1 / 16

I.1 Diffraction Stack Modeling

d (x) = (x |x’) m(x’) dx’. G. Integral Equation:. ò. model. data. Model Space. I.1 Diffraction Stack Modeling. 1. Forward modeling operator L. 2-way time. Forward Modeling. 2-way time. Forward Modeling: Sum of Weighted Hyperbolas. iw|x- x’ |/c . Phase. e .

thaddeus
Download Presentation

I.1 Diffraction Stack Modeling

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. d(x) = (x |x’) m(x’) dx’ G Integral Equation: ò model data Model Space I.1 Diffraction Stack Modeling 1. Forward modeling operator L

  2. 2-way time Forward Modeling

  3. 2-way time Forward Modeling: Sum of Weighted Hyperbolas

  4. iw|x-x’|/c Phase e |x-x’| Geom. Spread |x-x’| x x’ GREEN’s FUNCTION G(x|x’) =

  5.   iw|x-x’|/c Phase xx’ xx’ e -1 + O( ) |x-x’| Geom. Spread x x’ ASYMPTOTIC GREEN’s FUNCTION G(x|x’) = A(x,x’)

  6.  i xx’ R(x’)   x’ x’ reflectivity ASYMPTOTIC GREEN’s FUNCTION e G(x|x’) = A(x,x’)

  7. 1-way time Diffraction Stack Modeling = ZO Modeling

  8. 2-way time Diffraction Stack Modeling = ZO Modeling Dipping Reflector

  9. 1-way time Diffraction Stack Modeling = ZO Modeling If c for DS is ½ that for ZO Modeling

  10.    i i xx’ xx’ R(x’)   x’ x’ reflectivity Fourier Transform: F  e  (t- ) xx’  (t- )  ~ F xx’ d(x) R(x’) A(x,x’) ASYMPTOTIC GREEN’s FUNCTION ~ e d(x) = A(x,x’)

  11. QUICK REVIEW FOURIER TRANSFORM  ( - t) xx’ i   e  d  t  Cos( 4 t ) + +    (t- ) Cos( 2 t ) xx’ +  Cos( 3 t )  cancellation cancellation Cos( t ) constructive reinforcement @ t=0  (t)

  12. x’ time Sum over reflectivity Spray energy along hyperbolas  (t-  ) xx’  (t- )  xx’ Forward Modeling Operator d(x,t) = R(x’) A(x,x’)

  13. W  x’ time CANCELLATION REINFORCE  (t- )  xx’ Forward Modeling Operator d(x,t) = R(x’) A(x,x’)

  14. SUMMARY  x’ d(x) = (x |x’)m(x’) dx’ G Integral Equation: reflectivity ò Source wavelet Geom. spreading model data Model Space W (t- )  xx’ 1. Exploding Reflector Modeling = Diffraction Stack Modeling R(x’) d(x,t) = Sum over reflectors Data variables A(x,x’) 2. High Frequency Approximation (i.e c(x) variations > 3* ) 3. Approximates Kinematics of ZO Sections, but not Dynamics

  15. x’ d(x,t|x’,0) = R(x’) W (t- )  xx’ MATLAB Exercise: Forward Modeling 1. To account for the source wavelet W(t), we convolve data with W(t)(recall (t-  )*W(t)= W() ) so that modeling equation becomes (neglect A) A). Execute MATLAB program forw.m to generate synthetic data for a point scatterer and a 30 Hz wavelet. B). Execute MATLAB program forwl.m to generate synthetic data for a dipping layer model C). Execute MATLAB program forw.m to generate synthetic data for a syncline model. Note diffractions and multiple arrivals. Adjust for new models. Why the second time derivative?

  16. x’ d(x,t) = R(x’) Loop over traces Loop over x in model Loop over z in model Traveltime { W (t- )  R(x’) xx’ MATLAB Exercise: Forward Modeling for ixtrace=1:ntrace; for ixs=istart:iend; for izs=1:nz; r = sqrt((ixtrace*dx-ixs*dx)^2+(izs*dx)^2); time = 1 + round( r/c/dt ); data(ixtrace,time) = migi(ixs,izs)/r + data(ixtrace,time); end; end; data1(ixtrace,:)=conv2(data(ixtrace,:),rick); end; * Src Wave

More Related