Mondaic - Full Waveform Solutions

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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 $\mathbf{q} = \mathbf{q}(\mathbf{u}) = \mathcal{D} \mathbf{u}$ that applies a linear operator $\mathcal{D}$ involving spatial derivatives to the displacement field $\mathbf{u}$ .

Let's start with a simple $L_2$ -misfit based on a measurement of such a derived field

\chi(\mathbf{u}) := \frac{1}{2}\| \mathbf{q}(\mathbf{u}) - \mathbf{d}_{\text{obs}} \|^2,

where $\mathbf{d}_{\text{obs}}$ denotes an observation of the derived field, and thus has the same dimension as $\mathbf{q}$ . We can formally deduce the directional derivative with respect to $\mathbf{u}$ for a direction $\delta \mathbf{u}$ as

\begin{aligned} \left(\frac{\partial}{\partial \mathbf{u}} \chi, \delta \mathbf{u} \right) &= \left(\frac{\partial}{\partial \mathbf{q}} \chi, \frac{\partial }{\partial\mathbf{u}}\mathbf{q}(\mathbf{u})\,\delta \mathbf{u} \right) = \left(\frac{\partial}{\partial \mathbf{q}} \chi, \mathcal{D}(\delta \mathbf{u}) \right)\\ &= \biggl( (\mathbf{q}(\mathbf{u}) - \mathbf{d}_{\text{obs}}), \mathcal{D}(\delta \mathbf{u}) \biggr)\\ &= \biggl( \mathcal{D}^\dagger (\mathbf{q}(\mathbf{u}) - \mathbf{d}_{\text{obs}}), \delta \mathbf{u} \biggr). \end{aligned}

The adjoint $\mathcal{D}^\dagger$ is impractical in many situations, in particular, when $\mathcal{D}$ contains spatial derivatives. However, the variational form of the spectral-element method enables the use of the second line in the equations above, where $\mathcal{D}$ is applied to the test function instead. For some specific choices of $\mathcal{D}$ , it will be more convenient to isolate the strain or gradient operator within $\mathcal{D}$ 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 $\mathbf{v}$ , a 6-component moment tensor $\mathbf{M}$ or a gradient source $\mathbf{G}$ consisting of 9 individual components. Then the weak form of the source terms is given by

\begin{aligned} \left(\mathbf{M}, \varepsilon(\mathbf{v}) \right), \quad\text{and}\quad \left(\mathbf{G}, \nabla\mathbf{v} \right), \end{aligned}

respectively.

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 $\mathcal{D}\mathbf{u} = \varepsilon(\mathbf{u})$ or $\mathcal{D}\mathbf{u} = \nabla \mathbf{u}$ , respectively. In the case of the $L_2$ misfit, we thus obtain adjoint sources of the form

\mathbf{q}(\mathbf{u}) - \mathbf{d}_{\text{obs}},

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 $\mathbf{e}$ , this can be written as

\mathcal{D}\mathbf{u} = \mathbf{e}^T \varepsilon(\mathbf{u}) \, \mathbf{e}

In this case, it is more convenient to isolate the strain operator on the right-hand side

\begin{aligned} \left(\frac{\partial}{\partial \mathbf{u}} \chi, \delta \mathbf{u} \right) &= \biggl( (\mathbf{q}(\mathbf{u}) - \mathbf{d}_{\text{obs}}), \mathbf{e}^T \varepsilon(\delta \mathbf{u}) \, \mathbf{e} \biggr)\\ &= \biggl( (\mathbf{q}(\mathbf{u}) - \mathbf{d}_{\text{obs}}) \mathbf{e} \mathbf{e}^T, \varepsilon(\delta \mathbf{u}) \biggr). \end{aligned}

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 $\mathbf{e} \mathbf{e}^T$ and scaled by the scalar residual $\mathbf{q}(\mathbf{u}) - \mathbf{d}_{\text{obs}}$ .

The divergence of the displacement field is equal to the trace of the strain tensor, i.e.,

\mathcal{D}\mathbf{u} = \text{tr}(\varepsilon(\mathbf{u})).

Similar to the previous example, the scalar measurement leads to a moment tensor adjoint point source given by

\begin{aligned} \left(\frac{\partial}{\partial \mathbf{u}} \chi, \delta \mathbf{u} \right) &= \biggl( (\mathbf{q}(\mathbf{u}) - \mathbf{d}_{\text{obs}}), \text{tr}(\varepsilon(\delta\mathbf{u})) \biggr)\\ &= \biggl( (\mathbf{q}(\mathbf{u}) - \mathbf{d}_{\text{obs}})\mathcal{I}_{3\times 3}, \varepsilon(\delta \mathbf{u}) \biggr), \end{aligned}

where $\mathcal{I}_{3\times 3}$ is the identity matrix. Hence, the adjoint source results in an explosive moment-tensor point source.

Using

\left(\frac{\partial}{\partial \mathbf{u}} \chi, \delta \mathbf{u} \right) = \left(\frac{\partial}{\partial \mathbf{q}} \chi, \mathcal{D}(\delta \mathbf{u}) \right),

the extension to misfit functionals other than the $L_2$ norm is straightforward. In fact, the equation above readily provides the recipe for general misfit functional acting on $\mathbf{q}$ , for instance, cross-correlation time-shift, time-frequency phase misfits, or others. All that is required is injecting the derivative of $\chi$ with respect to the derived quantity $\mathbf{q}$ , which is no different than replacing $\mathbf{u}$ with $\mathbf{q}$ in a conventional misfit based on displacements.