Modify the calculation of the steepest descent direction in src/tomography/get_sd_direction.f90

[ZQiwen]

As a discrete version of the formulas (e.g. eq. (16)) in Tromp et al. (2005), should the steepest descent direction be model_dbeta(i,j,k,ispec) = - kernel_beta(i,j,k,ispec) * volumel instead of model_dbeta(i,j,k,ispec) = - kernel_beta(i,j,k,ispec) ? where as the volumel is the volume associated with GLL point of index (i,j,k,ispec)

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[danielpeter]

that is not correct. the equation you point out is to evaluate the misfit.

the model update is governed by the ln(vp), ln(vs), ln(rho) expression only. there is no need for a volumetric consideration of each GLL point. you can see it this way: the model provides material properties at different positions x, i.e., defines m(x). this is independent of how close these points would be located. updating the model properties is thus only an operation on these points, not a volumetric update.

the misfit evaluation you pointed out, i.e., chi(m), would require a volumetric computation of the product between model update and corresponding kernel expressions. this is for example done in the benchmarks EXAMPLES/BENCHMARK_CLAERBOUT_ADJOINT to validate the kernels and compare the misfit evaluation above with the measured data misfit.

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