Novel measurements based on strain, spatial gradients or rotational components are becoming more and more popular, especially in the context of distributed acoustic sensing.
Using those measurements for full-waveform inversion requires the derivation of the corresponding adjoint sources. Fortunately, the variational form of the wave equation, which is readily a key ingredient of the spectral-element method, facilitates constructing those adjoint sources consistently.
For a unified treatment of strains, gradients and related quantities, we consider a derived quantity that applies a linear operator involving spatial derivatives to the displacement field .
Let's start with a simple -misfit based on a measurement of such a derived field
where denotes an observation of the derived field, and thus has the same dimension as . We can formally deduce the directional derivative with respect to for a direction as
The adjoint is impractical in many situations, in particular, when contains spatial derivatives. However, the variational form of the spectral-element method enables the use of the second line in the equations above, where is applied to the test function instead. For some specific choices of , it will be more convenient to isolate the strain or gradient operator within and shift only the remaining terms onto the left-hand side of the inner product. This results in moment-tensor or gradient sources for the adjoint equation. Before giving a few concrete examples, we first recall the definition of a moment-tensor and gradient point sources in variational form.
Consider a vector-valued test function , a 6-component moment tensor or a gradient source consisting of 9 individual components. Then the weak form of the source terms is given by
Using the relations above, point-wise measurements of the full strain tensor or gradient of the displacement field are straightforward to include in adjoint computations. These cases result in or , respectively. In the case of the misfit, we thus obtain adjoint sources of the form
which either carries 6 or 9 components, respectively, and is injected as moment tensor or gradient point source into the adjoint simulation.
Applications of distributed acoustic sensing consider single-component measurement of the strain tensor in axial direction. Given a direction vector , this can be written as
In this case, it is more convenient to isolate the strain operator on the right-hand side
Hence, the adjoint source can be interpreted as a moment-tensor point source, where the 6-components are extracted from the upper triangular part of and scaled by the scalar residual .
The divergence of the displacement field is equal to the trace of the strain tensor, i.e.,
Similar to the previous example, the scalar measurement leads to a moment tensor adjoint point source given by
where is the identity matrix. Hence, the adjoint source results in an explosive moment-tensor point source.
the extension to misfit functionals other than the norm is straightforward. In fact, the equation above readily provides the recipe for general misfit functional acting on , for instance, cross-correlation time-shift, time-frequency phase misfits, or others. All that is required is injecting the derivative of with respect to the derived quantity , which is no different than replacing with in a conventional misfit based on displacements.