# An Analytic Solution to a Coupled System of Equations for Modeling Photoacoustic Trace Gas Sensors and a Full Waveform Inversion Approach to Microseismic Source Estimation

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## Abstract

We discuss two different problems in applied mathematics. The first is related to the modeling of a certain class of trace gas sensors. The second is the geophysical inverse problem of estimating the source of a microseismic event.

Quartz-enhanced photoacoustic spectroscopy and resonant optothermoacoustic detection are two promising techniques used in trace gas sensing. Both methods use a quartz tuning fork and modulated laser source to detect trace gases. We discuss a coupled system of equations for the pressure, temperature, and velocity of a fluid that accounts for both thermal effects and viscous damping. We derive an analytical solution to a pressure-temperature subsystem of the Morse-Ingard equations in the special case of cylindrical symmetry. We solve for the pressure and temperature in an infinitely long cylindrical fluid domain with a source function given by a constant-width Gaussian beam that is aligned with the axis of the cylinder. In addition, we surround this cylinder with an infinitely long annular solid domain, and we couple the pressure and temperature in the fluid domain to the temperature in the solid. We show that the temperature in the solid near the fluid-solid interface can be an order of magnitude larger than that computed using a simpler model in which the temperature is governed by the heat equation. We also verify that the temperature solution of the coupled system exhibits a thermal boundary layer. These results suggest that for computational modeling of resonant optothermoacoustic detection sensors, the temperature in the fluid should be computed by solving the Morse-Ingard equations rather than the heat equation.

In the second problem, we use full waveform inversion to estimate the full spatial and temporal description of a microseismic source which includes not only the location and origin time of the source but also the waveform itself. Beginning with the simplifying assumption of two-dimensional acoustic wave propagation, we compute the gradient via the adjoint-state method for both the spatial radiation pattern and the temporal waveform of the source. This approach identifies multiple sources, handles extremely low signal-to-noise ratio data, and produces accurate results in the absence of a good initial estimate.

Encouraged by the promising results of the two dimensional acoustic case, we apply the approach to the case where wave propagation is modeled via the velocity-stress formulation of the elastic wave equation in three dimensions. A change of variables applied to the velocity-stress formulation ensures that the system is self-adjoint. Thus full waveform inversion can be effectively tailored to use this transformed velocity-stress system to estimate microseismic events with limited modifications to the forward wave solver. The inversion produces either a spatial source description in the form of a volume in which each point in the domain indicates the magnitude of stress perturbation and therefore deformation of the medium or a temporal source description which gives the time evolution of the deformation. The inversion does not require any a priori assumptions about the form of the source and does not require a good starting guess for accurate source recovery.