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Starting from Kirchhoff-Huygens representation and Duhamel's principle of time-domain wave equations, we propose novel butterfly-compressed Hadamard integrators for self-adjoint wave equations in both time and frequency domain in an inhomogeneous medium. First, we incorporate the leading term of Hadamard's ansatz into the Kirchhoff-Huygens representation to develop a short-time valid propagator. Second, using Fourier transform in time, we derive the corresponding Eulerian short-time propagator in the frequency domain; on top of this propagator, we further develop a time-frequency-time (TFT) method for the Cauchy problem of time-domain wave equations. Third, we further propose a time-frequency-time-frequency (TFTF) method for the corresponding point-source Helmholtz equation, which provides Green's functions of the Helmholtz equation for all angular frequencies within a given frequency band. Fourth, to implement the TFT and TFTF methods efficiently, we introduce butterfly algorithms to compress oscillatory integral kernels at different frequencies. As a result, the proposed methods can construct wave field beyond caustics implicitly and advance spatially overturning waves in time naturally with quasi-optimal computational complexity and memory usage. Furthermore, once constructed the Hadamard integrators can be employed to solve both time-domain wave equations with various initial conditions and frequency-domain wave equations with different point sources. Numerical examples for two-dimensional wave equations illustrate the accuracy and efficiency of the proposed methods. 

In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that a time-harmonic elastic wave equation may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, motivated by our recent work [Hadamard- Babich ansatz for point-source elastic wave equations in variable media at high frequencies, Multiscale Model Simul. 19/1 (2021) 46–86], we propose a new truncated Hadamard-Babich ansatz based globally valid asymptotic method, dubbed the fast Huygens sweeping method, for computing Green’s functions of frequency-domain point-source elastic wave equations in inhomogeneous media in the high-frequency asymptotic regime and in the presence of caustics. The first novelty of the fast Huygens sweeping method is that the Huygens-Kirchhoff secondary-source principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics can be treated automatically. This yields uniformly accurate solutions both near the source and away from it. The second novelty is that a butterfly algorithm is adapted to accelerate matrix-vector products induced by the Huygens-Kirchhoff integral. The new method enjoys the following desired features: (1) it treats caustics automatically; (2) precomputed asymptotic ingredients can be used to construct Green’s functions of elastic wave equations for many different point sources and for arbitrary frequencies; (3) given a specified number of points per wavelength, it constructs Green’s functions in nearly optimal complexity O(N logN) in terms of the total number of mesh points N, where the prefactor of the complexity depends only on the specified accuracy and is independent of the frequency parameter. Three-dimensional numerical examples are presented to demonstrate the performance and accuracy of the new method. 

We have developed a Liouville partial-differential-equation (PDE)-based method for computing complex-valued eikonals in real phase space in the multivalued sense in attenuating media with frequency-independent qualify factors, where the new method computes the real and imaginary parts of the complex-valued eikonal in two steps by solving Liouville equations in real phase space. Because the earth is composed of attenuating materials, seismic waves usually attenuate so that seismic data processing calls for properly treating the resulting energy losses and phase distortions of wave propagation. In the regime of high-frequency asymptotics, the complex-valued eikonal is one essential ingredient for describing wave propagation in attenuating media because this unique quantity summarizes two wave properties into one function: Its real part describes the wave kinematics and its imaginary part captures the effects of phase dispersion and amplitude attenuation. Because some popular ordinary-differential-equation (ODE)-based ray-tracing methods for computing complex-valued eikonals in real space distribute the eikonal function irregularly in real space, we are motivated to develop PDE-based Eulerian methods for computing such complex-valued eikonals in real space on regular meshes. Therefore, we solved novel paraxial Liouville PDEs in real phase space so that we can compute the real and imaginary parts of the complex-valued eikonal in the multivalued sense on regular meshes. We call the resulting method the Liouville PDE method for complex-valued multivalued eikonals in attenuating media; moreover, this new method provides a unified framework for Eulerianizing several popular approximate real-space ray-tracing methods for complex-valued eikonals, such as viscoacoustic ray tracing, real viscoelastic ray tracing, and real elastic ray tracing. In addition, we also provide Liouville PDE formulations for computing multivalued ray amplitudes in a weakly viscoacoustic medium. Numerical examples, including a synthetic gas-cloud model, demonstrate that our methods yield highly accurate complex-valued eikonals in the multivalued sense.