelength respe tively.

In this thesis work, a Ri ker wavelet has been used as a sour e with a entral

frequen y of 41.5 Hz and a time sampling rate of 2 ms. So as to suppress the arti� ial

re�e tions from the boundaries of the model 200 m thi k tapers were applied ex ept

for the top of the model domain where a free surfa e was set (simulating an air-

water interfa e). The grid size was set to 0.3 m, small enough to avoid numeri al

dispersion and assure stability of results. However, in the mega-ripples experiment

(smaller dimensions dunes) the grid size was de reased to 0.12 m for an adequate

representation of the small topographi al hanges. Velo ities and model dimensions

were varying regarding the needs of ea h experiment and will be introdu ed at ea h

hapter separately.

3.4 Frequen y �ltering

In order to strengthen the S holte wave signal a frequen y �ltering has been used in

some of our models. S holte waves have low frequen y ontent (dominant frequen y

usually below 20 Hz). To enhan e the dispersion plots in this range, a low-pass �lter

of approximately less 30 Hz has been applied in order to separate the dispersive

surfa e waves from the P-wave events. As a onsequen e we get higher S/N ratios

that ensure a ura y in the al ulated dispersion urves and thus more reliable

inverted Vs pro�les. Figure 3.1 shows the low-pass �lter applied on a shot gather

of a 2-layer model, whereas �gure 3.2 shows the same shot gather before and after

applying �ltering.


Figure 3.1: Amplitude spe trum of an un�ltered ommon shot gather of a 2-layer

model with sour e and re eivers at 2 m above sea�oor. The green line shows the

low pass �lter (<30 Hz).

Figure 3.2: Top: Example of an un�ltered shot gather for a 2-layer model (same as

3.1). Bottom: The same shot gather after a low pass �lter is applied


3.5 Dispersion urves: Geopsy

The Geopsy® pa kage has been used in order to transform the data to the (f, k)

domain and produ e the dispersion plots in the (f, v)domain.

In this work we assume that the fundamental mode of the surfa e waves dominates

the re orded wave�eld and thus the higher modes are ignored. The a ura y in the

pi king of the dispersion urve is very ru ial in obtaining the orre t subsurfa e

model after inversion. For this reason the automati pi king has been used in order

to avoid potential mistakes and subje tive pi king. For displaying reasons all the

plots are normalized by the total spe trum power. In Figure 3.3 an example of a

dispersion plot with the automati ally pi ked fundamental mode is shown.

Figure 3.3: Example of a dispersion plot and automati ally pi ked dispersion urve,

for the 2-layer model of �gure 3, where with sour e and re eivers are pla ed at 2 m

above sea�oor.

3.6 Inversion: Dinver

In order to retrieve the V S pro�le of the subsurfa e the pi ked fundamental modes

need to be inverted. The software �Dinver� part of the Geopsy® pa kage was used

to perform the inversion. The software is based on the neighbourhood algorithm, a

sto hasti approa h whi h addresses the problem by generating a range of plausible


solutions. The user needs to spe ify a range of values for several input parameters

(V S, V P , density and Poisson's ratio) for the onstru tion of the initial model.

Di�erent ranges has been tested for P and S wave velo ities with a narrow ± 50 m/s

and a broad range of ± 200 m/s. Shear-wave velo ities for marine sediments usually

vary from 50 m/s to 400 m/s. Therefore a range of ± 100 m/s looks more suitable for

both P and S wave velo ities. In order to have omparable inversion results between

the models, the same range has been used in all inversions. A range of 0.2-0.5 has

been used for a Poisson ratio. Xia et al (1999) showed that a variation of 25% in

density auses small perturbation of the fundamental mode, usually less than 10%.

Therefore, �xed values were used for densities, orresponding to the input values in

the forward model.

Note that no water layer an be in luded both in �Geopsy� and �Dinver�. This

means that S holte waves are treated as Rayleigh waves in the pro essing as well as

in the inversion.

Ea h dispersion urve is inverted with 10 runs. For ea h run 2.500 models are

generated and investigated, for a total amount of 25.000 models for ea h shot. The

goodness of a model is spe i�ed by the mis�t fun tion a ording to the following

equation (Wathelet et al., 2004):

misfit =

√

√

√

√

nF∑

i=1

(xdi − xci)2

σ2i nF

(3.9)

where xdi is the velo ity of the data urve for a frequen y fi, xci is the velo ity of the

al ulated urve for a frequen y fi, σi is the un ertainty of the onsidered frequen y

samples, and nF is the number of the frequen y samples.

3.7 Full Velo ity Spe trum inversion: winMASW

Contrary to Geopsy that only uses the fundamental mode in the dispersion plot, a

program alled winMASW uses the full velo ity spe trum; winMASW is developed

a ording to Dal Moro, 2006. The Full-Velo ity Spe trum approa h allows for the

full exploitation of all the modes present in data and the amplitude variation. This

results in a well onstrained inversion pro edure that will eventually lead to a more

robust subsurfa e Vs model. Attenuation e�e ts are also onsidered during the

omputation. The use of the full velo ity spe trum, however, is omputationally

intensive.

The used inversion pro edure in winMASW is based on so alled geneti al-

gorithms (GA) (Sambridge & Drijkoningen, 1992). This s heme performs several


preliminary �parallel� runs (�rst step) and a �nal global run using the previously

determined �ttest models as starting population. This kind of approa h gives more

reliable results ompared to the traditional inversion methods. The a ura y of the

inversion then an be estimated by determining the mean value and standard devi-

ations of ea h onsidered variable through MPPD (Marginal Posterior Probability

Density) when a su� ient population of models is available. The omputation of

MPPD overs the entire sear h spa e so that any velo ity value an still be eval-

uated during the �nal run even if belonging to models that they were labelled as

not parti ularly �t during the preliminary runs (Dal Moro, 2006) as an be seen in

�gure 3.4.

Figure 3.4: Inversion results of a 10-layer model. Top left panel: dispersion plot;

the bla k lines indi ate the velo ity spe trum that is produ ed in the inversion

pro edure. Bottom left panel: mis�t evolution for both �ttest model (blue line) and

mean model (red line). Right panel: Inferred Vs pro�le. Grey lines represent the

entire population of the model during inversion whereas blue and red line are the

�ttest and mean model, respe tively.

Several input parameters are needed to onstru t the initial model. These are

a range of V S, thi kness and Poisson's ration. A range of V P does not need to be

spe i�ed. The inversion pro edure takes approximately one hour on a Thi kCentre

PC (7 ores i7 and 32 GB Ram), ontrary to �Dinver� that requires a ouple of

minutes on a less powerful ma hine.

Sin e the software be ame available at the latest stages of this thesis work, only

a sele ted number of models have been analysed using winMASW.

Chapter 4

Variation of re eiver height

4.1 Motivation

The prerequisite to e� ient generation of dispersed S holte waves is that the lo a-

tion of the sour e should be at the sea�oor or very lose to it. Additionally previous

studies showed that the thi kness of the water layer has an in�uen e to the propaga-

tion velo ity of S holte waves. S holte wave velo ity is slightly di�erent (lower) from

the Rayleigh-wave velo ity and hanges with the surfa e wave wavelength (λ) to wa-

ter depth (h) ratio (Fig. 4.1). A ording to Park et al. (2000) this in�uen e of the

water layer is more signi� ant for wavelengths shorter than several times the water

depth (�deep water� ondition). As the wavelength be omes larger than the water

depth, the in�uen e is no longer signi� ant (�shallow water� ondition). Kaufmann

et al (2005) showed in their experiment that there is a relative redu tion in phase

velo ities due to the water layer. The e�e ts gradually in rease as the thi kness of

the water layer in reases.

Figure 4.1: Approximate relationship between S holte wave velo ity (Vsch) and the

Rayleigh wave velo ity (VR)in omparison to the Shear-wave velo ity (VS)(Stokoeet al. 1994).

13

CHAPTER 4. VARIATION OF RECEIVER HEIGHT 14

In order to be able to pro ess and orre t the in�uen e on propagation of S holte

waves when sea�oor topography is present, whi h is the main goal of this resear h,

the best way of dealing with the presen e of the water needs to be investigated. A

time shift is expe ted when a �oating able is used due to longer travelling time as

the thi kness of the water body in reases. However, the question is if there is also a

frequen y shift together with the time shift. For this purpose, the �rst experiment

has been performed by varying the re eiver height.

4.2 Modeling

In order to test the e�e t of the water layer 7 shots were modeled with re eiver

height ranging from 0 m to 8 m above sea�oor for a �xed water depth of 10 m. A

�rst test is for a simple laterally homogeneous model (see Fig. 4.2). The sour e was

pla ed at 210 m in the x dire tion of the model and was moving relatively to the

re eivers (from 0 m to 8 m) for ea h shot.

Figure 4.2: On the left panel the properties of the simple laterally homogeneous

model are indi ated, while on the right panel the graphi of the model is shown.

The model has a dimension of 450 x 450 m.

4.3 Results

In �gure 4.3, two raw shot gathers are plotted for the two extreme ases: for sour e

and re eivers at the sea�oor (left) and for sour e and re eivers at 8 m above sea�oor

(right). It is observed that the amplitude of the S holte waves de reases as the

distan e between sour e/re eivers and sea�oor is in reased (see also Allou he, 2011).

This is to be expe ted, be ause S holte-wave amplitudes de ay exponentially with

in reasing distan e from the sea�oor.


Figure 4.3: Comparison between two shot gathers for the simple laterally homo-

geneous model. On the left pi ture, sour e and re eivers are pla ed at the sea�oor,

while on the right one they are pla ed 8 m above the sea�oor.

The raw shot gathers already show that the e�e t of water between the seabed

and sour e/re eivers is not only a time shift due to the longer travel time. The

e�e ts of the water layer are even learer in the phase-velo ity/frequen y plots. In

�gure 4.4 the dispersion plots and the pi ked fundamental modes of all the 7 ases

are shown. It is worth noti ing that by moving further away from the sea�oor there

is a big e�e t on frequen ies, as the higher frequen ies part is no longer observed;

espe ially in the ase of sour e and streamer at 8 m above sea�oor, the fundamental

mode extends only up to 24 Hz. This in�uen es the information for the shallower

part of the subsurfa e as higher frequen ies have small depth of penetration.


Figure 4.4: Dispersion plot and auto-pi ked fundamental mode for all the 7 shots for

the simple laterally homogeneous model starting with sour e and re eivers pla ed

at sea�oor on the top left panel and ending up with sour e/streamer at 8 m above

sea�oor on the bottom left panel. X axis orresponds to frequen ies [Hz℄ while the

y axis shows phase-velo ities [m/s℄.

In �gure 4.5 the auto-pi ked dispersion urves for di�erent values of re eiver

height are plotted in one graph. In addition to the loss of higher frequen ies with

the growth of water body, small di�eren es in phase velo ities an be also observed

in the lower frequen y range (f<15 Hz).


Figure 4.5: Pi ked fundamental modes for di�erent sour e/re eiver height over the

sea�oor in the ase of a laterally homogeneous model.

A ording to the inversion results (�gure 4.6 ) there is a good agreement between

the models of sea�oor and 1 m above sea�oor and the referen e one. The shear-

wave velo ities are well resolved and the di�eren es in thi knesses are approximately

within an a eptable range of 1 m. Even though this dis repan y remains within

nearly the same limits for the other models, the resolution of the shallower sub-

surfa e is de reasing with respe t to shear-wave velo ities due to the loss of higher

frequen ies.


Figure 4.6: Inversion results from Geopsy pa kage depending on the variation of

re eiver height for the laterally homogeneous model.

4.4 Con lusions

Klein et al. (2005) showed that the displa ement amplitude of S holte waves strongly

de rease with in reasing distan e of re eivers from the sea�oor. The overall amp-

litude and its de rease depend on frequen y (higher frequen ies are attenuated more

than lower frequen ies). Con erning the results from �gure 4.6 a maximum distan e

of 5 m above sea�oor is suggested to be used. Also the sour e should be pla ed

as lose as possible to the sea�oor to assure the e�e tive generation of dispersive

interfa e waves (Allou he, 2011).

Chapter 5

Variation of verti al heterogeneity

5.1 Motivation

Previously, a simple model with one layer and an underlying half spa e has been

implemented. This s enario is far from reality. In fa t, the subsurfa e an be har-

a terized by prominent lateral and verti al inhomogeneity. The fundamental mode

of the dispersion plot is inverted in a one-dimensional approa h. In other words, the

subsurfa e is assumed as horizontally layered where P- and S-wave velo ities as well

as the density an vary only with respe t to depth.

In this hapter, the sensitivity in verti al heterogeneity is tested, implementing

multi-layered models. For this purpose three di�erent models were used in order

to evaluate the e�e t. Sour e and re eiver height was kept always onstant at 3 m

above the sea�oor, a height su� ient enough to generate and a quire S holte waves

a ording to the on lusions of se tion 4.4.

5.2 Input models

5.2.1 Three-layer model

A simple model with two layers and underlying half spa e has been investigated. As

the maximum depth of interest is approximately around 30m, the �rst two layers

are set relatively thi k, 12 and 15 m respe tively (Figure 5.1).

19

CHAPTER 5. VARIATION OF VERTICAL HETEROGENEITY 20

Figure 5.1: Three- layer model: On the left panel the table with the properties of

ea h layer is shown while at the right panel the relevant graphi is presented.

5.2.2 Five-layer model

As a next step, the previous two layers are divided into two sub-layers. By keeping

the total maximum depth of around 30 m onstant, the thi knesses of ea h layer

therefore de reased. The input parameters of the forward model are listed in table

3 of �gure 5.2.

Figure 5.2: On the left panel velo ities and thi kness of ea h layer are indi ated in

table 3 for a 5-layer model. On the right panel, the graphi for the same model is

shown.

5.2.3 Ten-layer model

Pushing things to the limits, in this model a more ompli ated s enario of 9 layers

and a half spa e has been designed (table 4 in �gure 5.3). The velo ities are hosen

to be in reasing by 50 m/s steadily from layer to layer, while a small growth in

densities has been de ided sin e has a small in�uen e on S holte waves propagation.

The thi kness has been kept onstant at 3 m for ea h layer.


Figure 5.3: On the left panel the table with the input parameters for a 10-layer

model is present, whereas on the right panel the orresponding graphi is shown.

5.3 Results

In �gure 5.4 the dispersion plots for the 3, 5 and 10 layers ase are shown. For all

of the ases the fundamental mode is well resolved, with less sharpness though in

the 3-layer model. Moreover, shape and amplitude of higher modes hanges as a

onsequen e of the di�erent geologi al stru tures. Di�eren es are dete ted in the

range of 20-30 Hz regarding the dispersion urve between the �rst model and the

other two, while the urve is very alike for the �ve and ten layered model.


Figure 5.4: Pi ked fundamental models; on the top left pi ture the 3-layer model is

represented, on the top right the 5-layer model and on the pi ture below the 10-layer

model.

Figure 5.5 shows the inversion results for the three evaluated models. A general

pattern of in reasing shear-wave velo ities with depth is observed. The velo ities are

plotted up to 80 m, even though the depth of interest is up to 40-50 m (a ording to

the approximate al ulation of the wavelength from the formula c = λf). The la k of

�t is not surprising for the three-layer model that has a large dis ontinuity, as these

kinds of hanges are di� ult to be aptured with traditional inversion methods.


Figure 5.5: Inversion results obtained from Geopsy-dinver for the 3-, 5- and 10- layer

model.

A ru ial issue in dispersion urve inversion is the number of layers. If no ad-

ditional data are available, subsurfa e layering is basi ally unknown and its re on-

stru tion is the main goal of any non-invasive geophysi al investigation. O am, a

s holasti medieval philosopher, advo ated a reform both in method and ontent,

the aim of whi h was simpli� ation; �If in doubt, smooth�. The O am's inversion

has been implemented by Constable et al. 1987 in the inversion phase of EM data.

When hoosing a model to interpret, this model should be as simple, or smooth, as

possible in order to redu e the temptation to over-interpret the data and to elim-

inate arbitrary dis ontinuities in simple layered models. In �gure 5.6, the 3 layers

model has been inverted with an initial model of 10 layers with smooth hanges in

velo ities.


Figure 5.6: Three-layer model inverted with an initial model of 3 layers on the left

panel, and with an initial model of 10 layers on the right panel. Both inversions

were obtained by Geopsy-dinver. The right panel shows a gradual transition from

lower to higher velo ity values, while when an initial of 3 layers has been used (left)

the model is for ed to have a sharp transition at the depth of 12 m.

5.4 Con lusions

This se tion was set out to determine if verti al heterogeneity has an e�e t and an

be dete table in the inversion pro edure. All three experiments showed a general

pattern of in reased velo ities with depth; a good trend is observed up to a depth of

20 m. Nonetheless, the inversion performed well when smoother models were used.

Chapter 6

Combination of water layer and

layered subsurfa e

6.1 Motivation

In hapter 4, the e�e t of the water-layer thi kness has been investigated for a

simple 2 layer model. In hapter 5, the sensitivity in verti al heterogeneity has been

investigated by keeping the sour e and re eivers at a onstant depth.

In this hapter, the water-layer e�e t in ombination with a multi-layer subsur-

fa e is onsidered; this s enario is loser to reality. In order to a hieve this, the

10-layer model from hapter 5 has been used and examined in ombination with

variable re eiver height (note that sour e is moving relative to the re eivers). It

is expe ted that some loss of higher frequen ies will be obtained as the distan e

between sour e/re eivers and sea�oor is in reased. Additionally, it is assumed that

higher modes might appear in the dispersion plots, as layering disperses S holte

waves. No sea�oor topography has been implemented yet; the sea�oor is still �at.

6.2 Modeling

For all of the tests the sour e was pla ed at 210 m in x dire tion of the model. Five

di�erent shots were modelled with sour e and re eivers height ranging from 0m to

8m above the sea�oor, while the sour e was moving relatively with the re eivers.

6.3 Results

Figure 6.1 shows the in�uen e of water-layer thi kness on the dispersion urves.

The fundamental mode is well resolved for all the shots with signi� ant sharpness.

25

CHAPTER 6. COMBINATION OFWATER LAYER AND LAYERED SUBSURFACE26

As expe ted, by in reasing the thi kness of the water layer, information on the

higher frequen y part is lost. For instan e, for the shot with re eivers pla ed at 8

m above the sea�oor, the fundamental mode ontains information to a maximum

of approximately 18 Hz. In ontrast in the ideal ase of sour e and re eivers on

the sea�oor the fundamental mode rea hes frequen ies up to 35-40 Hz. It is also

remarkable that higher modes emerge, whi h be ome more noti eable as the water

body in reases. This e�e t, however, was not observable when the simple 2-layer

model was tested in hapter 4, as more layers are needed to generate higher modes.

It is interesting though, that those higher modes are not observed in the ase of

sour e and re eivers pla ed at sea�oor, meaning that is an e�e t of the water layer

but it is not a ounted for in the inversion.

Figure 6.1: Dispersion plots with the auto-pi ked fundamental modes for all the �ve

shots starting from the top left with sour e and re eivers pla ed at the sea�oor and

ending to the bottom right where they are pla ed at 8 m above the sea�oor. Input

model: 10 layers of 3 m thi kness ea h and with in reasing Vs velo ity (range 250

to 800 m/s, see table 4).


Figure 6.2: Auto-pi ked dispersion urves for �ve shots of in reasing distan e

between sea�oor and sour e/re eivers. Input model: 10 layers of 3 m thi kness

ea h and with in reasing Vs velo ity (range 250 to 800 m/s, see table 4).

In �gure 6.2 the auto-pi ked dispersion urves of all of the shots are presented.

Di�eren es in phase velo ities are observed in the lower frequen y range (f<13 Hz)

between the shot at sea�oor and the shots with some distan e from sea�oor. The

maximum di�eren e o urs in the range of 15-20 Hz with a relative in rease in phase

velo ities. Kaufmann et al. (2005) showed similar dis repan ies when evaluating

the e�e ts of a water body on S holte wave data with a ten layer model. However,

they observed a de rease in velo ities instead of an in rease as here. One possible

explanation might be the appearan e of the higher energies and the fa t that they

are not well separated from the fundamental mode.

A ording to the inversion results in �gure 6.3, the general trend of Vs distri-

bution in the subsurfa e is aptured for all the models. The observed thi knesses

di�er less than 3 m approximately from the referen e model, whereas the shear-wave

velo ities seems to be underestimated.


Figure 6.3: Referen e and inverted models for all of the �ve shots obtained from

Geopsy. Input model: 10 layers of 3 m thi kness ea h and with in reasing Vs velo ity

(range 250 to 800 m/s, see table 4).

6.4 Con lusions

The aim of this hapter was to assess the e�e t of the water-layer thi kness in

ombination with a strongly multi-layered subsurfa e. As expe ted from hapter 4,

an in rease in water body thi kness results in disappearan e of the higher frequen y

parts. Layering in the subsurfa e results in the appearan e of higher modes (see

hapter 5). The water layer a�e ts the higher modes both in shape and in allo ation

(Figure 6.1). However, onsidering the inversion results, a general trend of the

subsurfa e is aptured. This highlights the need of an inversion algorithm than an

take the water layer into a ount. Moreover, it is suggested to pla e the sour e and

re eivers as lose to the sea�oor as possible, preferably in a distan e less than 4 or

5 m.

Chapter 7

Variation of sea�oor topography

7.1 Motivation

In the previous hapters the in�uen e of water-layer thi kness as well as the sensitiv-

ity in verti al heterogeneity was tested. In real life situations, the sea-bed is not �at

and variations on the topography are present. They are expe ted to have an in�uen e

on the dispersion analysis, be ause of several e�e ts, su h as varying sour e-re eiver

height along the a quisition pro�le, within the spread length or between di�erent

shots. Also Zheng et al. 2013, showed in their experiment that S holte waves an

also easily generated by a variety of topographi features. In �gure 7.1 two of their

experiments are shown; on the left panel an example of irregular sea-�oor is shown,

whereas on the right panel a Gaussian hill model is tested. Time and spatial s ales

are bigger than the ones in this work. The entre frequen y for the Ri ker sour e

wavelet was lower (2 and 3 Hz) than the one in this resear h (41.5 Hz). However,

we assume that similar e�e ts will be observable here as well.

29

CHAPTER 7. VARIATION OF SEAFLOOR TOPOGRAPHY 30

Figure 7.1: Left panel: Random topography is model with sour e and re eivers

pla ed 10 m above sea�oor. In the middle pi ture the shot gather from the �nite-

di�eren e modelling is shown while on the bottom the shot gather from Boundary-

Element Method (BEM) is presented. Right panel: A Gaussian hill model; red star

is the sour e position and green dots are re eivers 10 m above sea�oor. On the

bottom pi ture the shot gather omputed using BEM is shown from Zheng et al.

(2013).

In the following se tion the in�uen e of bathymetry will be investigated. Rhythmi

bed forms, for instan e sand waves, are widespread in shallow sandy seas (e.g. North

Sea) and in river beds (e.g. Waal river, the Netherlands). For the investigation, two

di�erent models have been implemented to represent the sand waves at di�erent

spatial s ales. For the sea�oor regular sandwaves (se tion 7.2), the topography

variations have length s ales larger than the spread length of re eivers. For the

mega-ripples example (se tion 7.3), the topography variation is mu h smaller than

the spread of re eivers.

7.2 Regular sandwaves

7.2.1 Modeling

A large part of the seabed of the Dut h part of the North Sea is overed by sand

waves. Sand waves are bed forms o urring in oastal regions and hara terized by

non- ohesive (sandy) deposits. They usually form in shallow seas and are indu ed by

tidal urrents. Near Egmond aan Zee, the Netherlands (�gure 7.2), 50 km o�shore,

regular sand waves are present. They have a onsistent pattern, both in height

and alignment (Van Dijk et al., 2008). Over time, they migrate due to persistent

dire tional urrents. An example of a ross se tion of a typi al sand-wave area

is shown in �gure 7.3. It shows sand waves with a bathymetry variation of 3 m


over horizontal distan es of approximately 200 m. This number is signi� antly

bigger ompared to the spread length, resulting to sea-�oor topography be dete ted

as in lined interfa e for some shots su h as the shot at x=200 m and x=420 m.

Additionally, sand waves usually show a preferen e depending to the tidal urrent

meaning that the pro�le is steeper to some parts. The pro�le in �gure 7.3 has been

implemented in the 2-layer homogeneous model in order to test the e�e t of variable

sea-�oor topography. The input model is shown in �gure 7.5, left panel.

Figure 7.2: Bathymetry images of 1 km2 areas showing sand waves. a) Rotterdam

survey of 2002 indi ted by the red box on the map. b) Egmond survey of Mar h

2001 indi ated with the yellow box on the inset map. The sand waves presented on

at Egmond have been used in this work.

Figure 7.3: Cross-se tion of sea�oor topography with regular sand waves from right

panel of �gure 7.2. The x axis represents the horizontal distan e and the y axis

shows the bathymetry variation of the sand waves at the sea bottom.

In order to be able to study the in�uen e of sea�oor topography on the generation

and propagation of S holte waves, a simple lateral homogeneous model has been used

as a referen e (�gure 7.5, right panel). Both models onsist of a low-velo ity thi k

layer with an average thi kness of 3 m, overlaying a higher velo ity half spa e (table

5 in �gure 7.4) in a shallow marine setting (10 m water depth). For these models,

the minimum thi kness of the �rst layer was set to 3 m (in the troughs of the sand

waves) in order to have generation of S holte waves.


Figure 7.4: Table 5: Input parameters for sand waves and referen e model

The model domain had a dimension of 1118 x 500 m and the grid size was kept to

0.3 m. With the intention to ompletely evaluate the in�uen e of seabed topography

shots were lo ated along the x-axis between 200 m to 460 m. Sin e the sand waves

are not symmetri al (�gure 7.3), the shots have been modelled with three di�erent

setups:

1. Re eivers pla ed at the right side of ea h shot

2. Re eivers pla ed at the left side of ea h shot

3. Re eivers pla ed at sea�oor, following the seabed topography (OBC)

In total, 14 shots with 20 m spa ing have been modelled, numbered from left to

right and from right to the left as shown in �gure 7.5 to assess the e�e t of the angle

of the slope as well. Sour e and re eivers were at the same onstant depth of 5 m

below the sea surfa e. When pla ing re eivers were pla ed at sea�oor, following the

topography, the sour e was kept at the onstant depth of 5 m below sea surfa e.

Figure 7.5: Input model of sand-wave pro�le on the left panel and the orresponding

referen e �at pro�le on the right panel. The stars indi ate the shots positions. In

total 14 shots were modelled with a spa ing of 20 m between the shots.

7.2.2 Results

Setup 1: Re eivers pla ed on the right side of ea h shot

The goal of this experiment was the the evolution of the e�e t of sea�oor topography

ompared to a simple lateral homogeneous model with �at bathymetry. The per-


formed tests also show the degree of in�uen e in phase velo ities of S holte waves

when a varying topography is present.

For ea h of the 14 shots, a dispersion plot was reated. For the �rst setup, the

re eivers were pla ed at the right side of the sour e for all shots. The pi ked fun-

damental modes were then inverted in order to produ e the 1D shear wave velo ity

pro�le. In �gure 7.6 the dispersion plots are presented for the shots at 200, 300 and

420 m for the sand-wave ase are shown on the left panel, and the orresponding

shots for the �at ase on the right panel.


Figure 7.6: Left: Dispersion plots and the pi ked fundamental modes for the shots

at 200, 300 and 420 m for the regular sand-wave model (�gure 7.5, left, setup 1).

Right: Dispersion plots and pi ked fundamental modes for the orresponding shots

for the �at-topography model (�gure 7.5, right). Input parameters are listed in table

5 of �gure 7.4.


Starting with the simple model with �at topography (�gure 7.6, right panels),

it is expe ted that the three shots do not show a di�eren e. This is be ause of the

symmetry of the input model all shots are in fa t identi al. The fa t that there

are small di�eren es is due to automati pi king un ertainty. However, these small

di�eren es are expe ted to have no big impa t on the inversion phase. Contrarily, for

the sand-wave ase (�gure 7.6, left panels), there are noti eable di�eren es between

the shots at varying positions along the pro�le. The fundamental mode loses its

sharpness when the distan e between the streamer and seabed be omes larger as

for instan e for the shot at 300m relative to the shots at 200 and 420 m. For the

shots at 200 and 420 m, whi h re�e t the uppermost regions of the sand waves, the

fundamental mode is sharper. Thus,a longer and more a urate urve an be pi ked.

Additionally, the shot at 300 m for the �at model ase seems to be slightly better

resolved. It is expe ted that these di�eren es will have an in�uen e on the inversion

result.


Figure 7.7: 2D pro�les with the inversion results for setup 1; zero indi ates the

level of re eivers. Top: input model of sand waves, plotted as if topography is �at.

Middle: input model of sand waves, plotted with a tual bathymetry (green line), 1D

pro�les plotted at orresponding depths. Bottom: input model of �at topography.


All the shots were inverted separately resulting in 1D V S pro�les. For a better

understanding, a 2D plot was reated ombining all the 1D inversions overing the

whole tested pro�le. In �gure 7.7, the 2D pro�les for all ases are shown. On the top

panel, the V S pro�les of sand waves are plotted as if topography is �at. However, if

the sea�oor topography is known, as for instan e from multibeam data, it an lead

to better interpretation of the data, if this information is used in the plot as in the

middle panel of �gure 7.7. A range in velo ities and thi kness is observed ompared

to the input model and the simple model with �at topography as was expe ted. In

the area of 410 m to 500 m, the trend of the subsurfa e is aptured quite well both

in velo ities (∼200 m/s) and in thi kness (∼7 m). These results are very lose to the

input model (V S=200 m/s and thi kness in that area of 6 to 8 m). However, in the

area of 210 m to 300 m the subsurfa e looks poorly resolved. Lower velo ities are

observed (100-150 m/s) and mu h thinner thi kness is retrieved. The �gure shows

that a more or less onstant thi kness of 3 m was retrieved whi h is far from the

implemented input model (thi knesses in the range of 7-4 m). An explanation of

the di�eren e in ability to re over the layering in the various parts of the sand-wave

pro�le might be the topography of the area between 210 m and 300 m represents the

steepest side of the sand-wave slope; thus, the a ute angle at whi h the wavefront

impinges upon the interfa e di�ers to the two sides of the slope. The examination

of the e�e t of in iden e angle as in De Hoop et al. (1983) is outside the s ope of

this thesis. Instead, the e�e t of in iden e angle was investigated by hanging the

dire tion of the re eivers relative to the sour e (option 2 of setup). In this way, the

dire tion of the setup relative to the slope of the sand waves is hanged. The results

are des ribed in the next paragraph.

Setup 2: Re eivers pla ed on the left side of ea h shot

In setup 2, the re eivers were pla ed on the left side of ea h shot. Analogous to real

a quisition, the survey vessel would sail from right to left while a quiring hanging

in this way the dire tion of propagation. We expe t slight di�eren es in results.

Similarly to the previous ase for ea h of the 14 shots, dispersion plots were

produ ed. Figure 7.8 (left panels) represents the phase velo ity/ frequen y plots

with their pi ked fundamental mode for the shots of 200, 300 and 420 m respe tively.

For omparison, the dispersion plots for orresponding �at-topography input model

are shown in the right panel of �gure 7.8.



at 200, 300 and 420 m for the regular sandwave model (�gure 7.5, left, setup 2).



5 of �gure 7.4.


In ontrast with the previous a quisition s enario (setup 1, see �gure 7.6), dif-

feren es an be dete ted, espe ially for the shot at 200 m. There is an in rease in

the sharpness of the fundamental mode while the distribution of the higher energy

is hanged ompared to �gure 7.6. It is expe ted that this di�eren e will be seen

also in the inversion results.

Figure 7.9: 2D pro�les with the inversion results for setup 2; zero indi ates the

level of re eivers. Top: input model of sand waves, plotted as if topography is �at.

Bottom: input model of sand waves, plotted with a tual bathymetry (green line),

1D pro�les plotted at orresponding depths.

In �gure 7.9, the 2D plot with the retrieved models after inversion is shown.

There is an eye- at hing improvement in the retrieved velo ities and thi knesses for

the region 210 � 250 m ompared to the setup 1. Additionally, the results of the


middle area (300 � 350 m) are well related with the observed one when re eivers

were at the right side (see �gure 7.7).

Summing up these two experiments, it is evident that the retrievability of the

subsurfa e properties when topography is present depends on the in iden e angle

that the wavefronts impinge the interfa e. It is veri�ed, that a smoother slope leads

to more a urate results than a steeper one, whose e�e ts require further study.

Therefore, a onsequent re ommendation onsists of sailing in two dire tions so as

to the e�e ts of sea-�oor topography an be evaluated.

Setup 3: O ean Bottom Cable

Seismi data a quisition with O ean Bottom Cable is emerging rapidly espe ially for

deep-marine seismi monitoring. Many reasons are feeding this maturation su h as

�exibility of a quisition geometry, redu ed noise by eliminating able vibration and

better overage due to the elimination of able feather aused by urrents. Thus, this

survey set up represents an interesting option to eliminate the in�uen e of the water

layer (as re ommended in hapter 4), improve the inversion result and evaluate the

extent of in�uen e of topography.

Overall 14 shots were modelled, with re eivers pla ed at the sea�oor, while the

sour e was kept in the same positions as in the previous tests. For omparative

purposes, the same a quisition set up has been used to model the �at topography

ase as well. Dispersion plots were reated for ea h shot. In �gure 7.10, the auto-

pi ked fundamental modes for the shots at 200, 300 and 420 m for the sand-wave

ase are indi ated on the left panel, whereas on the right panel the same shots for

the �at ase are presented.



at 200, 300 and 420 m for the regular sand-wave model (�gure 7.5, left, setup 3).



5 of �gure 7.4.


From �gure 7.10, the appearan e of higher modes an be noted in all shots,

ontrary to what was observed for the model with a towed streamer. In the �at

sea-bed topography, the fundamental mode is well resolved in sharpness and amp-

litudes; proof of the bene� ial use of OBC. When the sand waves are present the

resolution of the fundamental mode de reases dramati ally relative in sharpness and

amplitudes. For example, onsider the shot at 420m: the pi ked urve an rea h

nearly the velo ity of 400 m/s in the frequen y range of around 10 Hz; meaning

insu� ient resolution for deeper depths. Moreover, it is remarkable that the same

pattern of in�uen e an be seen also in the higher mode. Likely, di�eren es an be

appre iated also in the inversion results sin e the relevant variation in shape of the

pi ked fundamental mode.


Figure 7.11: 2D pro�les with the inversion results for setup 3;Top: input model of

sand waves, plotted as if topography is �at (zero indi ates the level of the re eivers).

Middle: input model of sand waves, plotted with a tual bathymetry (green line),

1D pro�les plotted at orresponding depths. As re eivers are pla ed at sea bed,

green line also indi ates the position of the re eivers. Bottom: input model of �at

topography (zero is indi ates the level of the re eivers).


Figure 7.11 shows the inferred values of shear-wave velo ities for both sand-wave

pro�le (top) and �at-topography pro�le (bottom). In the �at-topography ase the

velo ities are well retrieved, even though there is an overestimation in thi knesses

(∼5 m instead of 3 m). This overestimation was anti ipated in the results of �gure

4.6 (variation of re eiver height) when sour e and re eivers were pla ed at sea�oor.

Many dis repan ies are present in the upper panel pi tures; as already expe ted from

the strong in�uen e of the bathymetry on the dispersion plots (�gure 7.10). As an

example, the shot at 420 m has a poorly resolved fundamental mode that results in

a strong underestimation of the shear-wave velo ity. There is a good agreement of

the results in the area of 280-320 orrelated with the �at topography model, as they

have approximately the same thi kness. Contrarily to �gure 7.7, there is a better

estimation of thi kness around x=200 m. All in all, it appears that the use of OBC

an provide slightly better results ompared to a towed streamer.

7.2.3 Con lusions

In this se tion, three di�erent a quisition set ups have been used in order to invest-

igate the in�uen e of sea-bed topography on surfa e waves has been veri�ed. When

re eivers are pla ed at the sea�oor simulating an OBC, onsiderable improvement in

the subsurfa e V S model are observed. Further investigation is required to evaluate

possible orre tion pro edures for the angles of in iden e of the wave�eld on the

sea-bottom. In addition, it would be interesting as a more advan ed level, not only

the water layer to be taken into a ount in the inversion s heme but also a variable

thi kness of it.

7.3 Mega ripples

7.3.1 Modeling

In se tion 7.2, typi al sea�oor sand waves and their in�uen e on the fundamental

and higher modes has been studied. In that ase, the wavelength of the sand-

wave feature was mu h longer than the re eiver spread. In this se tion, the ase is

investigated of an irregular topography with mu h shorter wavelength (10 m) and

amplitude (1 m depth variation). So, as a next stage the e�e t of smaller dimensions

topography alled mega ripples is investigated.

The sea-bottom pro�les used in this experiment (�gure 7.12) were taken from

data provided by the Hydrographi al O� e of the Royal Dut h Navy.


Figure 7.12: Bathymetri map showing mega ripples lo ated about 50 km o�shore

of the North Holland oast (red dot in map). Data provided from Hydrographi al

O� e of the Royal Dut h Navy.

Figure 7.13: Cross-se tion of mega ripples sea-�oor topography from �gure 31. The x

axis represents the horizontal distan e and the y axis shows the bathymetry variation

of the mega ripples at the sea bottom.

A ross se tion of the area (�gure 7.13) has been implemented as sea-bottom

topography in the 2-layer model, as in the previous experiments. In horizontal

s ale, the mega ripples are smaller (10 m distan e between the top points) than the

spread length (47 m). They are also lower in height (variation within 1 m) ompared

to the sand-waves (variation of 3 m).

The input model has the same velo ity as the regular sand-wave model (see table

5 of �gure 7.4), but a slightly thi ker �rst layer (4 m). The model domain had a

dimension of 150 x 300 m, whereas the grid size was de reased to 0.12 m for an

adequate representation of the small topographi al hanges. In total, 11 shots with

5 m spa ing have been modelled, numbered from left to the right (see �gure 7.14).


Sour e and re eivers were pla ed at the same onstant depth of 5 m below the sea

surfa e, at an average height of 5 m above sea-�oor.

Figure 7.14: Input model of mega ripples pro�le on the left panel and the or-

responding referen e �at pro�le on the right panel. The stars indi ate the shots

positions. In total 11 shots were modelled with a spa ing of 5 m between the shots.

7.3.2 Results

Analysing the dispersion plots (�gure 7.15), a strong in�uen e of the mega ripples

is noti ed. Contrary to the sand-wave ase, the presen e of mega ripples generates

higher modes. These higher modes are not present in the orresponding �at topo-

graphy ase. One possible ause for the appearan e of higher modes might be related

to the smaller amplitude and wavelength of the ripples. Furthermore, the dispersion

plots of all shots vary less ompared to the ones of the regular sand-waves. For most

of the shots, as for example the shot at 55 and 75 m, the fundamental mode is broad

and poorly resolved.



at 35, 55 and 75 m for the mega-ripples model (�gure 7.14, left). Right: Dis-

persion plots and pi ked fundamental modes for the orresponding shots for the

�at-topography model (�gure 7.14, right). Input parameters are listed in table 5 of

�gure 7.4.


The 2D inversion results are shown in �gure 7.16. For the �at topography, even

if the velo ities are well onstrained for both �rst layer and halfspa e, there is a

strong underestimation of the thi kness. When the mega ripples are present there

is a large overestimation in both velo ity and thi kness. This is likely due to the

appearan e of higher modes on the mega-ripples pro�le; a possible solution ould

be to take them into a ount in the inversion pro ess. All in all, in order to avoid

enormous interpretational errors laterally inhomegeneous models need to be in luded

in inversion.

Figure 7.16: 2D inversion results for the mega-ripples pro�le and the orresponding

�at-topography model; zero indi ates the level of re eivers. Top: input model of

mega ripples, plotted as if topography is �at (bathymetry variation of mega ripples

is too small to be inserted in the plot as in the previous �gures). Bottom: input

model of �at topography.


7.3.3 Con lusions

In this se tion, the in�uen e of mega ripples on S holte waves has been investig-

ated. Surprisingly, smaller variation of topography has bigger in�uen e on S holte

waves as it is lari�ed by the appearan e of higher modes in all dispersion plots.

However, during the inversion, we observed onsiderable overestimation of velo ity

and thi kness of the �rst layer. This ontrasts with the orresponding �at topo-

graphy ase where an underestimation is observed. As mega ripples seem to have

bigger in�uen e on S holte waves than the regular sand waves,in order to be sure

that is a plysi al in�uen e and not a numeri al artifa t, two similar experiments

were made. Therefore, those models were tested with mega ripples of wavelength

of 7.5 m and of 12.5 m. Higher modes were observed in both ases, justifying the

in�uen e of smaller dimensions topography. Thus, it would be very interesting to

run similar experiments by in reasing the amplitude of ripples while keeping a on-

stant wavelength and testing intermediate s ales dunes/ripples. Furthermore, it is

to be expe ted that this kind of topography generates S holte waves as Zheng et al.

(2013) showed. These S holte waves ontribute to the �nal inferred V S pro�les even

though they are not seen as di�erent s ales are used. Therefore next step should

be the observation of those by using di�erent time and spatial s ales, and the pro-

essing of them. Finally, it is of great importan e to be examined if higher modes

an be in luded in the inversion s heme leading to more reliable V S pro�les of the

subsurfa e.

Chapter 8

Full-Velo ity-Spe trum inversion: the

winMASW software with examples

and omparisons

8.1 Introdu tion

As mentioned in hapter 3, towards the end of this study, an alternative inver-

sion software for surfa e waves be ame available. WinMASW, based on the Full-

Velo ity-Spe trum (FVS) inversion method, it allows to use amplitude variations

in the (v, f)-spe trum and is therefore more general than only using the velo ity-

dispersion urve (as in Geopsy). Several models were hosen for a omparison test

between winMASW and Geopsy. The models that were onsidered were the 10-layer

model ( hapter 5.2.3), the sand-wave model ( hapter 7.2) and the mega-ripple model

( hapter 7.3).

When onsidering the full velo ity spe trum it is essential to onsider only the

information related to the surfa e waves. This is a hieved by:

� Frequen y �ltering

� Removing useless data from the shot gather (P-wave events), hoosing a spe-

i� group of datasets for further analysis

� Down-sampling in order to avoid uselessly-higher omputational times, sug-

gested down-sampling to 2 ms

For the inversion pro edure, several input parameters need to be spe i�ed: a range

of values for V S and thi kness for ea h of the input layer. For onsisten y, we

used the same ranges of shear wave velo ities as in Geopsy (± 100 m/s). For the

50

CHAPTER 8. FULL-VELOCITY-SPECTRUM INVERSION: THEWINMASW SOFTWAREWITH EXAMPLES AND COMPARISONS51

thi kness, a variation of 1-5 m has been set for the 10-layer model and a variation

of 1-10 m for the model with sandwaves. Additionally, the number of layers as well

as the Poisson's ratio need to be spe i�ed. A Poisson's ratio between 0.35 and 0.5

is suggested to be used espe ially for soft soils; the value of 0.35 was hosen for all

of the inversions.

8.2 Results

8.2.1 Ten-layer model

Figure 8.1 shows the 10-layer velo ities pro�le after inversion for the winMASW best

model (the purely geneti ally reated model) (left) and the winMASW mean model

(right). For omparison, the input model and the retrieved by Geopsy are plotted.

The winMASW result (full velo ity spe trum) shows better identi� ation of subsur-

fa e stru ture, up to the maximum depth of 30 m ompared to Geopsy result. The

winMASW result follows better the trend of the real model. In this ase, the mean

model is loser to the input model than the �ttest one. The number of layers and

thus depths were properly identi�ed and velo ity inversion was re onstru ted with

minor deviations from the input model. This highlights that the big omplexity of

the problem requires a statisti al approa h in order to provide meaningful solutions

with un ertainty evaluation (as standard deviations are not provided from Geopsy).


Figure 8.1: Inversion results for the 10-layer model (table 4) from both Geopsy pa k-

age and winMASW. Left winMASW best subsurfa e model together with Geopsy

result and input model. Right: WinMASW mean subsurfa e model together with

Geopsy result and input model.

8.2.2 Regular sandwaves

For the omparison with winMASW the setup 1 was used, where re eivers are po-

sitioned to the right of the shots (se tion 7.2.2.1). All 14 shots were inverted using

winMASW and a �ttest and mean model were produ ed. The results are visualised

in a 2D pro�le (�gure 8.2). For most shots (example in �gure 8.2, top panel), the

inversion result shows a V S standard deviation of 20 %, whi h is onsidered to be

a statisti ally a urate result. However, for 3 out of 14 shots the inversions showed

a V S standard deviation more than 20 % (�gure 8.2, bottom panel). Therefore, for

visualisation the purely GA- based solution (�ttest model) is shown in the 2D pro�le

(�gure 8.3).


Figure 8.2: Representation of winMASW result for sand waves by the software

provided. Top: winMASW inversion result for shot at 220 m, with a Vs standard

deviation of the 1st layer of only 5%. Bottom: WinMASW inversion result for shot

at 360 m, with Vs standard deviation of the 1st layer of 22%.

In �gure 8.3 the resulting 2D pro�le is shown. For most of the onsidered shots

the depth and thi kness of the 1st layer were re onstru ted with small deviations

from the input model. However, omparable to the results in �gure 7.7 of the Geopsy

inversion, similar de� ien ies appear in the vi inity of 200-250 m. As before, these

an be explained by the di�erent angles of the slopes of sandwaves: di�erent angles

of in iden e might a�e t the a ura y of the retrieved results. Nonetheless further


investigation is required on this topi and de�ning how the waves travel along the

dunes. The use of the whole velo ity spe trum in the inversion pro edure seems to

provide a better estimation of the subsurfa e stru ture.

Figure 8.3: 2D inversion plot of the sandwaves pro�le by using the �ttest model

from the full velo ity spe trum method. On the top panel the pro�le is shown with

�at topography, while on the bottom panel the bathymetry is indi ated with the

green line.

8.2.3 Mega ripples

The dispersion plots along the pro�le of the mega-ripples were all very similar (�gure

7.5). Therefore, not all shots were inverted using winMASW, but only two shots

were sele ted. These were the shot at 35 m (fundamental mode very sharp) and the


shot at 75 m (fundamental mode smeared and modes not well separated). Figure

8.4 shows the winMASW result for both shots. For both ases, the �ttest and the

mean model are in very good agreement.

Figure 8.4: Representation of winMASW result for mega ripples by the software

provided. Top: for shot at 35 m. Bottom: for shot at 75 m.

In �gure 8.5, the di�eren es between the two inversion pro edures are evaluated.

Using the winMASW pro edure, there is a huge improvement in retrieving the velo-

ities and depth of the interfa e between �rst layer and the halfspa e. This stresses

the advantage of full exploitation of information in the dispersion plot when higher


modes are present to a hieve a well onstrained inversion result.

Figure 8.5: Inversion results for the mega-ripples model (�gure, left panel) from both

Geopsy pa kage and winMASW. Left: winMASW subsurfa e model together with

Geopsy result and input model for the shot at 35 m. Right: winMASW subsurfa e

model together with Geopsy result and input model for the shot at 75 m.

8.3 Con lusions

From the omparison of winMASW (full velo ity spe trum) and Geopsy (pi ked

dispersion urve) for the three examples, it is lear that the exploitation of the

entire spe trum in reases the a ura y of the results. It provides more reliable

inversion results ompared to the inversion of pi ked dispersion urve. Even though

winMASW is omputationally more demanding than Geopsy, it is re ommended to

use winMASW when dealing with omplex subsurfa e stru tures or with real data.

Chapter 9

Dis ussions

The evaluation of the e�e t of a water layer on S holte waves has been possible by

modelling S holte waves propagation in a ase of a simple 1D model with variable

water layer thi kness. We showed that the lo ation of sour e/re eivers ompared to

the sea bottom prevents the dete tion of the higher frequen ies of S holte waves in

the seismi re ords. The maximum frequen y de reases gradually with the in rease

of the water layer thi kness rea hing a plateau of half redu tion when the biggest

distan e (8 m above sea�oor) was tested.

When a 1D multi-layered model (10 layers) was used to perform the same ex-

periment, di�eren es also o urred in the range of 15 to 20 Hz where a relative

in rement in phase velo ities was observed ompared to the simple 1D model. This

highlights the need of an inversion algorithm that in orporates the water layer in

the forward model to a hieve a good re onstru tion of the subsurfa e. Besides that,

there is no doubt that when a quiring S holte waves sour e and re eivers should

be as lose as possible to sea�oor, at least 5 m a ording to our modelling results.

However, it may be said that su h a short distan e of a powerful, low frequen y,

pressure sour e ould ause a relevant threat for the marine fauna in the vi inity of

the water-sediment interfa e and the fauna in the water olumn as well. A solution

for that would be the usage of marine vibrators instead of airguns, as they an also

provide a better ontrol of frequen ies and for e.

One of the main issues when a quiring S holte waves is the in�uen e of the

variation of topography. The test with regular sand waves and sour e and re eivers at

5 m above the seabed (minimum) showed a strong underestimation in terms of shear-

wave velo ities and thi knesses along the steepest part of the sand-waves slope. The

strong deviation from the input model is likely due to the di�erent in iden e angles at

whi h the seismi wave�eld impinges on the interfa e. The inversion result was mu h

better when re eivers were adjusted to follow the bathymetry su h as in an OBC.

57

CHAPTER 9. DISCUSSIONS 58

However, using streamers for a quisition is mu h more pra ti al than using OBC.

Additionally, on erning our tests of di�erent setups, re eivers positioned at the left

and right side of the sour e respe tively, it is re ommended data a quisition to be

done via sailing in two dire tions. In this way, evaluation of sea-�oor topography

and laterally varying subsurfa e stru tures an be a hieved. Therefore, an in rease

in the understanding of the e�e t of in iden e angles along slopes by numeri al

al ulations together with laterally varying media in the inversion will probably

lead to an improved inversion result.

In the ase of small s ale topography (mega ripples), a di�erent in�uen e of

topography was pointed out. It seems that ripples give rise to higher modes, even

though modes are not well resolved and separated. As expe ted, it has serious impa t

on the inversion result: using Geopsy erroneous results are produ ed, espe ially for

depth determination. However, when the entire spe trum is used (winMASW),

more a urate results are obtained. All in all, it is lear that sea-�oor topography

in�uen es to a ertain degree the propogation and dete tion of S holte waves. Thus,

further investigation for in lusion of laterally inhomegeneous models in the inversion

needs to be undertaken.

Prior knowledge of sea�oor topography is useful for logisti s su h as safety, qual-

ity and se urity issues. No damage of the equipment and personal safety of the rew

needs to be assured. Therefore, it is important after the streamer is deployed, depth

monitors alled birds to be atta hed to regulate the height of the re eivers above

the sea�oor.

Taking everything into a ount the water layer should be taken into a ount.

The in lusion of this information an follow two di�erent strategies, depending on

the two extreme topographies that were tested. For example, an average water

depth ould be taken into a ount for the entire inversion; a suitable solution for

the mega-ripples example sin e small bathymetry variations are present. On the

other side, with layer bathymetri al hanges, su h as sand waves, it is of great need

an average water depth to be taken into a ount for ea h of the shots to be inverted.

Last but not least, the use of the full velo ity spe trum during inversion pro ess

instead of a pi ked dispersion urve was proven to be bene� ial. Our tests, with the

multi-layered model and the seabed topography, showed a onsiderable improvement

in the inversion result, roughly omparable to the starting model. The negative

aspe t of this approa h is the omputation time required by inversion. Nevertheless,

as long as it provides more reliable robust subsurfa e V S model is suggested when

omplex stru tures are studied.

Chapter 10

Con lusions

The intention of this resear h was to investigate the in�uen e of sea�oor topography

on S holte-wave propagation and inversion by implementing s enarios lose to real-

ity. An elasti Finite-Di�eren e s heme was used for the modelling of propagation

of S holte waves with di�erent subsurfa e on�gurations. In addition, several ex-

periments have been performed in order to evaluate:

� The in�uen e of the water-layer thi kness for a 1D simple and a 1D multi-

layered model

� The sensitivity to verti al heterogeneity

� The in�uen e of S holte-wave propagation when sea-bed topography is present

Regarding the e�e t of a water layer, a 1 D simple model was investigated with

a varying sour e-re eiver height ranging from 0 m to 8 m above the sea�oor. We

observed that the water layer damps the higher frequen ies, in a ordan e with Klein

et al. (2005). Determining the shallow depths of penetration the higher frequen ies,

if lost, auses limited information for the shallow part of the subsurfa e. As a

onsequen e re eivers must be pla ed as lose as possible to the sea�oor (< 5 m) to

apture the shallow subsurfa e trend.

Three di�erent models have been used to assess the e�e t of verti al heterogen-

eity: several models with 3 to 10 layers were tested. For all ases, the retrieved V S

pro�les showed a good general trend up to a depth of approximately 20 m. Nonethe-

less, the inversion performed well when smoother hanges o urred (10-layer model)

instead of sharpe hanges (as in the 3-layer model). Additionally, higher modes were

noti ed, whi h implies the sensitivity of S holte waves to sediment layering.

Given the good performan e of the inversion s heme when the model was strongly

multi-layered, a 10 layer model was hosen to test the e�e t of variable water-layer

59

CHAPTER 10. CONCLUSIONS 60

model, starting from 0 to 8 m. Considering the dispersion plots, dis repan ies in

the fundamental mode are shown in the range of 15-20 Hz. In the inversion results,

the general trend of the subsurfa e is aptured even though less pre ise when a

thi ker water body is present. In any of the used inversion s hemes (Geopsy or

winMASW) the water layer is not taken into a ount. Part of the dis repan ies

is probably related to this omission. We expe t that the re onstru tion of the

subsurfa e stru ture will be more a urate when the water layer is in luded in the

inversion algorithm.

The last set of tests regarded the analysis and inversion of S holte waves for

variable bathymetry. Two sea-bed topographies were implemented with di�erent

dimensions: regular sand waves and mega ripples. Regular sand waves are kind of

bedforms that are usually present in shallow seas and indu ed by tidal urrents. The

s ale of sand waves is mu h larger (100 m) than the re eiver-spread length (47 m).

Three a quisition set ups have been tested for the sand waves: re eivers at the right

side of ea h shot, re eivers at the left side (streamer) and re eivers pla ed on the

topography (OBC). Higher modes were present only when re eivers were pla ed at

the sea �oor simulating an OBC, thus obtaining a better retrieval of V S and depths,

espe ially for spe i� lo ation (area around x=200 m). However, in all of the tests

a relevant underestimation of velo ities and depth of the interfa e between the �rst

layer and the halfspa e was observed when the entre of sour e-re eiver spread was

above the steepest part of the sand waves. An explanation might be the di�eren e

in in iden e angles and the way that S holte waves travel inside and along the �rst

layer. Further resear h needs to be done for the numeri al al ulation of the di�erent

in iden e angles along the slopes of sand waves and how lateral heterogeneity an

be allowed in the forward model so as to a better representation of the subsurfa e

an be provided.

The se ond type of sea-bed topography onsiders mega ripples, whi h are small

s ale morphologi al features forming under the a tions of water �ow as well. Only

one a quisition setup has been used with a towed streamer laying at the right side

of ea h shot. Surprisingly, these small-s ale features give rise to higher modes.

However, a strong overestimation both in terms of velo ities and thi knesses for

the �rst layer is obtained from the inversion (using Geopsy). All things onsidered,

it has been demonstrated that ea h type of topography in�uen es S holte-wave

generation and a quisition and therefore the inversion result. However, the best

way of in luding laterally-varying modelling of the wave propagation is in need of

further investigation. Sin e the water layer was not in luded in either of the inversion

s hemes used in this resear h, the suggestions on how to ope with topography are

CHAPTER 10. CONCLUSIONS 61

not implemented here.

In the MASW method the a ura y of the inversion is largely depending on the

a ura y of pi king dispersion urves. As a last stage of this thesis, an inversion

based on the entire velo ity spe trum was used. As expe ted, the utilization of

the all information available in the spe trum and modes lead to a more robust

subsurfa e Vs model in all tested models. Regardless of the fa t that this approa h

is omputationally intensive, it is re ommended when dealing with real data or

omplex subsurfa e stru tures.

Bibliography

[1℄ Allou he, N., Drijkoningen, G. G., Versteeg, W., Ranajit, G., 2011, Converted

waves in a shallow marine environment: experimental and modelling studies,

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Appendix A

Su� ient re onstru tion of wave�eld

In order to obtain a urate results, the wave�eld needs to be adequately re-

onstru ted both in time and spa e; it means that the Nyquist riterion should be

full �lled. The Nyquist sampling theorem provides a pres ription for the nominal

sampling interval required to avoid aliasing. It might be stated as follows: �The

sampling frequen y should be at least twi e the highest frequen y ontained in the

signal�.

Or in mathemati al terms:

f s = 2fmax (10.1)

where f s is the sampling frequen y (how often samples are taken per unit of time

or spa e) and fmax is the highest frequen y ontained in the signal.

In spa e, a ording to the same riterion, at least two sampling points per min-

imum wavelength are needed. In other words, the distan e between two re eivers

∆x must ful�ll the riterion:

∆x ≤cs(min)

2fmax

(10.2)

where cs(min) is the minimum shear wave velo ity.

65

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