A two-dimensional effective sound speed parabolic equation model for infrasound propagation with ground topography.

A terrain capable parabolic equation (PE) propagation algorithm for long range infrasound propagation modeling has been implemented using Padé approximations for the various operator valued functions that arise in PE algorithms. In this work, the influence of the winds are captured by the effective sound speed approximation and propagation is restricted to the range-altitude plane. The ground topography is included by the addition of an impenetrable fluid below the ground surface. The impedance condition at the ground is handled explicitly, including both vertical and radial components. It is found that including terrain can have a large influence on long range propagation. In particular, reflections from a sufficiently steep slope can change the inclination angle enough to move the propagation path from one atmospheric duct to another.

Characteristics of thermospheric infrasound predicted using ray tracing and weakly non-linear waveform analyses.

Infrasonic signals refracted by thermal gradients in the rarefied upper atmosphere are modeled using a combination of ray tracing and weak shock theory to develop an understanding of thermospheric infrasound signals produced by energetic, transient sources. Canonical arrival structures in the form of u-wave signatures are identified for returns refracted at lower altitudes within the thermosphere, and possible multi-pathing produced by effective sound speed inflections are investigated to elucidate more complex arrival structures, which are found to be spatially localized. Variability in the source characteristics is investigated and it is found that whereas some waveform phase information is lost due to finite amplitude effects, arrival characteristics are strongly dependent on the peak overpressure near the source. Variability in the propagation path is considered using archived atmospheric specifications and implies that despite uncertainties related to the dynamic and sparsely sampled nature of the atmosphere, thermospheric signatures might be useful in estimating the yield for explosive sources. Last, thermospheric arrivals from a failed rocket launch, as well as those from several large chemical explosions, are analyzed and it is found that qualitative trends match those predicted, and analyses here provide additional insight into such signatures.

Probabilistic inversion for submerged source depth and strength from infrasound observations.

In seismology, the depth of a near-surface source is hard to estimate in the absence of local stations. The depth-yield trade-off leads to significant uncertainties in the source's depth and strength estimations. Long-range infrasound propagation from an underwater or underground source is very sensitive to variations in the source's depth and strength. This characteristic is employed in an infrasound based inversion for the submerged source parameters. First, a Bayesian inversion scheme is tested under the variations of the number of stations, the signal's frequency band, and the signal-to-noise ratio (SNR). Second, an ensemble of realistic perturbed atmospheric profiles is used to investigate the effect of atmospheric uncertainties on the inversion results. Results show that long-range infrasound signals can be used to estimate the depth and strength of an underwater source. Using a broadband signal proved to be a fundamental element to obtain the real source parameters, whereas the SNR was secondary. Multiple station inversions perform better than one-station inversions; however, variations in their position can lead to source strength estimations with uncertainties up to 50%. Regardless of the number of stations, their positions, and SNRs, all of the estimated depths were within 10% from the real source depth.

Modeling and observations of an elevated, moving infrasonic source: Eigenray methods.

The acoustic ray tracing relations are extended by the inclusion of auxiliary parameters describing variations in the spatial ray coordinates and eikonal vector due to changes in the initial conditions. Computation of these parameters allows one to define the geometric spreading factor along individual ray paths and assists in identification of caustic surfaces so that phase shifts can be easily identified. A method is developed leveraging the auxiliary parameters to identify propagation paths connecting specific source-receiver geometries, termed eigenrays. The newly introduced method is found to be highly efficient in cases where propagation is non-planar due to horizontal variations in the propagation medium or the presence of cross winds. The eigenray method is utilized in analysis of infrasonic signals produced by a multi-stage sounding rocket launch with promising results for applications of tracking aeroacoustic sources in the atmosphere and specifically to analysis of motor performance during dynamic tests.

Modeling the refraction of microbaroms by the winds of a large maritime storm.

Continuous infrasonic signals produced by the ocean surface interacting with the atmosphere, termed microbaroms, are known to be generated by a number of phenomena including large maritime storms. Storm generated microbaroms exhibit axial asymmetry when observed at locations far from the storm due to the source location being offset from the storm center. Because of this offset, a portion of the microbarom energy will radiate towards the storm center and interact with the winds in the region. Detailed here are predictions for the propagation of microbaroms through an axisymmetric, three-dimensional model storm. Geometric propagation methods have been utilized and the predicted horizontal refraction is found to produce signals that appear to emanate from a virtual source near the storm center when observed far from the storm. This virtual source near the storm center is expected to be observed only from a limited arc around the storm system with increased extent associated with more intense wind fields. This result implies that identifying the extent of the arc observing signal from the virtual source could provide a means to estimate the wind structure using infrasonic observations far from the storm system.

Modal expansions for infrasound propagation and their implications for ground-to-ground propagation.

The use of expansions in vertical eigenmodes for long range infrasound propagation modeling in the effective sound speed approximation is investigated. The question of convergence of such expansions is related to the maximum elevation angles that are required. Including atmospheric attenuation leads to a non-self-adjoint vertical eigenvalue problem. The use of leading order perturbation theory for the modal attenuation is compared to the results of numerical solutions to the non-self-adjoint eigenvalue problem and conditions under which the perturbative result is expected to be valid are obtained. Modal expansions are obtained in the frequency domain; broadband signals must be modeled through Fourier reconstruction. Such broadband signal reconstruction is investigated and the relation between bandwidth, wavetrain duration, and frequency sampling is discussed.

A wide-angle high Mach number modal expansion for infrasound propagation.

The use of modal expansions to solve the problem of atmospheric infrasound propagation is revisited. A different form of the associated modal equation is introduced, valid for wide-angle propagation in atmospheres with high Mach number flow. The modal equation can be formulated as a quadratic eigenvalue problem for which there are simple and efficient numerical implementations. A perturbation expansion for the treatment of attenuation, valid for stratified media with background flow, is derived as well. Comparisons are carried out between the proposed algorithm and a modal algorithm assuming an effective sound speed, including a real data case study. The comparisons show that the effective sound speed approximation overestimates the effect of horizontal wind on sound propagation, leading to errors in traveltime, propagation path, trace velocity, and absorption. The error is found to be dependent on propagation angle and Mach number.

The stratospheric arrival pair in infrasound propagation.

The ideal case of a deep and well-formed stratospheric duct for long range infrasound propagation in the absence of tropospheric ducting is considered. A canonical form, that of a pair of arrivals, for ground returns of impulsive signals in a stratospheric duct is determined. The canonical form is derived from the geometrical acoustics approximation, and is validated and extended through full wave modeling. The full caustic structure of the field of ray paths is found and used to determine phase relations between the contributions to the wavetrain from different propagation paths. Finally, comparison with data collected from the 2005 fuel gas depot explosion in Buncefield, England is made. The correspondence between the theoretical results and the observations is shown to be quite good.

Acoustic detection, tracking, and characterization of three tornadoes.

Acoustic data recorded at 1000 samples per second by two sensor arrays located at ranges of 1-113 km from three tornadoes that occurred on 24 May 2011 in Oklahoma are analyzed. Accurate bearings to the tornadoes have been obtained using beamforming methods applied to the data at infrasonic frequencies. Beamforming was not viable at audio frequencies, but the data demonstrate the ability to detect significant changes in the shape of the estimated power spectral density in the band encompassing 10 Hz to approximately 100 Hz at distances of practical value from the sensors. This suggests that arrays of more closely spaced sensors might provide better bearing accuracy at practically useful distances from a tornado. Additionally, a mathematical model, based on established relationships of aeroacoustic turbulence, is demonstrated to provide good agreement to the estimated power spectra produced by the tornadoes at different times and distances from the sensors. The results of this analysis indicate that, qualitatively, an inverse relationship appears to exist between the frequency of an observed peak of the power spectral density and the reported tornado intensity.

Impulse propagation in the nocturnal boundary layer: analysis of the geometric component.

On clear dry nights over flat land, a temperature inversion and stable nocturnal wind jet lead to an acoustic duct in the lowest few hundred meters of the atmosphere. An impulsive signal propagating in such a duct is received at long ranges from the source as an extended wave train consisting of a series of weakly dispersed distinct arrivals followed by a strongly dispersed low-frequency tail. The leading distinct arrivals have been previously shown to be well modeled by geometric acoustics. In this paper, the geometric acoustics approximation for the leading arrivals is investigated. Using the solutions of the eikonal and transport equations, travel times, amplitudes, and caustic structures of the distinct arrivals have been determined. The time delay between and relative amplitudes of the direct-refracted and single ground reflection arrivals have been investigated as parameters for an inversion scheme. A two parameter quadratic approximation to the effective sound speed profile has been fit and found to be in strong agreement with meteorological measurements from the time of propagation.

Atmospheric waves and global seismoacoustic observations of the January 2022 Hunga eruption, Tonga.

The 15 January 2022 climactic eruption of Hunga volcano, Tonga, produced an explosion in the atmosphere of a size that has not been documented in the modern geophysical record. The event generated a broad range of atmospheric waves observed globally by various ground-based and spaceborne instrumentation networks. Most prominent was the surface-guided Lamb wave (≲0.01 hertz), which we observed propagating for four (plus three antipodal) passages around Earth over 6 days. As measured by the Lamb wave amplitudes, the climactic Hunga explosion was comparable in size to that of the 1883 Krakatau eruption. The Hunga eruption produced remarkable globally detected infrasound (0.01 to 20 hertz), long-range (~10,000 kilometers) audible sound, and ionospheric perturbations. Seismometers worldwide recorded pure seismic and air-to-ground coupled waves. Air-to-sea coupling likely contributed to fast-arriving tsunamis. Here, we highlight exceptional observations of the atmospheric waves.

The extended Fourier pseudospectral time-domain method for atmospheric sound propagation.

An extended Fourier pseudospectral time-domain (PSTD) method is presented to model atmospheric sound propagation by solving the linearized Euler equations. In this method, evaluation of spatial derivatives is based on an eigenfunction expansion. Evaluation on a spatial grid requires only two spatial points per wavelength. Time iteration is done using a low-storage optimized six-stage Runge-Kutta method. This method is applied to two-dimensional non-moving media models, one with screens and one for an urban canyon, with generally high accuracy in both amplitude and phase. For a moving atmosphere, accurate results have been obtained in models with both a uniform and a logarithmic wind velocity profile over a rigid ground surface and in the presence of a screen. The method has also been validated for three-dimensional sound propagation over a screen. For that application, the developed method is in the order of 100 times faster than the second-order-accurate FDTD solution to the linearized Euler equations. The method is found to be well suited for atmospheric sound propagation simulations where effects of complex meteorology and straight rigid boundary surfaces are to be investigated.

A direct method for measuring acoustic ground impedance in long-range propagation experiments.

A method is reported for determining ground impedance in long-range propagation experiments by using the definition of impedance directly. The method is envisioned as way of measuring the impedence at multiple locations along the propagation path, using the signals broadcast during the experiment itself. In a short-range (10 m) test, the direct method was in good agreement with a more conventional model-based least-squares method. The utility of the direct method was demonstrated in a 400 m propagation experiment in a agricultural field. The resulting impedance was consistent with the impedance measured previously in the same field.

A theoretical treatment of the long range propagation of impulsive signals under strongly ducted nocturnal conditions.

On clear nights, over flat land, a sound duct develops in which sound can carry to great distances. As is the case with all ducted propagation, there is strong dispersion so that a broadband signal undergoes severe distortion as it propagates. The signal received at long ranges from an impulsive source is a wave train, of much greater duration than the initial impulse, consisting of a series of arrivals. The long range ground to ground propagation of an impulsive signal in a typical nocturnal duct is studied and the natures of the various arrivals are explained. A direct connection is drawn between the meteorological and ground conditions and the structure of the propagated signal.

Observation of low-frequency acoustic surface waves in the nocturnal boundary layer.

A natural terrain surface, because of its porosity, can support an acoustic surface wave that is a mechanical analog of the familiar vertically polarized surface wave in AM radio transmission. At frequencies of several hundred hertz, the acoustic surface wave is attenuated over distances of a few hundred meters. At lower frequencies (e.g., below approximately 200 Hz) the attenuation is much less, allowing surface waves to propagate thousands of meters. At night, a low-frequency surface wave is generally present at long ranges even when downward refraction is weak. Thus, surface waves represent a ubiquitous nighttime transmission mode that exists even when other transmission modes are weak or absent. Data from recent nighttime field experiments and theoretical calculations are presented, demonstrating the persistence of the surface wave under different meteorological conditions. The low-frequency surface wave described here is the "quasiharmonical" tail observed previously in nighttime measurements but not identified by S. Kulichkov and his colleagues (Chunchuzov, I. P. et al. 1990. "On acoustical impulse propagation in a moving inhomogeneous atmospheric layer," J. Acoust. Soc. Am. 88, 455-461).

The near-ground structure of the nocturnal sound field.

The near-ground behavior of the low-frequency (100 Hz to 500 Hz) sound field in the nocturnal sound duct is studied theoretically and experimentally. In the first few meters of the atmosphere, narrow-band sound fields are found to have a characteristic vertical structure. The sound field is the superposition of a "surface mode," whose magnitude decreases monotonically with altitude, with a sum of "higher modes," each of whose magnitudes has a pronounced minimum a few meters from the ground at approximately the same height. The surface mode attenuates to negligible levels after a few hundred meters from the source. Consequently, more than a few hundred meters from a narrow-band source, there is a "quiet height" at which the sound level is reduced by 10 to 15 dB relative to its value on the ground. The narrow-band quiet height is shown to be a robust feature of nocturnal sound propagation.

A vertical eigenfunction expansion for the propagation of sound in a downward-refracting atmosphere over a complex impedance plane.

The propagation of sound in a stratified downward-refracting atmosphere over a complex impedance plane is studied. The problem is solved by separating the wave equation into vertical and horizontal parts. The vertical part has non-self-adjoint boundary conditions, so that the well-known expansion in orthonormal eigenfunctions cannot be used. Instead, a less widely known eigenfunction expansion for non-self-adjoint ordinary differential operators is employed. As in the self-adjoint case, this expansion separates the acoustic field into a ducted part, expressed as a sum over modes which decrease exponentially with height, and an upwardly propagating part, expressed as an integral over modes which are asymptotically (with height) plane waves. The eigenvalues associated with the modes in this eigenfunction expansion are, in general, complex valued. A technique is introduced which expresses the non-self-adjoint problem as a perturbation of a self-adjoint one, allowing one to efficiently find the complex eigenvalues without having to resort to searches in the complex plane. Finally, an application is made to a model for the nighttime boundary layer.