ORIGINAL RESEARCH PAPER

Seafloor elastic parameters estimation based on AVO inversion

Yangting Liu1 • Xuewei Liu1

Received: 8 January 2015 / Accepted: 24 July 2015 / Published online: 30 July 2015

� Springer Science+Business Media Dordrecht 2015

Abstract Seafloor elastic parameters play an important

role in many fields as diverse as marine construction,

seabed resources exploration and seafloor acoustics. In

order to estimate seafloor elastic parameters, we perform

AVO inversion with seafloor reflected seismic data. As a

particular reflection interface, the seafloor reflector does

not support S-waves and the elastic parameters change

dramatically across it. Conventional approximations to the

Zoeppritz equations are not applicable for the seafloor

situation. In this paper, we perform AVO inversion with

the exact Zoeppritz equations through an unconstrained

optimization method. Our synthetic study proves that the

inversion method does not show strong dependence on the

initial model for both unconsolidated and semi-consoli-

dated seabed situations. The inversion uncertainty of the

elastic parameters increases with the noise level, and

decreases with the incidence angle range. Finally, we

perform inversion of data from the South China Sea, and

obtain satisfactory results, which are in good agreement

with previous research.

Keywords Seafloor � AVO inversion � Elasticparameters � Optimization method � The South China Sea

Introduction

Seabed elastic parameters play an important role in many

fields as diverse as marine construction, seabed resources

exploration and seafloor acoustics. Significant research work

based on different wave fields has been previously carried

out. Riedel and Theilen (2001) and Riedel et al. (2003)

conducted AVO investigations of shallow marine sediments,

and inverted AVO data to provide estimates and uncertain-

ties of the viscoelastic physical parameters using a Bayesian

approach. By exploiting the dispersive properties of Scholte

waves, Klein et al. (2005) investigated the potential to reveal

the elastic parameters of the shallow seabed subsurface and

Wilken et al. (2009) inferred the shear-wave velocity

structure. The features of converted waves in a shallow

marine environment were studied by Allouche et al. (2011).

Weemstra et al. (2013) used ambient seismic noise to con-

strain subsurface attenuation by seismic interferometry. In

geoacoustic inversion, there is also significant work that is

based on the acoustic field rather than the seismic wave field

(e.g., Dosso and Holland 2006; Dettmer et al. 2007, 2011;

Dettmer and Dosso 2008).

In this paper, we estimate the seabed elastic parameters

by AVO response of reflected seismic data. AVO inversion

is an effective method to estimate lithology parameters.

Classical AVO inversion is based on approximations to the

Zoeppritz (1919) equations. Most approximations are based

on the weak property contrast assumptions or that the

properties across the interface obey certain laws (e.g.,

Bortfeld 1961; Aki and Richards 1980; Shuey 1985; Vedanti

and Sen 2009; Alemie and Sacchi 2011; Zhu and McMe-

chan 2012). However, the properties across the seafloor

interface vary dramatically. Furthermore, no S-wave gen-

erates from the seafloor interface when the incident P-wave

travels through seawater. Therefore, the assumption of those

& Yangting Liu

yangting.lau@gmail.com

Xuewei Liu

liuxw@cugb.edu.cn

1 School of Geophysics and Information Technology, China

University of Geosciences (Beijing), No. 29 Xueyuan Road,

Haidian District, Beijing 100083, China

123

Mar Geophys Res (2015) 36:335–342

DOI 10.1007/s11001-015-9260-1

linearized approximate equations leads to big errors when

dealing with the seabed situation. Research considering a

high contrast interface (e.g., Zheng 1991; Yang and Zhou

1994; Yin et al. 2013) has shown, however, that any

approximation breaks down at a sufficiently large incidence

angle. Liu et al. (2015) have derived an approximation to

Zoeppritz equations specified for the seafloor situation and

the two-step inversion method has overcome the angle limit

of the approximation. In this paper, we performed AVO

inversion directly with the exact function of P-wave reflec-

tion coefficient variation with incidence angle derived from

the Zoeppritz equations. Our synthetic study proves that the

inversion method does not show strong dependence on the

initial model. The influence of noise level and incidence

angle range on inversion result is also investigated. The

uncertainty in the elastic parameters resulting from the

inversion increases with the noise level, and decreases with

the incidence angle range. Finally, we perform an inversion

with seafloor reflected data from the South China Sea.

Methodology

AVO theory is based on the Zoeppritz (1919) equations,

which express the reflection and transmission coefficients

as a function of incidence angle and the elastic properties

across a planar interface. For the seabed situation, where

the upper medium (seawater) does not support S-waves,

there are only reflected and transmitted P-waves and a

transmitted S-wave (Fig. 1).

The fluid–solid interface must satisfy three boundary

conditions: the continuity of normal strain, the continuity

of normal stress, and that the shear stress vanishes at the

interface. Therefore, three equations are involved in

describing the reflection and transmission at the seafloor:

cosh1 cosh2 sinh30

a1a2

sin 2h2 �a1b2

cos 2h3

0 �a2q2a1q1

cos2h3 �b2q2a1q1

sin 2h3

26664

37775

Rpp

TppTps

24

35

cosh10

�1

24

35

ð1Þ

In Eq. (1), Rpp is the P-wave reflection coefficient; Tpp and

Tps are the transmission coefficients of the P-wave and S-

wave respectively; a1 and q1 are the P-wave velocity and

density of the seawater; a2, b2 and q2 are the P-wave

velocity, S-wave velocity and density of the seabed

respectively; h1 is the angle of incidence; h2 and h3 are thetransmission angles of the transmitted P-wave and con-

verted S-wave constrained by the Snell’s law:

sin h1a1

¼ sin h2a2

¼ sin h3b2

ð2Þ

From Eq. (1), the exact function Rpp can be obtained by

substituting Tpp and Tps:

Rpp ¼1þ N

1� Nð3Þ

where:

N ¼ �a1b2

cos 2h3sin 2h1

þ sin h3cos h1

a22q2

a1b2q1cos2 2h3sin 2h2

þ b2q2a1q1

sin 2h3ð4Þ

In the situation of seafloor AVO, the P-wave velocity

and density of the seawater are typically known, and the

seabed properties (a2, b2, q2, represented by m below) are

of interest. Consider a set of observed AVO data Riobs (P-

wave reflection coefficient as a function of incidence angle

hi) and the simulated data Risyn (calculated from Eq. 3 under

the incidence angle hi). Then, the classical least-square

misfit function is given by:

r mð Þ ¼ 1

2

Xni¼1

Robsi � R

syni

� �2 ð5Þ

where the simulated data Risyn is calculated with the given

seabed elastic parameters m. The summation is over all the

observed incidence angles. The goal of the inversion is to

obtain the model parameter m, which can be updated as

follows:

mnþ1 ¼ mn þ lndn ð6Þ

where ln is the step length in the nth iteration and dn cor-

responds to the updating direction, which can be obtained

Fig. 1 Reflection and transmission at the seafloor resulting from an

incident P-wave through the seawater

Table 1 Model parameters

Model a2 (m/s) b2 (m/s) q2 (kg/m3)

M1 1550 200 1600

M2 1800 400 2000

336 Mar Geophys Res (2015) 36:335–342

123

from the gradient of the misfit function with respect to the

model parameters. The steepest descent method is adopted

and the gradient is calculated as follows:

orom

¼Xni¼1

Robsi � R

syni

� � oRsyni

omð7Þ

The iteration calculation stops when each component of

the vector m begins to oscillate:

mk ¼ mk�2 ð8Þ

where k denotes the iteration number of the inversion. The

inversion result mf can be given as follows:

mf ¼ 1

2mk þmk�1ð Þ ð9Þ

Synthetic study

This section considers two kinds of seabed models to

investigate how well the method can work under different

sedimentary environments. Shallow marine environments

are often characterized by unconsolidated seabed sedi-

ments which typically have a P-wave velocity close to that

of the seawater, low S-wave velocity, and high attenuation

coefficients (e.g., Hamilton 1980). Semi-consolidated sed-

iments have higher values of elastic parameters but lower

attenuation coefficients. The influence of attenuation on the

reflection coefficients is only relevant near the critical

incidence angles (Riedel and Theilen 2001). In this paper,

we only deal with the reflection coefficients at small inci-

dence angles which are much smaller than the critical

angle, so it is safe to assume an elastic model.

The P-wave velocity and density of seawater are given

as 1500 m/s and 1000 kg/m3 respectively. The elastic

parameters adopted for the sedimentary environments (M1

for unconsolidated model, M2 for semi-consolidated

model) are listed in Table 1.

Reflection coefficient data are computed for 50 equally

spaced incidence angles ranging from 1� to 50� using

Eq. (3) (Fig. 2, solid lines). The apparent critical angles of

75.4� (unconsolidated) and 56.4� (semi-consolidated) are

out of the incidence angle range.

In the inversion procedure, the iteration step length is set

to 0.05. To investigate the effect of the initial model on the

inversion result, we perform the inversion with noise-free

data starting from three different initial models (Table 2)

and carry out the iteration calculation twenty thousand

times as the iteration calculation does not stop when

meeting the iteration termination condition of Eq. (8). The

calculation results of each inversion iteration step under the

given initial models are shown in Fig. 3.

From Fig. 3, we can see that the inversion results show

convergence to the true model parameters regardless of the

initial model. For the given iteration step length, the iter-

ation number at which the elastic parameters show con-

vergence mainly depends on the difference between initial

and true models. For the initial model S1, the inversion

convergence to M1 occurs after fewer iterations than for

that to M2. For the initial model S3, the inversion

Fig. 2 Reflection coefficient

variation with incidence angle.

The solid lines are noise-free

reflection coefficient calculated

from Eq. (3) with adopted

model parameters. The

scattered dots with two-SD

error bars are reflection

coefficients with additive

Gaussian-distributed random

errors (nl = 0.4 %, defined in

Eq. 10)

Table 2 Initial values of the elastic parameters

Initial model a2 (m/s) b2 (m/s) q2 (kg/m3)

S1 1500 150 1550

S2 1700 300 1800

S3 1900 450 2100

In Table 2, S1, S2, and S3 are three different sets of initial models.

The values of the elastic parameters in S1 are less than those in model

M1. The values of the elastic parameters in S3 are larger than those in

model M2. The values in S2 are between those in M1 and M2.

(S1\M1\S2\M2\S3)

Mar Geophys Res (2015) 36:335–342 337

123

Fig. 3 Parameter variations and

convergences through the

20,000 iterations. The red lines

denote inversion performed

under initial model S1. The

black lines denote inversion

performed under initial model

S2. The blue lines denote

inversion performed under

initial model S3

Fig. 4 Inversion results for

different incidence angle

ranges. The black line denotes

the inversion result for the

incidence angle range 0�–50�.The red line denotes the

inversion result for the

incidence angle range 0�–45�.The blue line denotes the

inversion result for the

incidence angle range 0�–40�.The green line denotes the

inversion result for the

incidence angle range 0�–35�.The three panels on the left

column are the inversion results

for M1, while the three panels

on the right column are for M2

338 Mar Geophys Res (2015) 36:335–342

123

convergence to M2 occurs after fewer iterations than for

that to M1. In general, the inversion requires higher num-

ber of iterations when the difference between initial and

true models is large, and vice versa. However, this is not

true for all situations because of the inherent nonlinearity

of the inversion. The differences are equal from S2 to M1

and S2 to M2 (for b2 and q2). However, the iteration

numbers at which the elastic parameters show convergence

are not exactly the same for M1 and M2.

With regard to the effect of the incidence angle range on

the inversion result we add independent Gaussian-dis-

tributed random errors (zero mean) of noise level

nl = 0.4 % to the noise-free reflection coefficient to pro-

duce the observed data (Fig. 2, scattered dots). The error

Fig. 5 Inversion results for

different noise levels. The black

line denotes the inversion

results with a noise level of

0.1 %. The red line denotes the

inversion results with a noise

level of 0.4 %. The blue line

denotes the inversion results

with a noise level of 0.7 %. The

green line denotes the inversion

results with a noise level of

1.0 %. The three panels on the

left column are the inversion

results for M1, while the three

panels on the right column are

for M2

Fig. 6 Approximate Location

of the Seismic Survey

Mar Geophys Res (2015) 36:335–342 339

123

bars in Fig. 2 are calculated as twice the SD of the noise

values. Noise level is defined as follows:

nl ¼

Pni¼1

jnoiseij

Pni¼1

jRsynij

ð10Þ

where nl denotes noise level, Rsyni

indicates the noise-free

reflection coefficient calculated from Eq. (3), noisei is the

noise values of the Gaussian-distributed random errors.

The inversion is performed for four different incidence

angle ranges (1�–50�, 1�–45�, 1�–40�, 1�–35�) of the

observed data. We perform the inversion from initial model

S2 and iterate until the calculation meets the iteration ter-

mination condition (Eq. 8). For each incidence angle range,

the inversion is performed 100 times under the given noise

level (0.4 %). The inversion results show the effect of the

incidence angle range clearly (Fig. 4).

The inversion results show that the uncertainty rises

with a decreasing incidence angle range, and vice versa.

The uncertainty in S-wave velocity is the highest among

the three inverted elastic parameters, while the uncer-

tainty in P-wave velocity and density is almost the same

for the same incidence angle range. For the same inci-

dence angle range, the uncertainty in the inversion of M1

is lower than that of M2. To reduce uncertainty in the

inversion caused by noise, we should use large incidence

angle range.

We also consider the effect of Gaussian-distributed

noise of different levels on the inversion result and a

similar method is adopted. Gaussian-distributed noise of

different levels at 0.1, 0.4, 0.7, 1.0 % was added to the

noise-free reflection coefficient (Fig. 2, solid lines) to

produce the observed data. One hundred experimental

inversions are performed for each noise level. All the

inversions are performed for the angle range of 1�–50�starting form initial model S2. The inversion results show

the effect of noise level clearly (Fig. 5).

The inversion results of M1 and M2 considering the

effect of noise show that the inversion uncertainty rises

with the noise level. The uncertainty in different parame-

ters can be compared by their relative values. The uncer-

tainty in S-wave velocity is higher than that of the other

two elastic parameters at the same noise level. The

uncertainty in P-wave velocity and density is almost the

same at a certain noise level. At the same noise level, the

inversion uncertainty of M1 is higher than that of M2.

Therefore, in order to reduce the inversion uncertainty, we

should adopt, whenever possible, low noise level data and/

or large incidence angle data.

Fig. 7 A migrated CRP gather from the seismic survey

Fig. 8 Reflection coefficient variation with incidence angle. The

solid line is calculated form Eq. (3) with the average of inverted

elastic parameters from the three initial models. The scattered dots

with two-SD error bars denote the observed reflection coefficient.

The error bars are calculated as twice the SD of the misfit between

the scattered dots and the solid line

Table 3 Inversion results from different initial models

Parameter a2 (m/s) b2 (m/s) q2 (kg/m3)

Inversion_1 1516.3959 363.3284 1638.9115

Inversion_2 1516.3893 363.3304 1638.9076

Inversion_3 1516.3867 363.3311 1638.9062

Average 1516.3906 363.3300 1638.9084

340 Mar Geophys Res (2015) 36:335–342

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Real data inversion: South China Sea

In this section we perform inversion with AVO data from a

seismic survey in the South China Sea (Fig. 6). High-res-

olution seismic reflection data were collected in 2014 by

Guangzhou Marine Geological Survey, using a 192-chan-

nel marine streamer and a tuned airgun array. The volume

of the tuned airgun array composed of 8 guns was

2622 cm3 with a dominant frequency of approximately

70 Hz. The acquisition parameters were set as follows:

25 m shot point interval, 1 ms sampling rate and 5 s record

length, 192 channels with interval of 12.5 m with a max-

imum offset of 2525 m.

The processing sequence applied to the observed data

mainly includes a bandpass filter, a spherical divergence

correction, source and receiver directivity correction

(Riedel and Theilen 2001). To achieve more accurate

imaging, amplitude-preserving prestack time migration is

also adopted (Mosher et al. 1996). The processed CRP

gather is shown in Fig. 7 (52 equally spaced incidence

angles ranging from 4� to 55�). By picking the amplitude of

the seafloor reflection from the CRP gather, we can obtain

the AVO data (Fig. 8).

The inversion is performed from the three models (S1,

S2 and S3) mentioned in Table 2. The P-wave velocity of

seawater is adopted as 1544 m/s according to the velocity

analysis procedure. The density of seawater is adopted as

1030 kg/m3. Figure 8 shows 20,000 times of iteration

calculations under a step length of 0.05. The inversion

results of the three elastic parameters are listed in Table 3

(Fig. 9).

The inversion results from different initial models are

very close to each other. The misfit of the results is less

than twice the step length. It is clear that the inverted

P-wave velocity of the seabed sediments is close to that of

the seawater, with a ratio as 0.98. Conventional oil

exploration does not focus on the seafloor, and therefore,

there is no dependent measurement (e.g., well log) of these

elastic parameters exactly at the location where the inver-

sion is performed. However, the inverted P-wave velocity

and density are in good agreement with previous research

(Pan 2003) in which contour maps of related parameters

measured in seafloor samples are reported. In Pan (2003),

the P-wave velocity of the nearest two samples is 1496 and

1542 m/s, and the density of the nearest two samples is

1550 and 1660 kg/m3. The S-wave velocity has been

Fig. 9 Variation and

convergence of parameters

through the 20,000 iterations.

The red lines denote inversion

performed under initial model

S1. The black lines denote

inversion performed under

initial model S2. The blue lines

denote inversion performed

under initial model S3

Mar Geophys Res (2015) 36:335–342 341

123

observed to range from 281 to 611 m/s (Lu 2005) in the

northern continental shelf where the sediments are similar.

Conclusions

Conventional approximate equations to the exact Zoeppritz

equations are not applicable for the seafloor situation. In

this paper, we performed AVO inversion based on the

exact Zoeppritz equations, through an unconstrained opti-

mization method. We consider an unconsolidated seabed

model and a semi-consolidated seabed model to test the

inversion method. From a synthetic study, we determine

that the inversion result does not show strong dependence

on the initial model. The uncertainty of the inversion in the

elastic parameters increases with the noise level, and

decreases with increasing incidence angle range. There-

fore, the lower the noise level, the smaller the uncertainty.

To reduce the inversion uncertainty caused by noise, a

large range in incidence angles should be used. Inversion

with AVO data from the Qiongdongnan Basin in the South

China Sea provides consistent inversion result and attests

to the validity of the method.

Acknowledgments Our project was funded by the International

Science and Technology Cooperation Program of China (Grant No.

2010DFA21630), and the National Basic Research Program of China

(973 Program, Grant No. 2009CB219505). Thanks go to the

Guangzhou Marine Geological Survey for providing the seismic data

and financial support.

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