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Misfit functionals involving strain and gradients

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

Let's start with a simple L2L_2-misfit based on a measurement of such a derived field

χ(u):=12q(u)dobs2,\chi(\mathbf{u}) := \frac{1}{2}\| \mathbf{q}(\mathbf{u}) - \mathbf{d}_{\text{obs}} \|^2,

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

(uχ,δu)=(qχ,uq(u)δu)=(qχ,D(δu))=((q(u)dobs),D(δu))=(D(q(u)dobs),δu).\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 D\mathcal{D}^\dagger is impractical in many situations, in particular, when D\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 D\mathcal{D} is applied to the test function instead. For some specific choices of D\mathcal{D}, it will be more convenient to isolate the strain or gradient operator within D\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 v\mathbf{v}, a 6-component moment tensor M\mathbf{M} or a gradient source G\mathbf{G} consisting of 9 individual components. Then the weak form of the source terms is given by

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


Example 1 - Strains and gradients

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 Du=ε(u)\mathcal{D}\mathbf{u} = \varepsilon(\mathbf{u}) or Du=u\mathcal{D}\mathbf{u} = \nabla \mathbf{u}, respectively. In the case of the L2L_2 misfit, we thus obtain adjoint sources of the form

q(u)dobs,\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.

Example 2 - Axial strain for distributed acoustic sensing

Applications of distributed acoustic sensing consider single-component measurement of the strain tensor in axial direction. Given a direction vector e\mathbf{e}, this can be written as

Du=eTε(u)e\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

(uχ,δu)=((q(u)dobs),eTε(δu)e)=((q(u)dobs)eeT,ε(δu)).\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 eeT\mathbf{e} \mathbf{e}^T and scaled by the scalar residual q(u)dobs\mathbf{q}(\mathbf{u}) - \mathbf{d}_{\text{obs}}.

Example 3 - Divergence measurement

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

Du=tr(ε(u)).\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

(uχ,δu)=((q(u)dobs),tr(ε(δu)))=((q(u)dobs)I3×3,ε(δu)),\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 I3×3\mathcal{I}_{3\times 3} is the identity matrix. Hence, the adjoint source results in an explosive moment-tensor point source.

Extensions to other misfit functionals


(uχ,δu)=(qχ,D(δu)),\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 L2L_2 norm is straightforward. In fact, the equation above readily provides the recipe for general misfit functional acting on q\mathbf{q}, for instance, cross-correlation time-shift, time-frequency phase misfits, or others. All that is required is injecting the derivative of χ\chi with respect to the derived quantity q\mathbf{q}, which is no different than replacing u\mathbf{u} with q\mathbf{q} in a conventional misfit based on displacements.