PSTM

Kirchhoff Migration (PSTM) tests

Testing forward and adjoint operators for Prestack Time Migration (PSTM)

The subsurface image is given by $m(z,x)$. The image is an estimator of the subsurface reflectivity. The data are given by $d(t,k)$ where $k$ is the trace number. Receiver and source positions are $xs(k)$ and $xg(k)$ which are deployed at $z=0$.

The code main.jl shows how to use the two operators

1. Demigration $d =W L m$ where $W$ is time convolution and $L$ is the de-migration operator:

d = Operator_Conv(Operator_PSTM(m,false; Param_PSTM...), false; Param_Conv...)

2. Migration $m' = L'W' d$ where $L'$ is the migration operator and $W'$ is the adjoint of time domain convolution equivalent to time domain cross-correlation:

mig = Operator_PSTM(Operator_Conv(d,true; Param_Conv...), true; Param_PSTM...)

3. In Least-squares Migration (LSM) we estimate the reflectivity by solving the following problem

$$m_{ls} = argmin_{m} {J(m)}$$ with cost $$J(m) = | WL m - d|_2^2 + \mu | m|_2^2$$ The cost function is minimized using the Conjugate Gradient method which requires the
operators $WL$ and $L'W'$. In the code, this is given by

m_ls,J = ConjugateGradients(d,[Operator_Conv, Operator_PSTM],[Param_Conv, Param_PSTM]; Niter=20,mu=0.01)

main.jl runs an example where I use the "invere problem crime" to model data and then retrieve the model via CGLS

dot_product_test.jl checks that $L$ and $L'$ pass the dot product test

The results are shown below. First, we show the data which is computed via the demigration operator $WLm$

Now, I show the migrated image which is computed via $L'W'd$

And finally, the least-squares migration after $20$ CG iterations:

I also show the cost $J$ vs iteration number. Clearly, the algorithm must converge because I am solving a linear problem by minimizing a quadratic cost.