CN109975868A - A kind of wave equation ghost reflection drawing method based on Taylor expansion - Google Patents

Abstract

The invention relates to a method for suppressing ghost waves in wave equations based on Taylor expansion, which is characterized by comprising the following steps: (1) expressing the seismic data containing ghost waves as the superposition of primary waves and ghost waves, and obtaining a ghost wave operator G (2) Based on Taylor expansion of ghost wave operator G, the first wave expression represented by wave field extension matrix F is obtained; (3) The wave field extension matrix F is obtained by calculation, and its Substitute into the primary wave expression in step (2) to obtain the primary wave after ghost wave suppression. The invention can be widely used in the field of ghost wave suppression of seismic data.

Description

A Method for Suppressing Ghost Waves of Wave Equation Based on Taylor Expansion

technical field

The invention relates to a method for suppressing ghost waves of wave equations based on Taylor expansion, and belongs to the field of seismic data processing.

Background technique

In the process of seismic data acquisition of offshore streamers, the streamers are generally placed at a certain depth below the sea level. Due to the strong reflection of the sea level, the strong ghost reflections in the streamer acquisition records lead to the notch effect of the seismic data spectrum, which greatly affects the In terms of data quality, seismic exploration generally refers to this kind of ghost reflection as ghost wave. Ghost waves are generally not well imaged, and will cause a large amount of migration noise in the migration profile. Therefore, ghost wave suppression is a key link in marine seismic exploration data processing.

In the past decade, in order to effectively suppress ghost waves and obtain broadband seismic data, a series of ghost wave suppression techniques have emerged. These technologies can usually be divided into two categories: (1) Related to the acquisition method, upper and lower source collection, upper and lower cable collection, dual detection collection, streamer three-component collection, submarine cable collection (OBC), variable depth cable collection, etc., mainly Through special acquisition methods and targeted processing methods, ghost waves can be effectively suppressed and the frequency range of seismic data can be broadened. These special collection methods are subject to many limitations in field operation conditions, such as the increased difficulty in controlling the source and cable during field collection, which leads to a substantial increase in collection costs; (2) related to processing technology, the commonly used method is to Wave suppression is regarded as an inversion problem and solved by least squares. Because there is a singularity problem in the direct inversion of the ghost wave operator, in order to solve the problem stably, the usual method is to carry out regularization constraints, but this method reduces the accuracy of ghost wave suppression due to the introduction of the regularization term, and the signal-to-noise ratio of the suppressed data is reduced. lower.

SUMMARY OF THE INVENTION

In view of the above problems, the purpose of the present invention is to provide a method for suppressing ghost waves in wave equations based on Taylor expansion. The method of the present invention can achieve high-precision suppression of ghost waves. Compared with traditional ghost wave suppression methods, the present invention has better results in suppressing ghost waves. Fidelity and higher signal-to-noise ratio.

To achieve the above object, the present invention adopts the following technical solutions: a method for suppressing ghost waves of wave equations based on Taylor expansion, which comprises the following steps:

(1) Express the seismic data containing ghost waves as the superposition of primary waves and ghost waves, and obtain the primary wave expression represented by the ghost wave operator G;

(2) Based on the Taylor expansion of the ghost wave operator G, the primary wave expression represented by the wave field extension matrix F is obtained;

(3) Calculate the wave field continuation matrix F, and substitute it into the primary wave expression in step (2) to obtain the primary wave after ghost wave suppression.

Further, in the step (2), the expression of the primary wave represented by the wave field extension matrix F is:

p=(1+F+F ² +...)d,

Among them, p is the primary wave, and d is the seismic data with ghost waves.

Further, in the described step (3), the method for obtaining the wavefield extension matrix F by calculation includes the following steps:

(3.1) Extend the recorded wave field p(r′) of the streamer surface Γ ₀ up to the sea level Γ ₁ along the Z axis, and calculate the wave field p(r) at r on the sea level Γ ₁ . The field p(r) is expressed in the integral form of the recorded wave field p(r') and its normal derivative;

(3.2) Assuming that the medium between the streamer surface and the sea level is uniform, the normal derivative of the recorded wavefield p(r′) at the streamer surface Γ ₀ can be approximately obtained;

(3.3) Express the Green's function in three dimensions, and obtain the three-dimensional expression of the Green's function;

(3.4) Substitute the normal derivative of the recorded wavefield p(r′) of the streamer surface Γ ₀ obtained in steps (3.2) and (3.3) and the three-dimensional expression of Green’s function into step (3.1), and get the integral equation of the wavefield p(r) at r at sea level with normal derivatives;

(3.5) Discretize the integral equation of the wave field p(r) at r at the sea level in step (3.4), and then the wave field extension matrix F can be obtained.

Further, in the step (3.1), the wave field p(r) at r on the sea level Γ ₁ is:

where G(r,r') is Green's function, Represents the normal derivative.

Further, in the step (3.2), the normal derivative of the recorded wave field p(r') is:

where θ is the angle between the boundary normal direction and the propagation path rr′, is the background wavenumber, c ₀ is the background velocity, and ω is the angular frequency.

Further, in the step (3.3), the three-dimensional expression of the Green's function is:

Further, in the step (3.4), the integral equation of the wave field p(r) at the sea level r without the normal derivative is:

Among them, r=|rr′|,

Because the present invention adopts the above technical scheme, it has the following advantages: the present invention adopts the wave equation based on Taylor expansion to represent the seismic data containing ghost waves, avoids the singularity problem when directly inverting the ghost wave operator, and simultaneously It avoids the shortcomings of low ghost wave suppression accuracy and low signal-to-noise ratio of data after suppression due to the introduction of a regularization term in the conventional method. Widely used in the field of ghost wave suppression technology.

Description of drawings

Figure 1 is the velocity model, the upward triangle in the figure represents the source, and the downward triangle represents the receiver;

Fig. 2a is a ghost wave recording synthesized by the wave equation;

Figure 2b is a ghost-free recording (as a reference) synthesized by the wave equation;

Figure 3a is the result of suppressing ghost waves by conventional methods;

Figure 3b shows the result of suppressing ghost waves by the method of the present invention.

Detailed ways

The present invention will be described in detail below with reference to the accompanying drawings and embodiments.

A method for suppressing ghost waves of wave equation based on Taylor expansion provided by the present invention comprises the following steps:

(1) Express the seismic data containing ghost waves as the superposition of primary waves and ghost waves, and obtain the primary wave expression represented by the ghost wave operator G.

The seismic data with ghost waves can be regarded as the superposition of the primary wave p and the ghost wave g, namely

d=p+g=(I+RF)p=Gp (1)

Among them, F represents the wave field continuation matrix, R represents the reflection coefficient of the sea level, which is usually assumed to be equal to -1, and G represents the ghost wave operator. In order to obtain a record without ghost waves, it is necessary to invert the ghost wave operator G, and there is a singularity in directly inverting the ghost wave operator G (the denominator is 0). The usual practice is to add a regularization term to the denominator, which is

where ε is a small positive number. Formula (2) is a conventional ghost wave suppression algorithm. This method overcomes the singularity problem and can achieve stable ghost wave suppression. However, because the operator of formula (2) is not exactly equal to the inverse of the ghost wave operator, so Ghost wave suppression accuracy is low.

(2) Based on the Taylor expansion of the ghost wave operator G, the primary wave expression represented by the wave field extension matrix F is obtained.

It can be seen from formula (1) that the inverse 1/(I-F) of the ghost wave operator G=(I-F) can avoid the singularity problem by the following Taylor expansion:

Substituting formula (3) into formula (1), the expression of the primary wave p represented by the wave field extension matrix F is obtained:

p=(1+F+F ² +…)d (4)

It can be seen that formula (4) is a stable inversion operator, and its solution needs to explicitly characterize the wavefield continuation matrix F.

(3) Calculate the wave field continuation matrix F, and substitute it into the primary wave expression in step (2) to obtain the primary wave after ghost wave suppression.

Specifically, it includes the following steps:

(3.1) Extend the recorded wave field p(r′) of the streamer surface Γ ₀ up to the sea level Γ ₁ along the Z-axis, and calculate the wave field p(r) at r on the sea level Γ ₁ , where, The Z axis represents the depth direction, and the X and Y axes represent the horizontal direction.

The wavefield p(r) is expressed in the integral form of the recorded wavefield p(r′) and its normal derivative, namely:

where G(r,r') is Green's function, Represents the normal derivative.

(3.2) Assuming that the medium between the streamer surface and the sea level (that is, the seawater layer) is uniform, the normal derivative of the recorded wavefield p(r′) at the streamer surface Γ ₀ can be approximated.

In order to calculate the wavefield u(r) at r, it is necessary to know the upper wavefield u(r′) of the streamer plane Γ ₀ and its normal derivative But usually we only record the wave field value p(r'), and the record of the normal derivative is missing. In order to use the equation (5) to carry out the wave field extension of the actual data, the equation (5) needs to be simplified. Ignoring the multiple reflections between the streamer surface Γ ₀ and the sea level Γ ₁ , the normal derivative of the recorded wavefield p(r') can be approximately expressed as:

where θ is the angle between the boundary normal direction and the propagation path rr′, is the background wavenumber, c ₀ is the background velocity, and ω is the angular frequency.

(3.3) Express the Green's function in three dimensions, and obtain its formula:

(3.4) Substitute the normal derivative of the recorded wavefield p(r′) of the streamer surface Γ ₀ obtained in steps (3.2) and (3.3) and the three-dimensional expression of Green’s function into step (3.1), and get The integral equation of the wavefield p(r) at r at sea level with normal derivatives:

Among them, r=|rr′|,

(3.5) Discretize the integral equation of the wave field p(r) at r at the sea level in step (3.4), and then the wave field extension matrix F can be obtained.

Discretizing the integral equation of the wave field p(r), we get:

p(r)=Fp(r′) (9)

Combining formulas (8) and (9), the wave field extension matrix F can be obtained.

The method of the present invention will be further introduced below in conjunction with specific embodiments.

1) Forward modeling of wave equation

As shown in Figure 1, it is the input half-space velocity model, in which the triangle with the upward direction represents the source, and the source function is the rake wavelet with the main frequency of 15Hz. The geophone (downward facing triangle in the figure) is located at a depth of z=50m below sea level. Selecting appropriate parameters for forward modeling, the seismic record shown in Figure 2 can be obtained, in which Figure 2a represents the seismic record with ghost waves, and Figure 2b represents the seismic record without ghost waves. It can be clearly seen from the seismic records that the sea surface ghost reflection (ghost wave) is superimposed on the primary wave. The purpose of the wave equation ghost wave suppression is to obtain the ghost wave-free seismic record shown in Figure 2b from the ghost wave-containing seismic record shown in Figure 2a.

2) High-precision ghost wave suppression based on formula (4)

As shown in Fig. 3a and Fig. 3b, the method of the present invention and the conventional method (based on equation (2)) are respectively used to perform ghost wave suppression on the seismic record containing ghost waves shown in Fig. 2a, wherein the ghost wave suppression result of the conventional method is As shown in Figure 3a, the ghost wave suppression result of the method of the present invention is shown in Figure 3b. By comparing with Fig. 2b, it can be seen that the ghost wave suppression method of the wave equation proposed by the present invention can perform high-precision ghost wave suppression, and the recorded records after suppression are almost identical to the ideal results. It can be seen from the profile that the conventional algorithm can recover the primary wave, but the output result has more noise, which is caused by the introduction of the regularization term.

The results of the numerical simulation experiments demonstrate the effectiveness and superiority of the wave equation ghost wave suppression technique based on Taylor expansion in this application.

The above-mentioned embodiments are only used to illustrate the present invention, and the structure, connection method and manufacturing process of each component can be changed to some extent. Any equivalent transformation and improvement based on the technical solution of the present invention should not be used. Excluded from the scope of protection of the present invention.

Claims (7)

1. a wave equation ghost wave suppression method based on Taylor expansion, is characterized in that comprising the following steps: (1) Express the seismic data containing ghost waves as the superposition of primary waves and ghost waves, and obtain the primary wave expression represented by the ghost wave operator G; (2) Based on the Taylor expansion of the ghost wave operator G, the primary wave expression represented by the wave field extension matrix F is obtained; (3) Calculate the wave field continuation matrix F, and substitute it into the primary wave expression in step (2) to obtain the primary wave after ghost wave suppression. 2. a kind of wave equation ghost wave suppression method based on Taylor expansion as claimed in claim 1, is characterized in that: in described step (2), described primary wave expression represented by wave field extension matrix F is as : p=(1+F+F ² +...)d, Among them, p is the primary wave, and d is the seismic data with ghost waves. 3. a kind of wave equation ghost wave suppression method based on Taylor expansion as claimed in claim 1, is characterized in that: in described step (3), calculate the method for wave field extension matrix F, comprises the following steps: (3.1) Extend the recorded wave field p(r′) of the streamer surface Γ ₀ up to the sea level Γ ₁ along the Z axis, and calculate the wave field p(r) at r on the sea level Γ ₁ . The field p(r) is expressed in the integral form of the recorded wave field p(r') and its normal derivative; (3.2) Assuming that the medium between the streamer surface and the sea level is uniform, the normal derivative of the recorded wavefield p(r′) at the streamer surface Γ ₀ can be approximately obtained; (3.3) Express the Green's function in three dimensions, and obtain the three-dimensional expression of the Green's function; (3.4) Substitute the normal derivative of the recorded wavefield p(r′) of the streamer surface Γ ₀ obtained in steps (3.2) and (3.3) and the three-dimensional expression of Green’s function into step (3.1), and get the integral equation of the wavefield p(r) at r at sea level with normal derivatives; (3.5) Discretize the integral equation of the wave field p(r) at r at the sea level in step (3.4), and then the wave field extension matrix F can be obtained. 4. a kind of wave equation ghost wave suppression method based on Taylor expansion as claimed in claim 3, is characterized in that: in described step (3.1), the wave field p (r) at r place on sea level Γ ₁ is: where G(r,r') is Green's function, Represents the normal derivative. 5. a kind of wave equation ghost wave suppression method based on Taylor expansion as claimed in claim 3, is characterized in that: in described step (3.2), the normal derivative of recording wave field p (r') is: where θ is the angle between the boundary normal direction and the propagation path rr′, is the background wavenumber, c ₀ is the background velocity, and ω is the angular frequency. 6. a kind of wave equation ghost wave suppression method based on Taylor expansion as claimed in claim 3, is characterized in that: in described step (3.3), the three-dimensional expression of Green's function is: 7. a kind of wave equation ghost wave suppression method based on Taylor expansion as claimed in claim 3 is characterized in that: in described step (3.4), the wave field p ( The integral equation of r) is: Among them, r=|rr′|,