Formulas for computing full and quasi-Newton steps in seismic full waveform inversion, specifically designed for pre-critical reflection experiments, are derived. The formulas are partly continuous and partly discrete. The discrete aspect of the problem is connected to the multiplicity of parameters, whereas the continuous aspect is connected to the distribution in space of the unknowns. We analyze the opportunities this formulation provides for forming quasi-Newton steps. There are two different kinds we can invoke, which we refer to as parameter-type and space-type. The parameter-type approximation appears to retain the ability of reflection FWI to correctly update one parameter when several are responsible for the amplitude content of the data. A third approximation can be created by invoking both simultaneously. All three are simple to implement, since they each amount to the setting of different, and well-defined, off-diagonal Hessian operator elements to zero.
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