## 1 Introduction

### 1.1 Background

[2] Infrasound generated on the ground or aloft can propagate thousands of kilometers in waveguides bounded by the Earth's surface and gradients in the atmospheric wind and temperature [*Gossard and Hooke*, 1976; *Georges and Beasley*, 1977]. Evanescent acoustic waveguides, akin to oceanographic SOFAR (SOund Fixing and Ranging) channels, can also occur aloft [e.g., *Arrowsmith et al*., 2007]. Although the spatiotemporal variability of these waveguides varies widely with location, season, and time of day [*Drob et al*., 2003, 2010], consensus within the infrasound research community is that available global atmospheric specifications from operational numerical weather prediction centers (e.g., NOAA Global Forecast System or ECMWF Integrated Forecast System) as well as atmospheric climate reanalysis activities (e.g., NASA Modern Era Retrospective Reanalysis for Research and Applications) provides adequate atmospheric specifications for detailed infrasound propagation calculations. This is particularly true for the tropospheric and stratospheric waveguides where well-resolved observationally based specifications of atmospheric temperature and wind profiles are available 24/7 for the entire globe below altitudes of 75 km. Two long-standing challenges for the infrasound research community are, however, (1) explaining frequently observed infrasound arrivals in the classical near-field “zones of silence” and (2) understanding waveform amplitudes and observed waveform signal durations. In solving these problems, the significance of small-scale fluctuations from the atmosphere's internal gravity wave spectrum has recently been realized [*Millet et al*., 2007; *Kulichkov et al*., 2010; *Chunchuzov et al*., 2011; *Green et al*., 2011].

[3] Owing to the ubiquity and nature of atmospheric gravity waves, much like turbulence, it is impractical to *deterministically* measure and resolve these waves in any comprehensive sense (i.e., for every arbitrary time and location). Doing so would be akin to tracking every crest and trough on the surface of the ocean, but in three dimensions at all times. Given current scientific understanding, even a directly measured local vertical profile of wind and temperature (e.g., from a nearby radiosonde, rocket sounding, or lidar measurement) is incomplete for resolving the two-dimensional internal gravity wavefield for the purpose of understanding the observed arrival structure or waveforms associated with any one particular infrasound event. For performing range-dependent infrasound propagation calculations, owing to the nature of the atmosphere's internal gravity wave spectrum, the phase relationships between the various waves making up the fine-scale structure will be completely different at distances as close as 25 km away from the measurement, or after a time of about 15 min. The ability to deterministically resolve the internal wavefield from a single radiosonde profile greatly diminishes as distances between the measurement location, source, and/or receiver increase. Also relevant is the fact that an isolated vertical sounding also does not provide any measure of the horizontal wind and sound velocity gradients for use in the Jacobian of the range-dependent acoustic ray-tracing equations.

[4] Furthermore, even if an individual measurement profile is available for a given time and geographic region of interest, in isolation, a single measurement is uncorroborated and thus subject to unchecked instrument calibration biases, which for the case of meteorological data analysis products is mitigated through the utilization of many different data sources from nearby satellite soundings, ground-based observations, and other in situ measurements, as well as the consideration of the well-known equations of geophysical fluid dynamics. In order to update today's operational numerical weather prediction data analysis fields, over 2 × 10^{6} fresh atmospheric observations are assimilated every 6 h [*Rienecker et al*., 2011].

[5] With respect to the specification of the background atmospheric fields for infrasound propagation calculations, it is important to note that any subgrid-scale phenomena are indeed filtered out during the data assimilation process to combine the diverse measurements to produce an initial atmospheric field which can be integrated forward in time to produce a long-range numerical weather forecast. This includes any atmospheric gravity wave perturbations with time scales of less than about 3 h and length scales of less than about twice the model horizontal grid point resolution, typically 0.5*°* × 0.5*°*. Fortunately below 65 km in the lower and middle atmosphere, the amplitude of these perturbations represents a small fraction of the overall variations of the average background state. Although the most prevalent sources of atmospheric gravity waves occur in the troposphere, such as the time-dependent flow winds over topography [*Lilly and Kennedy*, 1973; *Peltier and Clark*, 1979] and cumulous convection [*Stull*, 1976; *Clark et al*., 1986], initial wave amplitudes are relatively small, but as these gravity waves propagate upward, wave amplitude tends to grow as the result of exponentially decreasing air density. Thus, at certain locations and seasons in the upper mesosphere and lower thermosphere (above 75 km), gravity wave amplitudes can become comparable to, but generally not greater than, the dominant tidal variations.

[6] Thus for the purposes of infrasound propagation modeling calculations, the unresolved atmospheric gravity wave perturbations can be represented as a random noise field that is superimposed on the resolved average background state, much in the same way that turbulence is parameterized in aerodynamic drag calculations [e.g., *Jameson*, 1989]. A key issue is that the spectral characteristics of the wavefield should be correct; for example, the utilization of normalized Gaussian white noise would not be appropriate. It turns out that suitable subgrid-scale parameterizations have been developed and used by the numerical weather prediction and climate modeling communities for the past 30 years, because the energy and momentum carried and deposited aloft by unresolved small-scale waves dramatically influence the atmospheric general circulation. The fundamentals of atmospheric gravity wave physics can be found in the book by *Gossard and Hooke* [1976]. An extensive review of gravity wave theory, observations, and parameterizations is given by *Fritts and Alexander* [2003]. Recent developments in explicit stochastic parameterizations of non-orographic gravity wave drag for whole atmosphere numerical weather prediction systems are presented by *Eckermann* [2011].

### 1.2 Overview

[7] This paper presents a method for including unresolved gravity wave variability in infrasound propagation calculations. The model produces a four-dimensional stochastic field that can be superimposed on the larger-scale background of winds and temperature for a given location or time. The spatiotemporal scales relevant to infrasound propagation range from horizontal scales of hundreds of kilometers and time scales from 15 min to a few hours. An important aspect of the method is that the spectral characteristics of the wavefield are computed in such a way as to be self-consistent with the vertical structure of the atmosphere for that location and time. A novel aspect of the methodology is how gravity wave refraction is handled; this is done by a spectral ray-tracing technique, corrected for caustics, with the corresponding three-dimensional spatial solution obtained by a Fourier synthesis of the ray solutions. We do assume that the larger-scale background is horizontally uniform over this distance and steady over the wave propagation time periods. The resulting gravity wavefield, however, is three-dimensional and time dependent to altitudes of 150 km, adequate for ducted infrasound propagation calculations.

[8] Our approach is to specify a gravity wave spectrum for the troposphere, where gravity wave sources are strongest and where empirical spectral amplitudes are best constrained by several decades of measurements [see, for example, *Fritts and Alexander*, 2003 and references therein). We then ray trace the gravity wave spectrum upward, applying parameterizations for wave breaking and wave dissipation. In this sense, we are following the basic strategy of *Warner and McIntyre* [1996] in their development of a gravity wave drag parameterization for numerical weather models. They separate the wave propagation aspects from the more empirical wave-breaking and dissipation aspects. The former are handled by the ray tracing, while the latter are parameterized. This is the difference between our model and other gravity wave models for infrasound propagation where refraction is parameterized together with wave breaking and other processes in a Lagrangian framework, such as *Kulichkov et al*. [2010] and *Chunchuzov et al*. [2011].

[9] The spectral gravity wave ray-tracing technique implemented here is an adaptation of the method developed for mountain waves by *Broutman et al*. [2003, 2006, 2009] and compared with satellite observations by *Eckermann et al*. [2006] and *Alexander et al*. [2009]. Those studies are limited to stationary mountain waves (gravity waves of zero frequency relative to the ground), to altitudes below 50 km, and to conservative propagation. In this paper, we include a spectrum of frequencies, consider altitudes up to 150 km, and include dissipative processes. Extensions to higher altitudes are left to future work as they are not directly relevant to ducted infrasound propagation.

[10] In section 2, we derive the spectral ray-tracing equations for conservative gravity wave propagation. In section 3, we describe the parameterizations for the initial gravity wave source spectrum, as well as the upper atmospheric gravity wave dissipation mechanisms. In section 4, we provide algorithmic details for computational implementation. In section 5, we present two examples. The first illustrates the nature of the resulting gravity wavefield propagation physics by showing the vertical propagation of the resulting atmospheric gravity wavefields generated by a localized vertically oscillating source with a fixed intrinsic frequency, in idealized background atmospheres with and without winds. The second example illustrates application of the methodology to infrasound propagation modeling via a comparison of infrasound ray-tracing calculations with observed signal arrival times from a dense regional seismic network in the Western United States.