Seismic tomography, adjoint methods, time reversal, and banana-doughnut kernels

S U M M A R Y We draw connections between seismic tomography, adjoint methods popular in climate and ocean dynamics, time-reversal imaging and finite-frequency ‘banana-doughnut’ kernels. We demonstrate that Fréchet derivatives for tomographic and (finite) source inversions may be obtained based upon just two numerical simulations for each earthquake: one calculation for the current model and a second, ‘adjoint’, calculation that uses time-reversed signals at the receivers as simultaneous, fictitious sources. For a given model, m, we consider objective functions χ (m) that minimize differences between waveforms, traveltimes or amplitudes. For tomographic inversions we show that the Fréchet derivatives of such objective functions may be written in the generic form δχ = ∫V Km(x)δ ln m(x) d3x, where δ ln m = δm/m denotes the relative model perturbation. The volumetric kernel Km is defined throughout the model volume V and is determined by time-integrated products between spatial and temporal derivatives of the regular displacement field s and the adjoint displacement field s†; the latter is obtained by using time-reversed signals at the receivers as simultaneous sources. In waveform tomography the time-reversed signal consists of differences between the data and the synthetics, in traveltime tomography it is determined by synthetic velocities, and in amplitude tomography it is controlled by synthetic displacements. For each event, the construction of the kernel Km requires one forward calculation for the regular field s and one adjoint calculation involving the fields s and s†. In the case of traveltime tomography, the kernels Km are weighted combinations of banana-doughnut kernels. For multiple events the kernels are simply summed. The final summed kernel is controlled by the distribution of events and stations. Fréchet derivatives of the objective function with respect to topographic variations δh on internal discontinuities may be expressed in terms of 2-D kernels Kh and Kh in the form δχ = ∫ Kh(x)δh(x) d2x + ∫ FS Kh(x) · ∇ δh(x) d2x, where denotes a solid-solid or fluid-solid boundary and FS a fluid–solid boundary, and ∇ denotes the surface gradient. We illustrate how amplitude anomalies may be inverted for lateral variations in elastic and anelastic structure. In the context of a finite-source inversion, the model vector consists of the time-dependent elements of the moment-density tensor m(x, t). We demonstrate that the Fréchet derivatives of the objective function χ may in this case be written in the form δχ = ∫ t 0 ∫ †(x, T − t) : δm(x, t) d2x dt , where † denotes the adjoint strain tensor on the finite-fault plane . In the case of a point source this result reduces further to the calculation of the time-dependent adjoint strain tensor † at the location of the point source, an approach reminiscent of an acoustic time-reversal mirror. The theory is illustrated for both tomographic and source inversions using a 2-D spectral-element method.