1. I.1 Diffraction Stack Modeling I.1 Diffraction Stack Modeling 1. Forward modeling operator L 1. Forward modeling operator L d (x) = (x |x’) m(x’) dx’ ModelSpace G modeldata Integral IntegralEquation:

2. 2-way time Forward Modeling

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

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

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

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

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

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

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

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

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

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

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

14. SUMMARY W (t- )  xx’ d(x,t) =  x’ x’ 1. Exploding Reflector Modeling = Diffraction Stack Modeling A(x,x’) R(x’) reflectivity Geom. spreading Source wavelet Data Datavariables Sum over Sum overreflectors 2. High Frequency Approximation (i.e c(x) variations > 3* ) 3. Approximates Kinematics of ZO Sections, but not Dynamics d (x)x |x’)m(x’) d (x) = (x |x’) m(x’) dx’ ModelSpace G modeldata Integral IntegralEquation:

15. MATLAB Exercise: Forward Modeling W (t- )  xx’ d(x,t|x’,0) =  x’ x’ R(x’) 1. To account for the source wavelet W(t), we convolve data with W(t) (recall  (t-  )*W(t)= W(  ) ) convolve data with W(t) (recall  (t-  )*W(t)= W(  ) ) so that modeling equation becomes (neglect A) 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? second time derivative?

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