We compare different regularization techniques of the steepest-descent directions appearing in waveform inversion using a backpropagation technique. In the waveform inversion using the steepest-descent method, we can have better convergence to a true velocity model by regularizing the steepest-descent directions properly. The regularization can be done by using the diagonal of pseudo Hessian matrix instead of using the approximate Hessian matrix that appears in Gauss-Newton method but is too expensive to calculate. We can apply the regularization to inversion algorithms in two different ways. One is to regularize the steepest-descent direction at each frequency independently. The other is to regularize the steepest-descent direction summed over entire frequency band. The former plays a role of equally distributing a weight to the steepest-descent direction at each frequency. For the conventional waveform inversion, the former gives better results than the latter. We also applied the two regularization methods for the logarithmic waveform inversion, which gives better results than the conventional waveform inversion for the original Marmousi data. Numerical examples showed that the logarithmic waveform inversion is not sensitive to the regularization, because the logarithm makes the steepest-descent direction at each frequency commensurate with each other.


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