# 2D constant-density acoustic frequency-domain modeling, linearized modeling and imaging

This applications is available in the software release for both SINBAD consortium members and academic community.

The modeling code is based on a 9-point mixed-grid discretization of the 2D Helmholtz operator [1]. It solves the system in parallel over frequencies using direct factorization (Matlab's mldivide). Source injection and receiver sampling is done via cubic interpolation. The Jacobian is derived by linearizing the discretized system and its forward and adjoint action is calculated via the adjoint-state method.

## Dependencies

The modeling code uses the following packages, found in the tools part of the software release.

• SPOT - object oriented framework for matrix-free linear algebra.
• pSPOT - parallel extension of SPOT.

## Running & Parallelism

All the examples can be reproduced by running the scripts found in the software release under applications/Modeling/2DAcousticFreqModeling. Start matlab from that directory or run startup in that directory to add the appropriate paths.

The scripts can be run in serial mode but parallel mode is advised for the modeling and imaging examples. Use matlabpool open with the appropriate configuration and a divisor of 12 workers.

## Functions

The modeling code consists of 3 distinct packages which can be found in the tools part of the software release. The main components are listed below

algorithms/2DFreqModeling

• Helm2D - Construct Helmholtz matrix
• F - modeling operator
• DF - Jacobian
• G - modeling operator using analytic solution for constant and linear velocity profiles
• legendreQ - evaluate Legendre Q function (used for G).

operators/misc

• opLInterp1D - 1D cubic Lagrange interpolation
• opLInter2D - 2D linear Lagrange interpolation
• opExtension - Pads input with zeros or constant values
• opSmooth - 1D smoothing by convolution with triangular kernel
• opSpline1D - 1D cubic spline evaluation
• opSpline2D - 2D cubic spline evaluation
• opDFTR - FFT for real input, outputs only positive frequencies.

functions/misc

• grid2odn, odn2grid - convert grid vectors to [origin, increment, size] triplet and vice versa
• vec, invvec - vectorize multidimensional array and reshape vector into multidimensional array.

## Examples

A few examples are included here

• Some examples of the modeling capabilities are shown in modeling.m.
• Basic tests of the modeling code are presented in testing.m
• Some imaging examples are found in imaging.m

## References

[1] C-H Jo,* C. Shin,* and J.H. Suh, 1996. An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolator Geophysics 61(2), 529-537.