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

US 20080137477A1
Filed: 02/26/2007
Published: 06/12/2008
Est. Priority Date: 12/07/2006
Status: Active Grant

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1. A method for computing the impulse response [u_s(t) and reflection response [r(t)] of a plane acoustic reflector at zero offset, due to a point acoustic source described by the source time function s(t), wherein the said method comprising the steps of:

a. providing the equation;

$α (x_{3}) = - 8 c \int_{- \infty}^{\infty} \frac{\partial}{\partial ω} [\frac{u_{s} (ω)}{ω^{2}}] \exp {- 2  (\frac{ω}{c}) x_{3}} ω \partial ω,$ (in the prior art),whereinα

(x₃)=[c/(c+Δ

c)]²−

1;

u_s(ω

)=the Fourier transform of u_s(t), the impulse response;

ω

=angular frequency;

i=the imaginary unit;

x=co-ordinate;

c=reference wavespeed about which speed perturbation takes place;

b. substituting α

(x₃) as given in step (a) wherein, considering an infinite plane interface at x₃=h, across which the wavespeed changes from c to c+Δ

c which constitutes the reflector, equivalent to a special case whereα

(x₃)=α

₀H(x₃−

h), where H indicates Heaviside function,c. changing the variable according to t=2x₃/c, which along with the step (b) renders equation in step (a) to $α (t) = - 16 c π \cdot \frac{1}{2 π} \int_{- \infty}^{\infty} g (ω) \exp (- ω t) \partial ω, wherein$ $g (ω) \equiv \frac{\partial}{\partial ω} [\frac{u_{S} (ω)}{ω^{2}}] \cdot ω .,$ d. deriving from the equation in step (c) that −

α

₀/16cπ

H(t−

τ

)⇄

g(ω

), wherein, ‘

⇄

’

indicates that the members on the left and right sides of the symbol constitute a Fourier pair;

e. using the equations in steps (c) and (d) for obtaining the equation, $\frac{- α_{o}}{16 c π} [\frac{}{ω} + π δ (ω)] \exp (ωτ) = ω \frac{\partial}{\partial ω} [\frac{u_{S} (ω)}{ω^{2}}],$ f. using the equation in step (e) for further deriving the expression;

$\frac{{α}_{0}}{16 c π} [\exp (ωτ) {- \frac{1}{ω^{2}} + \frac{πδ (ω)}{ω}}] = \frac{\partial}{\partial ω} [\frac{u_{S} (ω)}{ω^{2}}]$ g. identifying the left hand side of equation in step (f) as the Fourier transform of (t−

τ

)H(t−

τ

), and deriving the equation $\frac{{α}_{0}}{16 c π} (t - τ) H (t - τ) \leftrightarrow \frac{\partial}{\partial ω} [\frac{u_{S} (ω)}{ω^{2}}],$ h. further deriving from equation as given in step (g), the following expression $\frac{{α}_{0}}{( t) 16 c π} (t - τ) H (t - τ) \leftrightarrow \frac{u_{S} (ω)}{ω^{2}},$ i. converting the equation in step (h) to the following equation, $\frac{- α_{0}}{16 c π} \frac{\partial^{2}}{\partial t^{2}} [\frac{(t - τ) H (t - τ)}{t}] \leftrightarrow u_{S} (ω) .$ j. modifying the equation in step (i) to the following equation as $u_{S} (t) = - \frac{α_{0}}{16 c π} \frac{\partial^{2}}{\partial t^{2}} [\frac{(t - τ) H (t - τ)}{t}],$ k. simplifying equation given in step (j) to $u_{S} (t) = - \frac{α_{0}}{16 c π} [\frac{δ (t - τ)}{τ} - \frac{2 τ H (t - τ)}{t^{3}}],$ wherein u_s(t) is the scattering response due to an impulse,l. removing the constraint of Born approximation inherent in the equation in step (k) by using the arguments based on the asymptotic theory of wave propagation as well as the principle of causality, and further deriving the equation $u_{s} (t) = \frac{R}{8 π h} [δ (t - τ) - \frac{2 τ^{2} H (t - τ)}{t^{3}}] .$ m. finally from the equation given in step (l), obtaining the following equation $r (t) = \frac{R}{8 π h} {s (t) - 2 τ^{2} \cdot s (t) * [\frac{H (t - τ)}{t^{3}}]}$ by using the principle of convolution, wherein,t=observation time that begins at the onset of the sources(t)=source time functionR=Δ

c/(2c+Δ

c);

c=reference wave speedΔ

c=change in wave speedh=shortest distance between the source and the reflectorτ

=2h/c; and

* denotes convolutionH=Heaviside function, denoted by H (t−

τ

) and is defined as $\begin{matrix} H (t - τ) = 1, t > τ \\ = 0, t < τ \end{matrix}$ n. computing impulse response of a plane acoustic reflector at zero offset due to a point acoustic source as claimed in step m wherein, the said 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 the equation claimed in step (l) of claim 1;

ii) obtaining impulse response u_s(t) of a plane acoustic reflector at zero offset due to a point acoustic source according to $u_{S} (t) = \frac{R}{8 π h} [δ (t - τ) - \frac{2 τ^{2} H (t - τ)}{t^{3}}] .$ 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 τ

=2h/c; and

* denotes convolution H=Heaviside function, denoted by H (t−

τ

) and is defined as $\begin{matrix} H (t - τ) = 1, t > τ \\ = 0, t < τ \end{matrix}$ 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 step (m) of claim 1;

$r (t) = \frac{R}{8 π h} {s (t) - 2 τ^{2} \cdot s (t) * [\frac{H (t - τ)}{t^{3}}]}$ wherein, R=Δ

c/(2c+Δ

c);

τ

=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 h=shortest distance between the source and the reflector o. validating a seismic numerical modeling algorithm (known method), wherein the said method comprising;

i) obtaining zero-offset reflection response of a plane acoustic reflector computed by seismic numerical modeling algorithm;

ii) obtaining zero-offset reflection response of a plane acoustic reflector computed by the method claimed in step (n); and

iii) comparing the data obtained in step (i) with the data obtained in step (ii) to validate the seismic numerical model algorithm.

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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.

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7 Claims

1. A method for computing the impulse response [u_s(t) and reflection response [r(t)] of a plane acoustic reflector at zero offset, due to a point acoustic source described by the source time function s(t), wherein the said method comprising the steps of:
- a. providing the equation;
  
  $α (x_{3}) = - 8 c \int_{- \infty}^{\infty} \frac{\partial}{\partial ω} [\frac{u_{s} (ω)}{ω^{2}}] \exp {- 2  (\frac{ω}{c}) x_{3}} ω \partial ω,$ (in the prior art),whereinα
  
  (x₃)=[c/(c+Δ
  
  c)]²−
  
  1;
  
  u_s(ω
  
  )=the Fourier transform of u_s(t), the impulse response;
  
  ω
  
  =angular frequency;
  
  i=the imaginary unit;
  
  x=co-ordinate;
  
  c=reference wavespeed about which speed perturbation takes place;
  
  b. substituting α
  
  (x₃) as given in step (a) wherein, considering an infinite plane interface at x₃=h, across which the wavespeed changes from c to c+Δ
  
  c which constitutes the reflector, equivalent to a special case whereα
  
  (x₃)=α
  
  ₀H(x₃−
  
  h), where H indicates Heaviside function,c. changing the variable according to t=2x₃/c, which along with the step (b) renders equation in step (a) to $α (t) = - 16 c π \cdot \frac{1}{2 π} \int_{- \infty}^{\infty} g (ω) \exp (- ω t) \partial ω, wherein$ $g (ω) \equiv \frac{\partial}{\partial ω} [\frac{u_{S} (ω)}{ω^{2}}] \cdot ω .,$ d. deriving from the equation in step (c) that −
  
  α
  
  ₀/16cπ
  
  H(t−
  
  τ
  
  )⇄
  
  g(ω
  
  ), wherein, ‘
  
  ⇄
  
  ’
  
  indicates that the members on the left and right sides of the symbol constitute a Fourier pair;
  
  e. using the equations in steps (c) and (d) for obtaining the equation, $\frac{- α_{o}}{16 c π} [\frac{}{ω} + π δ (ω)] \exp (ωτ) = ω \frac{\partial}{\partial ω} [\frac{u_{S} (ω)}{ω^{2}}],$ f. using the equation in step (e) for further deriving the expression;
  
  $\frac{{α}_{0}}{16 c π} [\exp (ωτ) {- \frac{1}{ω^{2}} + \frac{πδ (ω)}{ω}}] = \frac{\partial}{\partial ω} [\frac{u_{S} (ω)}{ω^{2}}]$ g. identifying the left hand side of equation in step (f) as the Fourier transform of (t−
  
  τ
  
  )H(t−
  
  τ
  
  ), and deriving the equation $\frac{{α}_{0}}{16 c π} (t - τ) H (t - τ) \leftrightarrow \frac{\partial}{\partial ω} [\frac{u_{S} (ω)}{ω^{2}}],$ h. further deriving from equation as given in step (g), the following expression $\frac{{α}_{0}}{( t) 16 c π} (t - τ) H (t - τ) \leftrightarrow \frac{u_{S} (ω)}{ω^{2}},$ i. converting the equation in step (h) to the following equation, $\frac{- α_{0}}{16 c π} \frac{\partial^{2}}{\partial t^{2}} [\frac{(t - τ) H (t - τ)}{t}] \leftrightarrow u_{S} (ω) .$ j. modifying the equation in step (i) to the following equation as $u_{S} (t) = - \frac{α_{0}}{16 c π} \frac{\partial^{2}}{\partial t^{2}} [\frac{(t - τ) H (t - τ)}{t}],$ k. simplifying equation given in step (j) to $u_{S} (t) = - \frac{α_{0}}{16 c π} [\frac{δ (t - τ)}{τ} - \frac{2 τ H (t - τ)}{t^{3}}],$ wherein u_s(t) is the scattering response due to an impulse,l. removing the constraint of Born approximation inherent in the equation in step (k) by using the arguments based on the asymptotic theory of wave propagation as well as the principle of causality, and further deriving the equation $u_{s} (t) = \frac{R}{8 π h} [δ (t - τ) - \frac{2 τ^{2} H (t - τ)}{t^{3}}] .$ m. finally from the equation given in step (l), obtaining the following equation $r (t) = \frac{R}{8 π h} {s (t) - 2 τ^{2} \cdot s (t) * [\frac{H (t - τ)}{t^{3}}]}$ by using the principle of convolution, wherein,t=observation time that begins at the onset of the sources(t)=source time functionR=Δ
  
  c/(2c+Δ
  
  c);
  
  c=reference wave speedΔ
  
  c=change in wave speedh=shortest distance between the source and the reflectorτ
  
  =2h/c; and
  
  * denotes convolutionH=Heaviside function, denoted by H (t−
  
  τ
  
  ) and is defined as $\begin{matrix} H (t - τ) = 1, t > τ \\ = 0, t < τ \end{matrix}$ n. computing impulse response of a plane acoustic reflector at zero offset due to a point acoustic source as claimed in step m wherein, the said 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 the equation claimed in step (l) of claim 1;
  
  ii) obtaining impulse response u_s(t) of a plane acoustic reflector at zero offset due to a point acoustic source according to $u_{S} (t) = \frac{R}{8 π h} [δ (t - τ) - \frac{2 τ^{2} H (t - τ)}{t^{3}}] .$ 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 τ
  
  =2h/c; and
  
  * denotes convolution H=Heaviside function, denoted by H (t−
  
  τ
  
  ) and is defined as $\begin{matrix} H (t - τ) = 1, t > τ \\ = 0, t < τ \end{matrix}$ 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 step (m) of claim 1;
  
  $r (t) = \frac{R}{8 π h} {s (t) - 2 τ^{2} \cdot s (t) * [\frac{H (t - τ)}{t^{3}}]}$ wherein, R=Δ
  
  c/(2c+Δ
  
  c);
  
  τ
  
  =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 h=shortest distance between the source and the reflector o. validating a seismic numerical modeling algorithm (known method), wherein the said method comprising;
  
  i) obtaining zero-offset reflection response of a plane acoustic reflector computed by seismic numerical modeling algorithm;
  
  ii) obtaining zero-offset reflection response of a plane acoustic reflector computed by the method claimed in step (n); and
  
  iii) comparing the data obtained in step (i) with the data obtained in step (ii) to validate the seismic numerical model algorithm.
- View Dependent Claims (2, 3, 4, 5, 6, 7)
- - 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 [_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.
  - 6. 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, characterized by step (b) wherein, substituting α
    - (x₃) in step (a) and considering an infinite plane interface at X₃=h, across which the wavespeed changes from c to c+Δ
      
      c which constitutes the reflector, equivalent to a special case where α
      
      (x₃)=α
      
      ₀H(x₃−
      
      h).
  - 7. 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, characterized by step (l) wherein, removing the constraint of Born approximation inherent in the equation in step (k) by using the arguments based on the asymptotic theory of wave propagation as well as the principle of causality, and further deriving the equation

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Current Assignee
Council of Scientific & Industrial Research
Original Assignee
Council of Scientific & Industrial Research
Inventors
Ghosh, Santi Kumar

Granted Patent

US 7,940,601 B2
Time in Patent Office

Days
Field of Search
US Class Current

367/38
CPC Class Codes

G01V 1/30 Analysis G01V1/50 takes pre...

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

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

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