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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.

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

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  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

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