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

2. c |r| = x + y + z 222 c Propagation time = What is a Green’s Function? t=1 t=2 t=3 t=4 Answer: Point Src. Response of Medium r Listening time Geometrical Spreading { (t-|r|/c) = 1 t=|r|/c ottherwise 0 |r| Answer: ImpulsePoint Src. Response of Medium Propagation time g(x,t |x,t )= s s o Listening & source coords.

3. What is a Green’s Function? Answer: Point Src. Response of Medium Answer: Impulse-Point Src. Response of Medium { 1 t =|r|/c 1 t-t =|r|/c ottherwise 0 |r| g(x,t|x,0 ) = s (t -|r|/c) = s G(x|x ) = s |r| i  r/c e Fourier  2 + k 2 []G(A|x) =- (x-A); 4

4. What is an Acausal Green’s Function? Answer: Point Src. Response of Medium Answer: Acausal Impulse-Point Src. Response of Medium Fourier  2 + k 2 []G*(A|x) =- (x-A); 4 { 1 t =-|r|/c 1 t-t =-|r|/c ottherwise 0 |r| g(x,t|x,0 ) = s (t+|r|/c) = s a G(x|x ) = s |r| -i  r/c e a g(x,-t|x,0 ) = G(x|x )* =

5. c |r| = x + y + z 222 c Propagation time = t=-1 t=-2 t=-3 t=-4 Answer: Point Src. Response of Medium Listening time Geometrical Spreading { (t+|r|/c) = 1 t=-|r|/c ottherwise 0 |r| Answer: Impulse-Point Src. Response of Medium Listening & source coords. What is an Acausal Green’s Function? r

6. What is a Green’s Function? Answer: Point Src. Response of Medium Answer: Impulse-Point Src. Response of Medium { 1 t =|r|/c 1 t-t =|r|/c ottherwise 0 |r| g(x,t|x,0 ) = s (t -|r|/c) = s G(x|x ) = s ~|r| i  r/c e Fourier d dr G(x|x ) = s ~|r| iwr/c e iw/c + O(1/r ) 2 derivative Far field approx k G(x|x ) = s |r| i  r/c e i  /c + O(1/r ) 2nnr

7. r 1 2 3 4 (2) Ae i(  -  t) G = G..c 2 2 satisfies except at origin Geometrical speading  kr = (kc ) r /c = Time taken along ray segment  Wave in Heterogeneous Medium Valid at high w and smooth media i i iiii

8. What is Reciprocity? G(x|x ) = s|r| i  r/c e = G(x |x ) s s x x

9. What is Reciprocity? G(x|x ) = s|r| i  r/c e = G(x |x ) s s x x G(x|x ) = = G(x |x ) ss Interchanging src & rec does not change value of r=|x – x|

10. What is Stationarity? sxx g(x,t|x,t ) = ss { 1 t - =|r|/c 1 t - t =|r|/c ottherwise 0 |r| (t – t -|r|/c) = s s g(x,t|x,t ) = ss g(x,t-t |x,0 ) s s s s Observed wave depends on difference between observation and start times, not on absolute start time

11. Summary G(x|x ) = s G(x |x ) s s x x = s x x3. 2. Ae i(  -  t) G(B|x) = Bx 1. G(B|x) = G(B|x)* a Causal & Acausal GF Source vs Sink Heterogeneous Hi-freq. Green’s function Reciprocity

12. Summary G(x|x ) = s G(x |x ) s s x x = s x x3. 2. Ae i(  -  t) G(B|x) = Bx 1. G(B|x) = G(B|x)* a Causal & Acausal GF Source vs Sink Heterogeneous Hi-freq. Green’s function Reciprocity s 4. g(x,t|x,t ) = ss g(x,t-t |x,0 ) s s Stationarity

13. Exercises

14. 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); Define Ricker and model variables Modeling

15. 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 Loop over srcs Compute traveltimes of primaries & multiples Loop over recs recs Seismogramcalculation Display CSG