RadonPy

Parabolic Radon Transform

Time domain Parabolic Radon transform via the Conjugate Gradient (CG) method with implicit forward and adjoint Radon operators. The Radon domain coefficients $m(\tau,p)$ are found by minimizing the following quadratic cost function

$$J =| Lm - d |_2^2 + \mu | m |_2^2,.$$ ​ The seismic gather is given by $d(t,x)$, and $L$ provides the Radon forward operator in implicit form. The scalar $\mu$ is the trade-off parameter of the problem. Notice that CG requires the operators $L$ (Forward) and $L'$ (adjoint or transpose operator). Check radon_forward and radon_adjoint in radon_lib.py. These two operators are functions that pass the dot product test.

The implicit form of the operators is given by

$$ d = L m \quad \equiv \quad d(t,x) = \sum_q m(t-q x^2,x) $$

$$ m' = L' d \quad \equiv \quad m'(\tau,q) = \sum_x d(\tau+q x^2,x) $$

The parameter $q x^2$ is replaced by $q (x/x_{max})^2$. Then, the variable $q$ is residual moveout at far offset and it is measured in seconds.

Running the program: python main.py. You will first need to install the following libraries numba, numpy and matplotlib.

You can modify the code to also compute Linear and Hyperbolic Radon Transforms. In those cases, the integration path $\tau + q x^2$ is replaced by $\tau + p x$ or $\sqrt{\tau^2 + (x/v)^2}$. Then we can estimate either $m(\tau,p)$ (Linear Radon Transform) or $m(\tau,v)$ (Hyperbolic Radon Transform also called Velocity Stack).

Reference: D Trad, T Ulrych, M Sacchi, 2003, Latest views of the sparse Radon transform: Geophysics 68 (1), 386-399.