1 / 19

Tutorial on Green’s Functions, Forward Modeling, Reciprocity Theorems, and Interferometry

Tutorial on Green’s Functions, Forward Modeling, Reciprocity Theorems, and Interferometry. P( x , T ). 5. T. 4. What is Huygen’s Principle?. Answer: Every pt. on a wavefront is a secondary src. pt. Common tangent kinematically defines next wavefront. T. 5. g( x ,t| x , T ). s. 5.

Download Presentation

Tutorial on Green’s Functions, Forward Modeling, Reciprocity Theorems, and Interferometry

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. Tutorial on Green’s Functions, Forward Modeling, Reciprocity Theorems, and Interferometry ..

  2. P(x,T ) 5 T 4 What is Huygen’s Principle? Answer: Every pt. on a wavefront is a secondary src. pt. Common tangent kinematically defines next wavefront. T 5

  3. g(x,t|x ,T ) s 5 s p(x,T ) 5 What is Huygen-Fresnel Principle? Answer: Every pt. on a wavefront is a secondary src. pt. Next wavefront is sum of weighted Green’s functions. T 5

  4. g(x,t|x ,T ) s 5 p(x,T ) 5 x t s s What is Huygen-Fresnel Principle? Answer: Every pt. on a wavefront is a secondary src. pt. Next wavefront is sum of weighted Green’s functions. T 5

  5. f(t) g(x,t|x ,t ) p(x ,t ) p(x,t) = s s s s Direct wave + g(x,t|x , 0) o p(x,t) p(x ,t ) s s x t s s Huygen’s Forward Modeling (Diffraction Stack Modeling) Scattered waves x o

  6. Convolution f(t) p(x,t ) p(x,t ) s s Direct wave + g(x,t|x , 0) o p(x,t) x t s s Huygen’s Forward Modeling (Diffraction Stack Modeling) Stationarity says g(x,t-t |x ,0 ) p(x,t) = s s Scattered waves x o

  7. Convolution f(t) p(x,t ) p(x,t ) s s Direct wave + g(x,t|x , 0) o p(x,t) x s Huygen’s Forward Modeling (Diffraction Stack Modeling) g(x, t |x ,0 ) p(x,t) = s s Scattered waves Problem: this formulation predicts waves going in both directions x o

  8. Problem with Huygen’s Principle? g(x,t|x ,T ) s 5 p(x,T ) 5 x t s s T 5

  9. Problem with Huygen’s Principle? ZOOM g(x,t|x ,T ) s 5 p(x,T ) 5 x t s s Solution: Add a Dipole Term T 5

  10. Solution: Add a Dipole Term ZOOM + -

  11. ZOOM p(x’,T ) p(x’,T ) + 5 5 - g(x,t|x’ ,T ) -g(x,t|x’,T ) s s 5 5 + dn - Solution: Add a Dipole Term = pdg/dn +

  12. F(w) p(x,t ) s Direct wave + G(x, |x) o P(x ) - G(x|x) Integration path d d dn dn Green’s Theorem Forward Modeling in w } 2 d x G(x|x) P(x ) P(x ) = } x o

  13. convolution f(t) p(x,t ) s Direct wave + g(x,t|x , 0) o x o d d dn dn Green’s Theorem Forward Modeling in t - } 2 p(x,t ) - d x g(x,t|x,0) p(x ,t) g(x,t|x,0) P(x,t ) = }

  14. x x p(x,t ) p(x,t ) s s g(x, t |x ,0 ) s s x x s s Primary Primary -> Multiple Ray Diagram Interpretation x Convolution between 2 short traces -> longer trace Redatuming short offset src-rec -> long offset src-rec

  15. 1. Huygen-Fresnel Modeling (Diffraction Stack) Scattered waves F(w) g(x,t|x ,t ) p(x ,t ) p(x,t) = s s s s } 2 d x G(x|x) P(x ) P(x ) = Direct wave Direct wave } + g(x,t|x , 0) + G(x|x ,) o o 2. Green’s Theorem P(x ) - G(x|x) d d 3.Primary Primary -> Multiple dn dn x t s s Summary

  16. 1. Huygen-Fresnel Modeling (Diffraction Stack) Scattered waves F(w) g(x,t|x ,t )* p(x ,t ) p(x,t) = s s s s } 2 d x G(x|x) P(x ) P(x ) = Direct wave Direct wave } + g(x,t|x , 0) + G(x|x ,) o o 2. Green’s Theorem * * 2iIm[ ] P(x ) - G(x|x) 3.Primary Multiple -> Primary d d dn dn x t s s Summary*

  17. A B A B * ? Exercise 1. Diffraction Stack Modeling Lab 2. Use Ray Diagrams to show how a 3rd -order multiple can be predicted from a primary and a 2nd-order multiple

  18. Define Ricker and model variables Modeling MATLAB Exercise % twod.m computes forward model of 2-layer model and % % --^----x-----------A------------B------ % | % d v1 % | % -v------------------------------------ % % v2 % % Define 2-layer model: % d - input - depth of layer % (v1,v2) - input- velocities top/bottom layers % (dx, L) - input- src-geo (interval, recording aperture) %(np,fr,dt) –input -(# pts wavelet, peak frequency, time interval) clear all % Model input variables v1=1.0;v2=2.0;d=.2725;L=1;dx=.005;x=[0:dx:L];nx=round(length(x)); np=100;dt=.006;fr=15; % Define Ricker Wavelet [rick]=ricker(np,dt,fr); % Compute synthetic seismograms 2-Layer model figure(1);[seismo,ntime]=forward(v1,v2,dx,nx,d,dt,np,rick,x,fr); % Correlate G(A|x) and G(B|x) and sum over x % to get G(A|B) figure(2);A=round(nx/3);B=round(nx/2); [GABT,GAB,peak]=corrsum(ntime,seismo,A,B,rick,nx);

  19. Loop over srcs Loop over recs Compute traveltimes of primaries & multiples Seismogram calculation MATLAB Exercise function [seismo,ntime]=forward(v1,v2,dx,nx,d,dt,np,rick,x,fr) r=(v2-v1)/(v2+v1);r1=r*r;r2=r1*r; for ixs=1:nx % Loop over sources xs=(ixs-1)*dx; t=round(sqrt((xs-x).^2+(2*d)^2)/v1/dt)+1; %Primary Time t1=round(sqrt((xs-x).^2+(4*d)^2)/v1/dt)+1; %1st Multiple Time t2=round(sqrt((xs-x).^2+(6*d)^2)/v1/dt)+1; %2nd Multiple Time s=zeros(nx,ntime); for i=1:nx; % Loop over recievers s(i,round(t(i)))=r/t(i); %Primary s(i,round(t1(i)))=-r1/t1(i); % 1st-order Multiple s(i,round(t2(i)))=r2/t2(i); % 2nd-order Multiple ss=conv(s(i,:),rick); % Convolve Ricker & Trace s(i,:)=ss(1:ntime); % Synthetic Seismograms seismo(ixs,i,1:ntime)=ss(1:ntime); end c=seismo(ixs,:,:);c=reshape(c,nx,ntime); imagesc([1:nx]*dx,[1:ntime]*dt,c');xlabel('X(km)');ylabel('Time (s)') title('Shot Gather for Two-Layer Model') pause(.1) end Display CSG

More Related