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

Forward Modeling 2-way time

Forward Modeling: Sum of Weighted Hyperbolas 2-way time

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

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

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

Diffraction Stack Modeling = ZO Modeling 1-way time

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

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

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

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

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

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

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

  d(x,t|x’,0) = MATLAB Exercise: Forward Modeling W (t- ) R(x’) 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) W (t- )  xx’ d(x,t|x’,0) =  x’ R(x’) 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?

{   d(x,t) = MATLAB Exercise: Forward Modeling W (t- ) R(x’) x’ Loop over x in model Loop over z in model Loop over traces Traveltime 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; data1(ixtrace,:)=conv2(data(ixtrace,:),rick); R(x’) { * Src Wave