A method for specifying atmospheric gravity wavefields for long-range infrasound propagation calculations


  • This article was corrected on 30 OCT 2014. See the end of the full text for details.


[1] Two important challenges in infrasound propagation physics are (1) to explain frequently observed infrasound signals in the classical near-field shadow zones and (2) to accurately predict observed waveform amplitude and signal duration. For these problems, the role that small-scale internal atmospheric gravity wave fluctuations play has recently been realized. This paper provides a methodology for representing small-scale internal gravity wave fluctuations which is suitable for infrasound propagation calculations. Adapted from the numerical weather prediction and climate modeling communities, the resulting stochastic gravity wave noise field model is three-dimensional, time dependent, and self-consistent with the atmospheric background state. To illustrate the methodology the resultant gravity wavefields are applied to ray-trace simulations of observed infrasound travel times for a dense seismic network in the Western United States which recorded infrasound signals from a large surface explosion.

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 × 106 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.

2 Conservative Propagation

[11] This section describes the mathematical formalism for gravity wave ray-tracing equations in the Fourier domain which forms the basis of the methodology provided. The vertical integration of both freely propagating and trapped waves in the spectral domain, expressed in terms of the vertical velocity spectrum, is considered. Expressions relating the temperature and horizontal wind perturbation spectra to the vertical velocity perturbation spectrum as a function of altitude are also provided.

2.1 Vertical Propagation Without Trapping

[12] The large-scale background (e.g., wind, temperature) of the atmosphere will be referred to as the ambient background. The ambient background is assumed to be horizontally steady and only varies with altitude. The gravity waves superimposed upon the ambient background are also refracted by it. In this section, we derive the equations for the refracted gravity waves. We start with inviscid vertically propagating gravity waves of a single frequency and then sequentially consider trapped waves, dissipative effects, wave-breaking parameterization, and a broadband spectrum of frequencies.

[13] The notation has horizontal coordinates (x, y), vertical coordinate z, and time t. The ambient background profile consists of the horizontal winds (U, V), air density ρ, buoyancy frequency N, and temperature T, as functions of altitude z. The internal gravity waves satisfy the anelastic dispersion relation [Fritts and Vadas, 2008, equation ((1))]:

display math(1)
display math(2)

where inline image is the intrinsic angular frequency (relative to the moving air), ω is the absolute frequency (relative to the ground), (k, l) are the zonal and meridional wave numbers, m is the vertical wave number, and kh = (k2 + l2)1/2 is the resultant horizontal wave number. The atmospheric density scale height is Hρ = −ρ(dρ/dz)−1 which varies with altitude, where ρ is the mass density. The vertical group velocity is inline image. An upward propagating wave group, cg > 0, has downward moving phases, m < 0, and vice versa [Gossard and Hooke, 1976; Fritts and Alexander, 2003].

[14] The net gravity wavefield is represented as the superposition of individual spectral components. The amplitude of each spectral component is completely specified by the altitude z and three of the parameters k, l, ω, inline image, and m. The relationship between k, l, ω, and inline image is given by ((1)) and with m by ((2)). The parameters k, l, ω are constants of propagation from an ambient background that only depends on altitude.

[15] We refer to k, l, ω as the spectral domain and use z as the parametric variable to follow the propagation. Each spectral component satisfies conservation of wave action, apart from the effects of dissipation introduced later. When expressed in the spectral domain, conservation of wave action means that the vertical flux of wave action is constant along the ray, i.e.,

display math(3)

where inline image is the wave action density. The wave energy density E is related to the vertical velocity inline image (not to be confused with inline image) of the spectral component by

display math(4)

[16] Note that conserving the vertical flux of wave action ((3)) in the spectral domain is not the same as conserving the vertical flux of wave action in the spatial domain. The latter would be equivalent to ignoring geometrical spreading in the horizontal coordinates x, y. Here, three-dimensional geometrical spreading in the spatial domain is accounted for by the Fourier superposition ((8)) below.

[17] In terms of inline image, wave action conservation ((3)) becomes

display math(5)

[18] Hence, an upward propagating wave (cg > 0) is represented by

display math(6)

where the 0 subscript denotes the value at a reference level z0, and the wave phase is

display math(7)

[19] The expression for gravity waves in the spatial domain is given by Fourier superposition:

display math(8)

[20] For future reference, the propagation time tp between heights z1 to z2 is

display math(9)

[21] Equation ((6)) applies to conservative upward propagation without reflection at turning points. We next consider extensions of ((6)) for trapped waves, then later viscous and thermal dissipation, as well as wavefield saturation.

2.2 Trapped Waves

[22] Trapped waves are reflected vertically at a height where the vertical wave number m and the vertical group velocity cg vanish. Here inline image in ((6)) diverges, indicating a caustic singularity that is corrected with an Airy function. The simplest case to treat analytically is where the trapped waves are bounded by a turning point above and by the ground below, not by another turning point below. That is, the waveguide for trapping reaches the ground.

[23] For trapped waves, we add to the solution ((6)) a wave reflected from the turning point, yielding

display math(10)

[24] The reflected wave is the second term in the square brackets above. The π/2 factor is the caustic phase shift resulting from reflection at the turning point. Defining zt as the height of the turning point, we have

display math(11)
display math(12)

[25] For downward ray propagation, our notation is that m > 0. Thus Φ and φr both decrease along the ray, noting that dz′ is negative in ((12)).

[26] For trapped waves, the ray solutions for the upward and downward propagating rays can be combined to form a uniform solution that is valid close to, and far from, the caustic at the turning point. This uniform solution is

display math
display math(13)

[27] The Airy function Ai has argument

display math(14)

in the propagating region below the caustic (zzt), where r ≤ 0. In the evanescent region above the caustic (z > zt), m is imaginary and r is given by ((14)) with the integration limits reversed and the sign outside the brackets switched from negative to positive.

[28] The term Sn in ((13)) is a way of including multiple reflections of the rays between the turning point and the ground. The uniform solution by itself, without the Sn term, represents the caustic corrected solution for a single reflection from the turning point, i.e., the sum of one incident ray and one reflected ray. However, the reflected ray will continue to reflect between the ground and the turning point. Each time the ray reaches the turning point, its phase is changed by 2Φ − π/2 relative to the previous time. The term 2Φ is, from ((11)), the contribution to the phase change from propagation, and the term − π/2 is the contribution from reflections, i.e., the sum of the π/2 phase shift at the caustic and the − π phase shift at the ground.

[29] Thus to incorporate multiple reflections between the ground and the turning point, the phase-shifted copies of the uniform solution for the first reflection from the turning point need to be summed. This is the purpose of the Sn term, defined by

display math(15)

[30] Here n is the total number of round trips for the each ray, starting from the source, propagating upward to the turning point, then downward to the ground, and then back upward to the source.

[31] The propagation time for each round trip is 2tp(0,zt), where tp is defined in ((9)). Rounding 2tp(0,zt)/t to the nearest integer and using that integer for n are sufficient for our purposes.

[32] Rays that at time t do not reach their turning point (n = 0) are treated as untrapped upward propagating rays, described by ((6)). This technique for treating multiple turning point reflections using the sum ((15)) is developed in Broutman et al. [2003, 2006] and applied to trapped mountain waves generated by Jan Mayen Island in Eckermann et al. [2006].

2.3 Solution for Other Variables

[33] The above solutions are for the vertical velocity w. For the other atmospheric state variables, the gravity wave polarization relation can be used to make the variable conversion in the spectral domain, and then inverse Fourier transform to obtain the spatial solution. For vertical displacement η and perturbation temperature T′, the polarization relations are

display math(16)
display math(17)

where ^ corresponds to the spectral domain representation, i.e., as a function of k, l, ω, z. Here Tz is the vertical temperature gradient of the unperturbed background atmosphere. These conversions apply both to the vertically propagating solution ((6)) and the vertically trapped solution ((13)).

[34] The wind velocity perturbations u, v have the polarization relations:

display math(18)

[35] These conversions apply only to the vertically propagating solution ((6)). For vertically trapped waves, the Airy function representation in ((13)) must be changed to one involving an Airy function derivative. Basically, the factor m in ((18)) corresponds to the ray approximation for − i ∂ z, which must be applied to the Airy function. That is, in the ray approximation

display math(19)

[36] The uniform solution thus becomes

display math(20)

where Ai′ is the derivative of the Airy function with respect to its argument r. This uniform solution ((20)) can also be obtained directly from the ray solutions for u, v; see section 10.1.3 of Kravtsov and Orlov [1999].

3 Parameterizations

[37] This section describes additional specifics for the parametric approximation of the gravity field to account for the physics of viscous and thermal damping in the thermosphere, characteristics of the initial source spectrum, and nonlinear wave amplitude saturation processes.

3.1 Viscous and Thermal Damping

[38] At altitudes above 100 km, gravity wave damping by molecular viscosity and thermal diffusivity becomes increasingly important as the result of exponentially decreasing air density [Pitteway and Hines, 1963; Yiğit et al., 2008]. Vadas and Fritts [2005] derive viscous and thermal damping terms for a ray-tracing formulation. They consider two options, incorporating the damping as either a complex intrinsic frequency or a complex vertical wave number. In the former, the wave amplitudes decay explicitly in time and implicitly in altitude. In the latter, the wave amplitudes decay explicitly in altitude and implicitly in time. Either option can be implemented, although the latter (complex vertical wave number) is slightly more convenient as the altitude z is the parametric variable for the ray tracing. For application to ducted infrasound propagation calculations limited to altitudes below approximately 120 km, gravity wave damping due to ion drag, which as noted by Vadas and Fritts [2005] is negligible for the higher-frequency waves that reach the lower thermosphere, can be ignored to first order. Introducing an imaginary component mi, such that the vertical wave number is m + imi, Vadas and Fritts [2005] find that the combined effects of viscosity and thermal diffusivity give

display math(21)

where ν = μ/ρ is the kinematic viscosity, with μ being the molecular viscosity. Several approximations have gone into the derivation of ((21)), including the Boussinesq limit and a Prandtl number of unity. See Vadas and Fritts [2005] for more discussion. For intrinsic frequencies less than about N/2, the dispersion relation ((2)) is approximately inline image, implying mi ∝ m4, as noted by Yiğit et al. [2008].

[39] For ν we follow many others [e.g., Yiğit et al., 2008] and use the parameterization of Banks and Kockarts [1973]:

display math(22)

where T is the mean temperature.

[40] For gravity wave propagation calculations and infrasound propagation modeling which extend well into the thermosphere, along with ion-drag dissipation, additional damping factors for nonlinear wave-wave interaction and Newtonian cooling (radiative damping) ignored in the present development should also be included. These factors were shown to be important for gravity wave dynamics in the thermosphere by Yiğit et al. [2008], who extended the gravity wave parameterization of Medvedev and Klaassen [1995] to later investigate the influence to mid-frequency gravity waves on the momentum and energy budgets of the thermosphere [Yiğit et al., 2009; Yiğit and Medvedev, 2009].

3.2 Source Spectrum

[41] A gravity wave source spectrum is specified at altitude z0, taken below to be any altitude between 10 and 20 km. Following many others, but most closely Warner and McIntyre [1996], a separable form for the wave energy spectrum inline image, where θ = tan− 1(l/k) is the angle of the horizontal wave number vector, can be used. Specifically,

display math(23)


display math(24)

[42] The spectrum is assumed to be independent of θ. The parameter inline image is the minimum value of inline image used in the source spectrum (cf. section 4). The parameter m determines where the spectrum peaks in vertical wave number. It peaks close to, but not exactly at m. We set m such that 2π/m ≈ 2.5 km, corresponding to a source altitude of 20 km. Equation ((23)) is the same as equation ((9)) of Warner and McIntyre [1996] when their choice of empirical constants is used.

[43] Transformed to spectral coordinates (k, l, ω), the energy spectrum becomes

display math(25)

where the factor inline image is the Jacobian of the transformation

display math(26)

[44] For further details on the transformation, see equations ((19))–((23)) of Warner and McIntyre [1996]. Thus to initialize the wave energy density, it is assumed that

display math(27)

with the initial vertical velocity amplitude inline image obtained from E using ((4)).

3.3 Wave Saturation

[45] There are several parameterizations for wave breaking based on various saturation criteria usually related either to convective instability, as in equation ((25)) of Warner and McIntyre [1996], or to an assumed form of a saturation spectrum, as in equation ((24)) of Warner and McIntyre [1996]. Either of these schemes can be implemented in the methodology, but for the present application, the criterion based on a saturation spectrum is chosen. As noted in Warner and McIntyre [1996, pp. 3218], there are unresolved questions about how realistic either scheme might be. An alternate approach is described by Medvedev and Klaassen [1995].

[46] From equation ((24)) of Warner and McIntyre [1996], the saturation spectrum is

display math(28)

[47] In spectral coordinates

display math(29)

as in the transformation ((25)).

[48] The saturation threshold is applied to each Fourier component, such that when

display math(30)

the wave energy E is clipped to the saturation value inline image. If there were no refraction, viscosity, or themal diffusion, inline image would grow in height as ρ/ρ0 = exp[(z − z0)/H], for scale height H ≃ 7.5 km, and would eventually exceed inline image. With refraction, there is still a strong tendency for exponential growth, and much of the spectrum saturates before 100 km altitude, where viscosity and thermal diffusion become important.

[49] The gravity wavefield in the atmosphere is complicated, being to some extent intermittent and nonlinear. Our model uses a statistically steady gravity wave spectrum and linear dynamics for propagation up to the altitude of saturation. To make our model more realistic, we have needed two other fixes that are not obviously justified in terms of dynamics.

[50] Note that critical layer filtering is handled in one of three ways. As the waves approach a critical layer, the vertical group velocity cg → 0, the intrinsic frequency inline image, and the wave energy density E → . Thus ultimately the saturation threshold is met. The propagation time to a critical layer tp(z, zc) is infinite in the ray approximation, where tp is defined in ((9)). So ultimately, the propagation time limit is exceeded. Typically, this time limit is set to 3–6 h, so that after this time the wave amplitude is exponentially damped with a time constant of 3 h. Finally, the vertical wave number m →  as the critical layer is approached, so frictional effects significantly damp the waves, especially if the critical layer is in the viscous region above about ∼100 km altitude.

4 Implementation

[51] The computational procedure is

  1. [52] Specify a discrete grid for the horizontal wave numbers k, l.

  2. [53] Specify the average background profiles U(z), V(z), ρ(z), N(z), H(z), etc.

  3. [54] Set the initial intrinsic frequency inline image between a minimum value of inline image and inline image.

  4. [55] Mask out any waves where kh > max(k, l) and C = N/m > Cmax (typically 90 m/s).

  5. [56] Determine the height of the turning point (where m = 0) and critical layers (where inline image) for each Fourier component. Compute the total phase change ((11)) between the ground and the turning point, the propagation time between the source to the turning point, and quantities needed for the trapped wave solution.

  6. [57] Initialize inline image of ((6)) with amplitudes derived from the source energy spectrum ((27)) and with random phases for φ0 in ((7)).

  7. [58] Integrate the spectrum of rays upward, applying the saturation threshold ((28)) and time cutoffs at each height.

  8. [59] Obtain the spatial solution at each height from the Fourier integral ((8)), approximated by fast Fourier transform.

  9. [60] Repeat steps 3–8 integrating over inline image from inline image to inline image by inline image.

[61] Physically meaningful results can be produced with a grid of 128 or 256 points for Nk, Nl, for the wave numbers in k, l, respectively. In physical space, Δx, Δy can range from 2 to 4 km with the respective spatial domain having periodic boundary conditions and a domain size from 512 × 512 km to 1024 × 1024 km. The ray solution is computed in vertical steps of Δz, typically 200 m. The numerical integration for the wave phase ((7)) and for the propagation time ((9)) is performed by the trapezoidal rule. All Fourier components are advanced upward together, with the same Δz. The solution for trapped waves includes downward moving rays reflected from the turning point. These can be computed while advancing the solution upward because Φ and n (see section 2.2) are calculated beforehand in step 4 of the computational procedure above. The source height z0 is generally taken to be 20 km, but other source heights can be used with the observed characteristic vertical wave number m being adjusted accordingly to observations [cf. Fritts and Alexander, 2003]. To fill in the solution at heights below z0, the initial condition to the ground can be back traced to the ground. The effects of refraction can be omitted in this region for simplicity, with the justification that the source spectrum represents an average over the region below the source.

[62] To obtain the correct wave density for the assumed empirical source spectrum, the wave number integration must also occur over intrinsic frequency inline image. This adds another dimension to the calculation, which contains inline image spectral grid points. For the present application, five evenly spaced values of intrinsic frequency between inline image and inline image are used, where f is the Coriolis frequency. Waves with frequency where inline image require additional terms in equation ((2)) and typically have large enough spatial scales to be resolved explicitly as part of the ambient background. Waves with frequency where inline image require the addition of acoustic terms in equation ((2)) and typically have small enough spatial scales to be neglected. To integrate over inline image, steps 3–9 are repeated for each value, summing the contributions at the end for the solution.

5 Results

5.1 Idealized Gravity Wave Propagation

[63] Here we illustrate the spectral gravity wave formulations ability to account for horizontal wind shear. We first consider gravity waves generated by a localized source oscillating vertically at fixed intrinsic frequency inline image. In a uniform background, at rest, the gravity wave rays lie on the surface of a cone, with angle inline image to the vertical.

[64] An example is shown in Figure 1. The source is located at z0 = 20 km, with inline image and N = 0.01s− 1. The initial wave phase is not random in this case (cf. step 6 in the computational procedure of section 4). Instead the initial wave phase φ and initial wave amplitude w0 in ((6)) are set by the deterministic method given in Broutman and Rottman [2004] for a localized source. The theoretical calculation for the spatial ray path is shown in the vertical cross-section (Figure 1, left panel) by the dashed line. The right panel shows the gravity wave solution at z = 50 km, a pattern of concentric rings, expanding with height.

Figure 1.

The solution for vertical velocity w (m/s) in a test case with uniform background and a single-frequency oscillating source. The source is centered at x = y = 0, z = 20 km.

[65] In Figure 2, a mean wind shear is added to the case of Figure 1. The mean wind is in the x direction and grows linearly from rest at z = 20 km to 40 m s− 1 at z = 100 km. Rays propagating upward to the right of the source refract toward critical layers. The horizontal cross-section (Figure 2, right panel) shows a horseshoe pattern, consistent with the results of Vadas and Fritts [2004].

Figure 2.

As in Figure 1, but with a background wind U(z) increasing linearly with height. The color scale represents vertical velocity w in units of m/s.

5.2 Comparison of Modeled Infrasound Travel Times With Seismic Network Data

[66] Although it has been acknowledged for at least a decade that synthetic acoustic propagation modeling that does not account for gravity waves fails to accurately reproduce observed infrasound waveforms, progress toward a solution has been slowed by a lack of data. A cornerstone of research in this field, the International Monitoring System (IMS) infrasound network, is too sparse for detailed investigations of the infrasound wavefield. Therefore, the IMS network cannot be used to investigate the finer details of the wavefield from an infrasound source, such as whether or not a station will be shadowed and unlikely to record a large signal. These details are of critical importance for basic research into atmospheric structure and practical applications such as hazard and nuclear monitoring.

[67] Infrasound signals are commonly recorded by seismometers due to coupling at the Earth's free-surface [e.g., Kanamori et al., 1991; Langston, 2004], and some recent studies have used dense seismic networks to map out the spread of the infrasound wavefields from well-constrained sources [e.g., de Groot-Hedlin et al., 2008; Hedlin et al., 2010; Walker et al., 2010]. Here we present a preliminary analysis of recordings of infrasound signals from a ground truth explosion in northern Utah made by stations in two seismic networks. The 400-station broadband seismic network USArray Transportable Array (TA) is gradually moving east across the continental U.S. through station redeployments from the trailing western edge of the network to the leading eastern edge. In June 2008, the network spanned mid-western states (including Utah) as shown in Figure 3. A second seismic network in the region the near 100-station High Lava Plains (HLP) Network was deployed on a radial line (at an azimuth 300°) from the Utah Testing and Training Range (UTTR). UTTR is the site of large surface explosions such as the Trident rocket motor ordinance disposal detonations that occur each summer. The UTTR blasts are known to occur in a small area near 41.13152°N, 112.89577°W, and shot times are known to within ~1 s (Relu Berlacu, personal communication). These reported times have been confirmed by our analysis of onset times of Pg crustal seismic waves recorded at nearby stations. The shots commonly have net explosive weight of 17,500 kg (also reported by Relu Berlacu) and routinely produce infrasound signals that are recorded seismically to a range of 800 km. By combining stations in the TA at this azimuth and the HLP network, we have recordings of UTTR blasts collected continuously from UTTR to past 1000 km.

Figure 3.

Seismic and infrasound stations operating in the vicinity of UTTR on 16 June 2008 are shown. The USArray Transportable Array (TA) comprises 400 broadband seismic stations installed on a Cartesian grid spanning 2,000,000 km2. Through individual station redeployments, the TA network is gradually moving to the East Coast and on this date was largely located to the east of UTTR. Also on this date, the very dense, 73-station, High Lava Plains network was located at an azimuth of 300° from UTTR. Stations in the TA and HLP networks in the corridor at an azimuth of 300° from UTTR that recorded a signal from the blast are highlighted in yellow. Other stations in both networks are represented by the small, hollow, circles. Three infrasound arrays (black triangles) were operating to the west of UTTR at this time, IS57 in the south, IS56 in the north, and NVIAR. The location of UTTR is marked by the red star.

[68] For this study, we have chosen a typical blast that occurred on 16 June 2008. The observations from the broadband seismic stations (Figure 3) are shown as a reduced travel time record section in Figure 4. The observed arrival branches of infrasound signals are uS, uS2, uS3, and uS4 (in order from earliest to latest, from closest to UTTR to most distant). These branches are stratospherically ducted with the subscript indicating the number of turns aloft [Hedlin et al., 2010]. The duration of the observed arrivals varies from a few seconds for the uS arrival near 200 km, to as much as 50 s for the uS2, uS3 arrivals at distances beyond 500 km from the source. Although the acoustic waveform from this event was not measured at close range (<10 km), data from four measured UTTR rocket motor ordinance disposal events in 2007 indicate peak amplitudes of greater than 40 Pa with pulse durations on the order of 1.5 s, measured at a distance of 10 km from the source [Stump et al., 2007].

Figure 4.

A recorded section from a UTTR event that occurred on 16 June 2008 is shown. All traces are from broadband seismic stations (see Figure 4). This figure also shows ray predictions shot through an unperturbed G2S model in green and rays shot through five realizations of gravity wave-perturbed G2S models in red. Although the gravity wave-perturbed rays fit the recorded data quite well (far better than rays shot through the baseline G2S model), we expect further research on the gravity wave statistics will improve the fit.

[69] To establish a baseline, two-dimensional range-dependent Hamiltonian acoustic ray tracing through a standard unperturbed but range-dependent, regional ground-to-space (G2S) data field was performed. The G2S fields were computed for this time period based on the NOAA Global Forecast System analysis fields below 35 km, the NASA MERRA analysis fields [Rienecker et al., 2011] between 25 and 75 km, and the NRL HWM07/MSISE-00 empirical upper atmospheric model above that [e.g., Drob et al., 2010]. The baseline G2S zonal wind profile at the source location for the time of the event is shown by the green profile in the left panel of Figure 5.

Figure 5.

This figure shows the unperturbed (green) and perturbed (blue) G2S zonal wind profile (left) at UTTR along with a north-south cross-section of the zonal wind gravity wave perturbations (right) for the 16 June 2008 event. The perturbations in the vertical profile (left) correspond to the location indicated by the vertical line in the 2D cross-section (right).

[70] For the baseline calculation, a 2D fan of 6001 rays at 0.01° elevation angle increments is traced. The resulting reduced travel times, i.e., the time and location where the individual rays strike the ground, are shown by the green dots in Figure 4. In general, the baseline G2S ray-tracing calculations predict the onset time for portions of the observed uS, uS2, uS3 branches, but the calculations clearly underpredict the starting point in range of each of the branches. The region between the first predicted arrival and the source is known as the classical shadow zones [e.g., Gutenberg, 1926]. In contrast to the observations, the baseline calculations also do not predict any spreading of the acoustic arrivals from the initial impulse to a dispersed wave train, which increases in duration away from the source.

[71] For comparison, two-dimensional range-dependent Hamiltonian acoustic ray tracing is performed through the same G2S atmospheric fields, but with the altitude and range-dependent perturbations from the gravity wave perturbation parameterization included (see Figure 5). As in the other acoustic-internal gravity wave interactions calculations [e.g., Gibson et al., 2010], a stationary phase approximation for the background field is assumed. In other words, although the gravity wave perturbation field is time dependent over time scales greater than a few minutes, the local gravity wavefield remains stationary with respect to the propagation of the acoustic waves.

[72] To improve and examine the statistical robustness of the results, the perturbed ray-tracing calculation, the 2D fan of 6001 rays is performed through five different random phase shift realizations of the gravity wave perturbation field. Each unique member of the gravity wavefield ensemble is generated by a circular shift of a single pre-computed 3D gravity wavefield with periodic boundary conditions. Alternatively an e− iωt temporal phase shift can be applied to the field to generate a new ensemble member with equivalent results, assuming the stationary phase approximation.

[73] In contrast to the baseline calculations, the arrival times of the rays traced through the range-dependent G2S fields that include gravity wave perturbations, shown as the red dots in Figure 4, accurately predict both the onset of each branch with range and the duration of the infrasound signals. Importantly, the qualitative characteristics (time of first arrival, time of last arrival, and overall ray density) for a given arrival branch (uS1, uS2, uS3) are qualitatively indistinguishable from member to member in the ensemble, although ray for ray, each is different. In reality for a given event, the actual acoustic wavefield will only encounter one specific gravity wavefield which results in the actual observed dispersion of wave energy, so in the context of the ensemble modeling approach, the idea is to only estimate the most probable characteristics of an infrasound wavefield in the absence of other information.

[74] As expected from wave theory, the acoustic wavefront elements are refractively scattered into the classical shadow zones predicted by the baseline calculations. The tail of the classical arrival branches is also extended to longer ranges. Furthermore, the gravity wavefield spreads the acoustic waveform in time; i.e., the effect of the gravity wave perturbations is to disperse the wavefield with a Gaussian-like distribution, such that there is a net delay in the majority of the wave components of the arrival times. This is not surprising from the standpoint of geometric acoustics as the various wavefronts which comprise the total waveform must on average follow slightly longer paths when subjected to subtle random perturbations by the internal gravity waves they encounter. While the average effect is to cause a slight net delay in arrival times as compared to the case with no gravity wave perturbations, there is also the possibility that some of the rays arrive earlier than would be predicted by an atmosphere without gravity waves. This must necessarily be the result of those rays having a slightly lower turning height and thus following a shorter path than in an atmosphere devoid of any internal perturbations.

[75] These results suggest that to first order, the general dispersion of a very narrow impulse observed at ranges greater than about 100 km can be explained in terms of geometric acoustics, i.e., in the high-frequency limit of ray theory. One important difference in our acoustic ray-tracing approach is that we account for the effects of vertical wind perturbation in the forward operator. The vertical wind perturbations are on an order of 1 m/s near the stratopause and in excess of 4 m/s near the lower thermosphere. Currently, the influence of a vertical wind component is not included in the Parabolic Equation (PE) methods utilized to also investigate the influence of fine-scale atmospheric inhomogeneities on infrasound propagation [Millet et al., 2007; Kulichkov et al., 2010; Chunchuzov et al., 2011; Green et al., 2011]. We note that we also included in-plane terrain variations in our calculation which will also result in additional dispersion through geometric focusing and spreading of already dispersed ray pairs.

6 Conclusion

[76] While available global atmospheric specifications from operational numerical weather prediction centers, as well as atmospheric climate reanalysis activities, provide adequate atmospheric specifications (particularly below altitudes of 75 km), these specifications do not explicitly resolve atmospheric gravity waves with length scales less than 50 km and time scales less than several hours. Recent research efforts have identified the necessity of accounting for the fluctuations from atmosphere's internal gravity wave spectrum in infrasound propagation calculations [Millet et al., 2007; Kulichkov et al., 2010; Chunchuzov et al., 2011; Green et al., 2011]. As it is impractical to deterministically resolve these internal gravity wave fluctuations everywhere over the globe at all times, we must rely upon a stochastic or parameterized wavefield approach with the proper spectral characteristics (cascade of length and time scales) in order to account for fine-scale atmospheric inhomogeneities much in the same way that turbulence is parameterized in aerodynamic drag calculations.

[77] We have presented a stochastic gravity wave model to account for the effects of fine-scale atmospheric inhomogeneities on infrasound propagation. It combines a ray formulation for conservative gravity wave propagation with various parameterizations for forcing and dissipation. The ray formulation takes into account refraction by the height-dependent background wind and temperature structure, full three-dimensional geometrical spreading and dispersion, correction for caustics at turning points, and wave transience. The parameterizations that are based on widely used schemes, and as necessary can be revised or amended as improved schemes [e.g., Yiğit et al., 2008; Eckermann, 2011], are developed. Comparison to other published models and application to other events is left for future work.

[78] In order to better understand and explain observed infrasound waveform characteristics (arrival time, signal duration, and amplitude), we are encouraged by the agreement between our theoretical calculations and the observations presented here. Given our results and the results of others in the scientific literature, we believe that the infrasound research community now has the fundamental understanding required to explain frequently observed infrasound signals in the classical near-field shadow zones, as well as related phenomena.


[79] The GEOS-5 data utilized in conjunction with other data sources in the NRL G2S atmospheric specification for the two examples were provided by the Global Modeling and Assimilation Office (GMAO) at NASA Goddard Space Flight Center through the online data portal in the NASA Center for Climate Simulation. The NOAA GFS, also utilized in the G2S specifications, was obtained from NOAA's National Operational Model Archive and Distribution System (NOMADS), which is maintained at NOAA's National Climatic Data Center (NCDC). We acknowledge support from the Office of Naval Research (ONR), National Nuclear Security Administration (NNSA) under contract DE-AI52-08NA28653 and DE-AC52-08NA28652, and the U.S. Army Space and Missile Defense Command (SMDC) project W9113M-06-C-0029. M.A.H. Hedlin would also like to acknowledge support provided by NSF under contract EAR-1053576.


  1. In the originally published version of this article, several instances of text were incorrectly typeset. These errors have since been corrected and this version may be considered the authoritative version of record. The details of the corrected text are as follows:In Paragraph 13 the first sentence after equation (2) should read, “The atmospheric density scale height is Hρ = −ρ(dρ/dz)−1 which varies with altitude, where ρ is the mass density”.In Paragraph 41, equation (23) should read:

    display math(23)

    In Paragraph 46, equation (28) should read:

    display math(28)