Method for computing an exact impulse response of a plane acoustic reflector at zero offset due to a point acoustic source

• US 7,940,601 B2
• Filed: 02/26/2007
• Issued: 05/10/2011
• Est. Priority Date: 12/07/2006
• Status: Active Grant

• Current Assignee: Council of Scientific & Industrial Research
• Original Assignee: Council of Scientific & Industrial Research
• Inventors: Ghosh, Santi Kumar
• Primary Examiner(s): Keith; Jack
• Assistant Examiner(s): BREIER, KRYSTINE E
• Application Number:
  US11/711,469
• Publication Number:
  US 20080137477A1
• Time in Patent Office: 1,534 Days
• Field of Search: 367/38, 367/50, 367/53, 702/14, 702/16, 702/17
• US Class Current: 367/38
• CPC Class Codes: G01V 1/30 Analysis G01V1/50 takes precedence

Abstract

Originating from a novel and an exact algebraic formula for the impulse response of a plane acoustic reflector at zero offset due to a point acoustic source the present invention provides a method for computing an exact impulse response of a plane acoustic reflector at zero offset due to a point acoustic source; and originating from the method, methods for testing and validating algorithms for numerical modeling of seismic reflection, seismic migration and seismic inversion; a method for testing the efficacy of ray-theoretical solution for a given source-reflector configuration; another method for computing zero-offset reflection response of a circular reflector at its central axis; yet another method for validating an interpretation of a reflector as a planar structure; still yet another method for estimating the seismic source-time function when the zero-offset reflection response of a plane reflector is given. Although the algebraic formula and the methods originating from it are, in a strict sense, valid for an acoustic earth and an acoustic source, these would also be of immense utility in the seismic industry where the earth is successfully approximated as an acoustic medium and a seismic source as an acoustic source.

Claims

1. A method for determining impulse response [u_s(t)] and reflection response [r(t)] of a plane acoustic reflector at zero offset, due to a point acoustic source by varying parameters consisting of reference wave speed about which speed perturbation takes place (c), change in wave speed across a reflector (Δ

c), plane-wave-reflection coefficient of a reflector for normal incidence (R), depth to the reflector from the source/receiver point (h) and source time function s(t), employing Fourier method, wherein perturbation parameter is substituted in terms of reference wave speed and change in wave speed at an infinite plane interface across which the wave speed changes from the reference wave speed to a sum of the reference wave speed and the change in wave speed, followed by removing the restrictions of Born approximation, the method being useful for seismological applications, the method comprising the steps of;

i) inputting distance of the reflector (h), reference wave speed (c), change in wave speed (Δ

c), source time function s(t) and truncation time of the impulse response of a reflector (T) in equation A;

u_s(t)=R/8π

h[δ

(t−

{hacek over (c)})−

2{hacek over (c)}²H(t−

{hacek over (c)})/t³] 

 

(Equation A);

ii) obtaining impulse response u_S (t) of a plane acoustic reflector at zero offset due to a point acoustic source according to

u_s(t)=R/8π

h[δ

(t−

{hacek over (c)})−

2{hacek over (c)}²H(t−

{hacek over (c)})/t³],

wherein;

R=Δ

c/(2c+Δ

c);

t=observation time that begins at the onset of the source;

δ

(t)=Dirac delta function;

c=Reference wave speed;

Δ

c=change in wave speed;

h=shortest distance between the source and the reflector;

{hacek over (c)}=2h/c;

H=Heaviside function, denoted by H (t−

{hacek over (c)}) and is defined as

H(t−

{hacek over (c)})=1,t>

{hacek over (c)}=0,t<

{hacek over (c)}

iii) obtaining reflection response [r(t)] of a plane acoustic reflector at zero offset due to a point acoustic source described by source time function s (t), from the expression as obtained in Equation B;

r(t)=R/8π

h{s(t)−

2{hacek over (c)}²s(t)*[H(t−

{hacek over (c)})/t³]} 

 

(Equation B);

wherein;

R=Δ

c/(2c+Δ

c);

{hacek over (c)}=2h/c; and

* denotes convolution;

H=Heaviside function;

t=observation time that begins at the onset of the source;

s(t)=source time function;

c=Reference wave speed;

Δ

c=change in wave speed; and

h=shortest distance between the source and the reflector, wherein a visualization of the transformed data is output.

2. A method for computing the impulse response [u_s(t)] and reflection response [r(t)] of a plane acoustic reflector as claimed in claim 1 wherein, the zero-offset reflection response of circular reflector is computed considering observations at its central axis and up to a limited span of time.

3. A method for computing the impulse response [u_s(t)] and reflection response [r(t)] of a plane acoustic reflector as claimed in claim 1 wherein, the accurate zero-offset seismic section is computed due to a dipping or horizontal planar reflector to validate seismic migration and inversion algorithms.

4. A method for computing the impulse response [u_s(t)] and reflection response [r(t)] of a plane acoustic reflector as claimed in claim 1, wherein the said method validates an interpretation of a reflector as a planar structure.

5. A method for computing the impulse response [u_s(t)] and reflection response [r(t)] of a plane acoustic reflector as claimed in claim 1, wherein the said method computes the seismic source time function s(t) by inputting the zero-offset reflection response of a plane reflector.