改进的角域CIGs在差分似然优化方法中的应用

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Differential Semblance Optimization Method with Angle domain CIGs

Fang Ren*, ZhenChun Li, Min Zhang and FeiXu Chen, m China University of Petroleum

Summary

Differential semblance optimization (DSO) is an approach

to inverse the velocity which avoids the severe convergence

associated with nonlinear least-squares inversion. The DSO

objective function measures the deviation of image gathers

and expected to have much better global convexity property

and suffer less from the local minima. Generally, ODCIGs

are used in the construction of objective function. In this

paper, we also use ADCIGs to construct the function which

estimate the correctness of velocity model directly and

reflect the coupling relation between velocity and depth

accurately. The result of Marmousi shows that angle

domain CIGs correct the velocity effectively and almost

have no image artifacts.

Introduction

An accurate velocity model and exact migration algorithms

are both essential for the result of migration and imaging,

usually, velocity plays a more important role, as the

advantage of the sophisticated migration algorithms is

evident only when the velocity is reliable. Since first

proposed in exploration seismology in the 1970s (Gardner,

1974; Sattlegger,1975),migration velocity analysis has

become increasingly important in seismic imaging and

velocity modeling.

According to different theories, seismic velocity analysis

methods can be divided into two groups: One aims at

minimizing the misfit in data domain, such as waveform

inversion (WI) (Tarantola, 1984; Woodward, 1992; Pratt,

1999; Virieux and Operto, 2009), while the other aims at

improving the quality in image domain, such as migration

velocity analysis (MVA) (Symes and Carazzone,

1991;Biondi and Sava, 1999; Sava and Biondi, 2004a,b;

Shen and Symes, 2008; Guerra et al., 2009).Several

advantages drive us to use the image space method

(WEMVA) instead of data-space method (Waveform

Inversion): first, the migrated image is often much cleaner

than the recorded wave fields; second, the objective

function is directly related to the final image. However, as

only transmission information is used in the image domain

method, the vertical resolution of inverted velocity model is

limited.

Differential Semblance Optimization (Symes and

Carazzone, 1991; Shen and Symes, 2008) is one of the

wave-equation migration velocity analysis methods that

utilize the difference between image gathers to form

objective function, the function has better convexity and

the gradient is smooth, which avoids the local minimum

when there lacks low frequency or long offset information

in the data. Several authors have since developed inversion

algorithms based on the DSO concept (Shen et al, 2003,

Chauris and Noble, 2001, Plessix, et al, 2000). DSO-MVA

uses (one-way) wave equation migration as the engine to

calculate gathers and update the velocity model, with the

gradient of the objective function being calculated by the

adjoint state method.

Theory

DSO objective function: offset & angle

The concept of DSO method was proposed by Symes

(1991), it introduces a extend axis, usually is offset or angle,

when the velocity is correct, the normed difference between

neighboring traces along the “redundant” axis in a prestack

common image gather should be minimal.

First, we start with the imaging condition as following:

( , ) ( , , ) ( , , ) ( , , )

s r r s

I x h G x h x G x h x d x x



  



  



(1)

where

is the image,

is the Green’s function,

is the

surface data,

is the subsurface offset,

and

are the

source and receiver coordinates respectively. Then we

construct the DSO objective function

|| ||

J PI

(2)

where

are the common image gathers,

is the DSO

operator that penalizes the singularity in common image

gathers when the velocity is accurate, and when the

velocity is inaccurate, the DSO operator can eliminate part

of the imaging errors. The image gathers

is called

redundant images.

As to offset domain CIGs, the objective function can be

written as

|| || || ( , ) ||

J P I hI x h

(3)

which

Ph

and

( , )I x h

are offset gathers which

effectively demonstrate the degree of focusing. In terms of

the image volume

( , , )I x z h

, the characteristic signature of

kinematically correct velocity is focusing or concentrating

0h 

. If there exists some energy away from

0h 

, then

( , , ) 0hI x z h 

. So the DSO operator annihilates the energy

0h 

and emphasizes the energy away from

0h 

when the objective function verges to zero, the energy

mainly focuses at one point near

0h 

, it thus provides a

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