## spherical earth and dinver

### spherical earth and dinver

Hi Marc,

I'm using dinver to invert phase velocities at longer periods (noise cross-correlation). From about 20 s period, it starts to have a significant effect on phase velocities whether they are computed from a flat or from a spherical earth model.

I was just wondering if it would be straight forwards to include the correction in the forward computation in dinver or if there is any option which I didn't discovered yet.

Thanks of your help!

Andreas

I'm using dinver to invert phase velocities at longer periods (noise cross-correlation). From about 20 s period, it starts to have a significant effect on phase velocities whether they are computed from a flat or from a spherical earth model.

I was just wondering if it would be straight forwards to include the correction in the forward computation in dinver or if there is any option which I didn't discovered yet.

Thanks of your help!

Andreas

In order to take into account the spherical earth, the model has to be sub-sampled in depth and the solution is found numerically. Could be too much effort to include that into the code and the inversion in dinver would finally become too slow.

However, there is also another approach: I will try to correct phase velocities before inverting with dinver (one could find a empirical relation and there is also work done by Bhattacharya, GJR, 1976). To check the correction one can recompute misfits with the Saito program after running dinver.

Andreas

1. Correct phase velocities empirically (don't have to change anything in dinver). Bhattacharyam, 1976 is showing a plot of that relation (Fig 1), but I also tried to find one by myself by simply generating random models (gpparam2model) and comparing the forwards computed velocities from dinver (gpdc) and Saitos program. Problem: Correction might not be that exact

2. Flattening transformation directly on the models: Bhattacharyam, "Computation of Surface waves ...",BSSA, 99(6), 2009 is a good reference here. Also the book of Kennett "Seismic Wave Propagation in Stratified Media" explains this transformation. The problem here: Body waves velocities and densities have to be corrected depending on the depths (or radius). That could be difficult to implement in dinver since simple layered models are not that simple any more.

3. Forward computation of phase velocities: no analytic solution, needs numerical integration, e.g. Saitos program: Takeuchi, H. and Saito, M., "Seismic surface waves", Methods in computational physics,1972. Problem: Computation time

I'm not sure what Herrmann did ...

I guess that discretization remains identical to the current flat implementation. Surface wave dinver module could serve as a bsis for a new plugin (e.g. Spherical Surface Wave). Parameterization is already implemented in a separate library that both plugin can access.

Marc

I compared the Saito (spherical) and dinver forwards computation. Difference in phase velocity is about 1% at 50 and 2% at 100 s. Computation time: Not sure if we can compare it directly since I'm calling saito in a script for each model separately, but it takes about 10 seconds for 100 models. Without all the written output, I expect that we can speed up the process.

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