Full-waveform inversion by source extension: Why it works

An extremely simple single-trace transmission example shows how an extended source formulation of full waveform inversion can produce an optimization problem without spurious local minima ("cycle skipping"). The data consist of a single trace recorded at a given distance from a point source. The velocity or slowness is presumed homogeneous, and the target source wavelet is presumed quasi-impulsive or focused at zero time lag. The source is extended by permitting energy to spread in time, and the spread is controlled by adding a weighted mean square of the extended source wavelet to the data misfit, to produce the extended inversion objective. The objective function and its gradient can be computed explicitly, and it is easily seen that all local minimizers must be within a wavelength of the correct slowness. The derivation shows several important features of all similar extended source algorithms. For example, nested optimization, with the source estimation in the inner optimization (variable projection method), is essential. The choice of the weight operator, controlling the extended source degrees of freedom, is critical: the choice presented here is a differential operator, and that property is crucial for production of an objective immune from cycle-skipping.