In our recent paper, the sensitivity of infrasound to the upper atmosphere is investigated using impulsive signals from the Tungurahua volcano in Ecuador. We reported on the coherent variability of thermospheric travel times, with periods equal to those of the tidal harmonics. Moreover, it was shown that the error in predicted thermospheric travel time is in accord with typical uncertainties in the upper atmospheric wind speed models. Given the observed response of the infrasound celerities to upper atmospheric tidal variability, it was suggested that infrasound observations may be used to reduce uncertainty in the knowledge of the atmospheric specifications in the upper atmosphere. In this paper, we discuss the estimation of upper atmospheric wind model updates from the infrasound data described in the aforementioned paper. The parameterization of the model space by empirical orthogonal functions is described; it is found that the wind model in the upper mesosphere and lower thermosphere can be described by a four-parameter model. Due to the small dimensionality of the model space, a grid search method can be used to solve the inverse problem. A Bayesian method is used to assess the uncertainty in the inverse solution given the a priori uncertainty in the data and model spaces and the nonlinearity of the inverse problem at hand. We believe that this is the first study in which such methods are applied to real infrasound data, allowing for a rigorous analysis of this inverse problem. It is found that the complexity of the a posteriori model distribution increases for a larger dimensional model space and larger uncertainties in the data. A case study is presented in which the nonlinear propagation from source to receiver is simulated using an updated wind model and nonlinear ray theory. As nonlinear propagation effects further constrain the propagation path, this is a way to check the physical self-consistency of the travel time inversion approach. We obtain excellent agreement between the simulated and observed waveforms.
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 Over the past decades, there have been many advances in measuring properties of upper atmospheric regions with ground-based optical remote sensing instruments, such as radio detection and ranging [Manson et al., 2002], light detection and ranging [Gardner et al., 1995; She et al., 2004], or probes on research satellites [e.g., Killeen et al., 2006; Shepherd et al., 1993; Hays et al., 1993]. Recently, the European Center for Medium-Range Weather Forecast and the National Aeronautics and Space Administration (NASA) have begun to produce specifications up to 75km altitude, based on satellite temperature soundings [Rienecker et al., 2008; Drob et al., 2010]. Furthermore, fully assimilative high-top global circulation models (GCMs) that extend into the lower thermosphere are currently under development by the upper atmospheric research community. These GCMs include the Navy Operational Global Atmospheric Prediction System [Eckermann et al., 2009], the National Oceanic and Atmospheric Administration (NOAA) Whole Atmospheric Model [Akmaev et al., 2008], and the Canadian Middle Atmosphere Model [Fomichev et al., 2002; McLandress et al., 2006]. In spite of this progress, databases with actual atmospheric measurements above the stratopause remain sparse, in comparison to the operational measurements available for the lower atmosphere. Any and all available atmospheric measurements (indirect or otherwise) are useful in further characterizing this region of the atmosphere. Furthermore, comparing data sets from multiple techniques allows for valuable cross-validation studies.
 Comprehensive observationally based specifications of wind, temperature, and composition in the mesosphere and lower thermosphere (MLT; approximately 50–110km altitude) are limited to semiempirical models such as the Horizontal Wind Model (HWM) [Hedin et al., 1996; Drob et al., 2008] and the Mass Spectrometer and Incoherent Scatter Radar Extended Model (MSIS-E) [Hedin, 1991; Picone et al., 2002]. Both HWM and MSIS-E extend from the ground to the exosphere. These models can represent the average climatology of the upper atmosphere reasonably well, but phenomena that occur on shorter time and spatial scales are not captured. The most recent edition of HWM dates from 2007 and represents over 50years of satellite, rocket, and ground-based wind observations; it is estimated that the horizontal winds in the MLT have standard deviations of at least 25ms−1[Drob et al., 2008]. Additionally, the uncertainties in horizontal wind specifications are considered to be large compared to uncertainties in temperature [Drob et al., 2010]. The dynamics of the MLT is strongly influenced by planetary waves, atmospheric tides, and gravity waves [Andrewes et al., 1987].
 Observational papers, such as Fritts and Isler , and first principles modeling work [i.e., Akmaev et al., 2008] show that the day-to-day variability of upper atmospheric winds due to the atmospheric tides over a specific region can be very significant. It has been observed that the modulation of the amplitude and phase of the migrating and nonmigrating tides have periodicities corresponding to planetary waves (10–30days). Therefore, the variability of the atmospheric tides has been ascribed to wave-wave interaction of tidal and planetary waves. Other proposed mechanisms include planetary wave filtering and modulation of small-scale gravity waves [Akmaev et al., 2008]. The findings of Fritts and Isler  and Akmaev et al.  are supported by infrasound data presented in our recent paper [Assink et al., 2012], in which we show that thermospheric travel times have a coherent variability with periodicities equal to the atmospheric tides. The travel times have been obtained using recordings from arrays that were deployed as part of the Acoustic Surveillance for Hazardous Eruptions (ASHE) [Garcés et al., 2007] project in Ecuador. The arrays are located in Riobamba (RIOE) and Lita (LITE), respectively at 37km and 250km from the Tungurahua volcano. The stratospheric arrivals that are observed during the same time are shown to have a relatively constant travel time when compared to the thermospheric travel times. It is also shown that the estimated error in thermospheric travel time is in accord with the estimated uncertainties of HWM07, in the case that thermospheric arrivals are predicted at LITE. More often, however, no thermospheric arrivals are predicted using acoustic propagation modeling techniques for the given atmospheric state. Conversely, the stratospheric arrivals are typically predicted within 1% error; this is not surprising as atmospheric specifications below the stratopause are known to be more accurate than above the stratopause [Assink et al., 2012]. Given the fact that lower atmospheric arrivals are typically much better predicted than upper atmospheric arrivals, we regard errors due to assumptions in the propagation codes to be negligible compared to the errors due to uncertainties in the atmospheric state. Given the observed response of the infrasound celerities to upper atmospheric tidal variability, it is suggested that infrasound observations may be used to further reduce uncertainty in the knowledge of the atmospheric specifications in the upper atmosphere. It is the objective of the current paper to estimate upper atmospheric wind model updates, based on the infrasound observations and the a priori wind database given by the HWM07 specifications. Thus, this paper is a direct continuation of the work developed in Assink et al. . Such wind model updates may be useful in reducing uncertainty in the knowledge of wind speed models in the region above the stratopause.
 Many other studies have demonstrated that infrasound is sensitive to the dynamics of the upper atmosphere. Donn and Rind [Donn and Rind, 1972; Rind, 1978] presented statistics of tidal variability in the thermosphere using continuous recordings of microbaroms [Posmentier, 1967] obtained over 10years. Moreover, the sensitivity of infrasound propagation to sudden stratospheric warmings (SSWs) [Charney and Drazin, 1961] was demonstrated. With the advent of global-scale atmospheric specifications and infrasound propagation models, it has become clear that theoretical acoustical predictions based on the atmospheric specifications are not always in accord with observations, both on seasonal and diurnal time scales [Le Pichon et al., 2005a; Matoza et al., 2011]. While many studies have reported on smaller discrepancies for infrasound arrivals that have propagated through the lower and middle regions of the atmosphere [i.e., Antier et al., 2007; Evers and Haak, 2007; Evers et al., 2012], other observational studies have indicated that significant misfits can be present [Assink et al., 2011a; Gainville et al., 2010; Hedlin et al., 2012]. Apart from the zeroth order structure, understanding the influence of small-scale atmospheric structure (e.g., due to gravity waves) on stratospheric arrivals is a current topic of interest [Kulichkov, 2010; Chunchuzov et al., 2011a; Hedlin et al., 2012; Drob et al., 2013].
 In addition to reducing uncertainty in the knowledge of upper atmospheric wind speed models, wind model updates are beneficial in the context of infrasound propagation studies, as the atmospheric specifications are based on spatially and temporally averaged measurements that do not (fully) explain infrasound observations. To further illustrate this, we apply the wind model updates to simulate nonlinear acoustical propagation effects for a thermospheric arrival in this paper. Nonlinear propagation effects can be used in the estimation of explosion yield [Waxler et al., 2012].
 Previously, inverse methods have been used in atmospheric acoustics (low audible frequencies) studies to estimate atmospheric specifications in the lower atmosphere [e.g., Vecherin et al., 2006, 2007; Blom and Waxler, 2012]. Relatively few studies have focused on the inverse problem of estimating wind speeds from infrasound data. In Le Pichon et al. [2005b], the authors minimize the misfits between simulated and observed bearing deviations from continuous volcano infrasound using an inversion procedure. A simple Gaussian correction factor is used to parameterize the model space. Deviations up to 50ms−1are estimated, but no uncertainty analysis is provided. In Drob et al. , a more comprehensive framework is described for estimating upper atmospheric wind parameters from an observational network. The authors propose a parameterization of the horizontal wind parameters using empirical orthogonal functions (EOFs). An iterative method based on the Levenberg-Marquardt algorithm, in combination with ray theory, is used to minimize the weighted orthogonal distance regression problem. To avoid local minima, the iterative method is restarted from various different regions in the parameter space. Lalande et al.  develop an alternative method to solving the inverse problem by linearizing the theory operator explicitly. The inverse problem is solved using an iterative least squares method and has been successfully applied to a synthetic problem. This method is limited to small perturbations in wind or temperature, but the method makes the inverse problem more tractable.
 Rather than the more generalized studies by Drob et al.  and Lalande et al. , we focus here more specifically on the inverse problem of estimating upper atmospheric wind speed models using the Tungurahua data set, such as described in Assink et al.  using the HWM07 database as a priori model space. We make use of a grid search method in combination with Bayesian statistics to analyze the uncertainties that are associated with solving the inverse problem. Such a mapping provides enhanced insight in the nature of the inverse problem, and no assumptions with respect to the nonlinearity of the inverse problem have to be made. However, this comes at a computational cost. Fortunately, grid search methods are manageable for model spaces of sufficiently small dimension, such as the model space considered in this work.
 The remainder of this paper is organized as follows. The next section introduces the inversion framework. In section 3, the parameterization of the vertical wind profiles by EOFs is described. Thereafter, we estimate wind speed updates based on the HWM07 climatology for a specific explosion from the Tungurahua data set. In section 5 we apply a nonlinear ray tracing propagation scheme to the updated atmospheric specifications. We find excellent agreement between the predictions and observations. Section 6 discusses and summarizes our findings.
2 Inverse Methods
 The problem of estimating upper wind model updates from infrasound data is an inverse problem that is similar to the seismic tomography problem [Romanowicz, 1991]. In this paper, we predominantly make use of reviews by Snieder and Trampert  and the book by Tarantola . Throughout this paper, the physical system will be referred to as the model, which is the common denotation in the field of inverse theory. We explicitly mention this, as the term model is somewhat ambiguous as it is also used to denote the physical theory (“forward model”) used in simulations. We will use the term theory to describe the latter throughout this paper. In the ideal case, an exact theory G predicts data d from a model m: d=G(m). The theory operator may either be linear or nonlinear. It is one of the objectives to find the set of models that explain the data by minimizing an objective function. This is called the estimation problem.
 Inverse problems are classified as linear or nonlinear, depending on whether the theory operator is a linear or nonlinear function of the model parameters. This nonlinearity is different from the nonlinear propagation effects in (infra)sound propagation. Propagation is said to be linear if the defining acoustics equations are linear. The ray theory operator is often linearized [Lalande et al., 2012] in order to use ray theory in a linearized inversion framework. Naturally, nonlinear problems are more complicated than linear problems and require a more careful analysis than linear problems do. These complications are due to a more complicated form of the objective function as a function of the model parameters. It may be that multiple minima occur, that the objective function is not defined, that the function is discontinuous, or that the function does not depend on the model parameters at all [Snieder and Trampert, 1999]. For example, discontinuities in the objective function may occur when stratospheric arrivals are used to invert for specifications at stratospheric altitudes using ray theory. Such discontinuities correspond to the sudden loss of stratospheric returns, for example, due to a decreasing stratospheric jet [Lalande et al., 2012].
 Geophysical inverse theory is intimately connected with an uncertainty analysis, as the presence of noise, incompleteness of theory, and limited imagination of the modeled physical system propagates into the inversion result. This causes most inverse problems to be ill conditioned and/or ill posed. The analysis of the errors associated with the estimated models is called the appraisal problem. In the case of an ill-conditioned problem, small variations in the data cause large errors in the obtained model. For ill-posed problems, there is typically not a unique solution, if there is a solution at all. Problems can be ill posed due to various reasons, such as the choice of parameterization or the choice of the theory operator. Some model parameters may be better constrained by the data than others which contribute to the ill-posedness. Because inverse problems may be ill posed and/or ill conditioned, there is a need to regularize both model space and data space. In a Bayesian framework, the regularization takes place by using covariance matrices that describe the a priori information [Snieder and Trampert, 1999].
 To solve the estimation problem, various methods exist. For truly linear problems, a system of equations is to be solved. Weakly nonlinear problems can be linearized; in the case of travel time tomography, one typically makes use of Fermat's principle to linearly relate perturbations in travel time to perturbations in medium velocity along a reference ray. This is valid for small velocity perturbations only; in general the ray position is dependent on the medium velocity, and larger perturbations require the solution of a nonlinear problem. Optimization algorithms, such as conjugate gradient and Levenberg-Marquardt iterative methods, can be used to minimize the objective function [Lalande et al., 2012; Snieder and Trampert, 1999]. For truly nonlinear problems, optimization algorithms may fail due to the complicated nature of the objective function [Sambridge, 1999]. Often, direct search methods are the only option to search the model space for potential solutions. Large populations of models can be generated, for example, using Monte Carlo methods [Mosegaard and Tarantola, 1995], to find solutions that fit the data. Generating populations can be computationally demanding, especially when the theory is numerically involved and/or the dimensionality of the model space is large. In this paper, we make use of an exhaustive search in which an N-dimensional grid of models is generated. The parameterization of the model space is described in the next section. We make use of a Bayesian formulation to address the appraisal problem.
 In the Bayesian formulation, the probabilistic prior information is combined with the data misfit in order to update the a priori model distribution to an a posteriori model distribution. The a posteriori model distribution is the posterior conditional model distribution φM given the a priori data ρD and model probability densities ρM. We can write this mathematically, following Bayes' theorem [Tarantola, 2005]:
 In equation (1), x represents the independent variables used in the theory. It is the goal to have the posterior model distribution more constrained than the prior distribution. The description allows one to find credible regions of the model parameters β given the data and use the posterior mean as model estimate with the standard deviation as the error. It is assumed in this paper that a priori data and model probabilities can be modeled using Gaussian distributions:
 Here and mprior are the estimated and prior wind model, respectively. CD and CM are the covariance matrices, representing the uncertainties in the data and model space, respectively. If the relation is linear, the posterior probability is Gaussian. The further the theory operator is from linear, the further φM(β) is from Gaussian [Tarantola, 2005]. As the maximum of likelihood is found at the maximum of the probability density function (pdf), the most probable model and the associated uncertainties are most readily found for linear inverse problems or for problems for which the theory operator can be linearized. For stronger nonlinearities, various regions of likelihood must be studied in further detail. In the case of the Gaussian additive noise assumption, the minimum least squares error solution is the same as the maximum-likelihood estimate.
3 Parameterization of the Model Space
 We will focus in this section on parameterization of the model space, which is represented by the relevant atmospheric specifications. In general, infrasound propagation is sensitive to variations in temperature and wind [Georges and Beasley, 1977]. Over regional distances, vertical gradients in temperature and wind typically dominate, and the atmosphere may be assumed to be range independent [Drob et al., 2003]. The approximation of range independence holds for the propagation from Tungurahua to the far-field array, located approximately at 250km north of the source.
 We further assume that the error in the infrasound predictions are due to the uncertainties in the horizontal winds only. This is typically justified by the argument that the vertical temperature profile is fairly well known with uncertainties on the order of 10–15K in the upper atmosphere [Drob et al., 2010]. Additionally, the effect of temperature variations δT on the adiabatic sound speed cT is relatively small, due to the mean temperature T0 being large compared to temperature variations δT. Given the uncertainty of 10–15K, the expected deviation from the sound speed in the mesosphere and lower thermosphere is estimated to be smaller than 5%. This can be understood from the following expansion:
 This analysis assumes that the ideal gas law holds in this region.
 In contrast, the expected uncertainty of the wind is much more significant (≫ 5%), with estimated values on the order of 30ms−1. However, as temperature and wind gradients are coupled, it is incorrect to keep the temperature fixed while perturbing the winds. Self-consistency between these quantities is to be preserved in the atmospheric specifications. In regions of the atmosphere where the geostrophic and hydrostatic balance applies, the thermal wind relation can be used to relate temperature and wind gradients. However, this balance does not hold for the region of interest, near the equator and in the upper atmosphere [Andrewes et al., 1987]. For those regions, rigorous methods are required to model the wind and temperature gradient relations as these quantities are dynamically related. Inverting for an effective sound speed would equally well introduce errors in terms of propagation modeling [Assink et al., 2011b]. In this paper, we choose to invert for horizontal wind perturbations only. We believe that such perturbations should be partially translated into temperature gradient perturbations in order to maintain consistency in the atmospheric specifications. However, we consider this a topic of future work.
 We base the parameterization of the model space on HWM07, as it is based on 50 years of upper atmospheric wind data [Drob et al., 2008] and thus adds to the confidence in the solution. In this study we are concerned with estimating deviations from HWM07, using infrasound observations. Similar to the studies by Drob et al.  and Lalande et al. , we make use of EOFs to parameterize the wind fields. The EOFs are formed based upon the statistical properties of the temporal and vertical variability of the wind fields. It will be shown later that a fairly limited number of EOFs is required to describe the Horizontal Wind Model, as the specifications in this region are semiempirical and thus repetitive. One could select a different parameterization, such as Gaussian correction factors [Le Pichon et al., 2005b], but such a parameterization does not involve statistical information of the wind fields.
 We form EOFs by performing a singular value decomposition (SVD, Golub and van Loan ) on the wind fields from the Ground-to-Space (G2S) database, as a function of time and altitude (M profiles with N levels). The wind fields in the G2S database are completely described by HWM07 in the MLT. For this study, we make use of atmospheric specifications retrieved from the 6-hourly G2S specifications [Drob et al., 2003; Drob et al., 2010] above Quito, Ecuador. The specifications are based on the 4× daily NOAA operational global forecast system (GFS) analysis products from 0 to 45km (1 hPa) [Kalnay et al., 1990], the 4× daily stratospheric analysis from 35 to 75km (10 to 0.01 hPa) from the NASA Goddard Space Flight Center, Modern-Era Retrospective Analysis for Research and Applications system [Rienecker et al., 2008], and above 75km the HWM07 empirical model [Drob et al., 2008].
 The SVD decomposition is described as
 Here M is the matrix of zonal or meridional wind data without mean mav(z) (M×N), U is a matrix of basis function coefficients (M×N), Σ is a matrix of singular values (N×N), and V is a matrix of singular vectors (N×N). The singular vectors, scaled by the proper singular value from Σ, are approximations to the orthogonal functions ψ(z). Every row in the coefficient matrix U contains the coefficients βj,q for a given wind profile q=1…M. Thus, a zonal or meridional wind profile mq(z) can be constructed by the following linear superposition of empirical orthogonal functions:
 In general, the sum can be truncated after a small number of basis functions, since only the largest singular values contribute to the sum. However, such truncations may introduce a bias in the model space, which propagates into the estimated solution [Snieder and Trampert, 1999]. In this work, we consider variations in the coefficients corresponding to the meridional EOFs of the highest order, while keeping the other coefficients fixed to the a priori value. Thus, the full set of coefficients is used to compute the wind field and no bias is introduced. Because the inverse problem consists in determining the expansion coefficients βj,q, it is important to make a choice for which expansion coefficients to invert, as the amount of infrasound observables is typically smaller than the number of coefficients necessary to reconstruct the full profile. This is certainly the case in the application of this method to the Tungurahua data set, as the far-field observables are limited to one station only. In this study we predominantly make use of geometrical constraints, travel time, bearing, and trace velocity measurements.
 We are interested in parametrizing the wind in the MLT, since the observations presented earlier [Assink et al., 2012] show that the misfit between predicted and observed travel time and receiver locations is significant for infrasound propagating through this region. In contrast, the stratospheric arrivals are relatively well predicted, with travel time prediction errors within 1%. As the wind specifications in the MLT are limited to the semiempirical HWM07 model, we can limit the G2S database to 1year of specifications, as HWM07 has no interannual variability.
 The parameterization of zonal (u) and meridional (v) winds, from 0 to 150km and from 80 to 150km, is shown in Figure 1. The eigenvalue spectra, obtained from Σu,v, are shown in Figures 1a and 1d, respectively. These frames show that while all of the variability in the 0–150km specifications is captured with about 32 basis functions, even fewer functions—about 12—are required to describe the specifications in the upper mesosphere and lower thermosphere (80–150km). This is consistent with the analysis from Drob et al. . An example EOF representation of G2S zonal and meridional wind profiles above Quito, Ecuador on 15 July 2006 is shown in Figures 1b and 1e. While the finer details are not fully captured by the EOF representation, the average structure is reasonably well resolved with eight functions. Figures 1c and 1f show an excellent agreement between the original profiles and the EOF representations with eight basis functions in the upper atmosphere, both for the zonal and the meridional winds. The basis functions that are used in Figures 1c and 1f are determined uniquely from HWM specifications; the corresponding eigenvalue spectrum is shown in Figures 1a and 1d with a blue line. This further illustrates that HWM is a semiempirical model of the upper atmosphere that can be described by a limited number of coefficients. The number of coefficients required to describe the model is much smaller than the number of coefficients required to describe G2S, which shows day-to-day variability below 65 km altitude.
 The meridional wind basis functions that correspond to the largest singular values are almost completely located in the upper atmosphere. In contrast, the zonal wind basis functions of the same order have appreciable structure around the stratopause in addition to the structure in the upper atmosphere. This is logical, since much of the zonal variance is associated with the behavior of the zonal wind field around the stratopause. This difference is reflected by the eigenvalue spectra of the G2S (0–150km) and HWM (80–150km) decompositions, shown in Figures 1a and 1d. The spectra have been normalized to the maximum G2S (0–150km) eigenvalue. The eigenvalue spectra for the meridional winds are approximately the same for the first five basis functions, after which the HWM spectrum decays quickly. In contrast, the HWM zonal wind eigenvalue spectrum is reduced overall, compared to the G2S zonal wind eigenvalue spectrum.
 Strong crosswinds can lead to horizontal refraction out of the azimuthal plane [Georges and Beasley, 1977]. However, variations in the raypath, travel time, and trace velocity are determined to first order by the in-plane specifications. As the propagation path from Tungurahua to LITE is approximately northward, this means that we seek variations in the meridional wind profiles to explain the propagation to LITE with the observed travel times. Zonal wind variations would predominantly influence the measured bearing. In a study by Le Pichon et al. [2005b], the authors optimize for zonal wind profile deviations based on bearing misfits. However, both variations in zonal and meridional model spaces could be accounted for to higher order. In this study, we keep the zonal wind field fixed to the a priori model.
 Figure 2 shows the meridional wind field above Quito, Ecuador during the second week of July 2006. Figures 2a–2c show the meridional wind field, parameterized by 2, 3, and 4 orthogonal functions from top to bottom. Figure 2d shows the original meridional wind field, as obtained from the G2S database. The thermospheric meridional wind structure, dominated by the semidiurnal and the diurnal tide, can be well represented by a limited number of EOF parameters.
4 A Search in G2S Model Space
 We focus in this section on the computation of posterior pdfs for various parameterizations of the meridional wind field and data uncertainties, following equation (1). Our full data set consists of a series of travel time, trace velocity, and bearing measurements from explosive eruptions emitted by the Tungurahua volcano measured at RIOE and LITE. RIOE is located 37km to the southwest of the Tungurahua at 2.7km altitude. LITE is located 9.3km east and 250.8km north of the source and is situated at 1.2km above mean sea level (MSL). The acoustic source is assumed to be at z0=5.1 km above MSL, which is a little above the crater rim. The uncertainties in source and receiver locations are estimated to be negligible. We have reported on 3 weeks of explosive activity in August 2006, January–February 2008, and May–June 2010 in our recent paper [Assink et al., 2012]. For the computation of the pdfs in this paper, we consider a series of explosive signals measured on 15 July 2006, which marked the beginning of an explosive episode of the Tungurahua volcano.
 Waveforms measured at RIOE and LITE are shown in Figure 3. The measurement at RIOE is assumed to be the direct arrival from Tungurahua; at LITE three different arrival groups can be distinguished. The travel time from Tungurahua to RIOE is estimated using a direct wave path and a nominal sound speed of 340ms−1, from which the signal onset time can be estimated. Once the onset time is known, the travel time from Tungurahua to LITE can be determined. The three different groups are assumed to be stratospheric (Is), mesospheric (Im), and thermospheric (It) signals. Note that the thermospheric travel time decreases with time of day, a characteristic that we described earlier in Assink et al. . The stratospheric waveform has a small amplitude and generally has an extended duration composed of several shorter period cycles. In contrast, the thermospheric arrival consists of a relatively clear, single waveform of fairly long period. The structure of the stratospheric arrival is indicative of interaction with small-scale structures in the stratosphere. The mesospheric arrival is observed to arrive in between the stratospheric and thermospheric arrivals. The waveform varies significantly compared to the thermospheric arrival, suggesting that the arrival is a return from mesospheric altitudes, possibly due to reflection from a localized shear layer [Kulichkov, 2010]. Vertical gradients in wind magnitude and direction due to shear in the middle atmosphere can be large enough to form reflective layers throughout the middle atmosphere [von Zahn and Widdel, 1985]. Such gradients are not captured by HWM, which explains why these arrivals are typically not predicted. Note that the mesospheric travel times also decrease with time of day, similarly to the thermospheric arrivals. This decrease is not as strong as that of the thermospheric arrivals; this is in line with the atmospheric tides being more strongly excited at greater altitudes due to the exponentially decreasing density in the atmosphere.
 We assume that the set of observables can be represented by pdfs. Here we represent the travel time, trace velocity, and bearing (deviation) pdfs by a Gaussian distribution, centered around the measured value with an uncertainty of σ. In this paper, we consider σt=1.0 s and σt=2.0 s for travel time, σc=5.0 ms−1and σc=25.0 ms−1for trace velocity, and σφ=0.5° for bearing. These uncertainties are typical for the array geometry and sample rate of the Ecuadorian arrays and may be estimated by measuring the deviation of an incoming signal from the plane wave model [Szuberla and Olson, 2004]. We choose two values of uncertainties for travel time and trace velocity, as we are interested in the dependence of the inverse solution on these uncertainties. We expect that the variability in these quantities are largest, due to the meridional propagation path. As a case study, we consider the thermospheric arrival associated with the first eruption at 0601 UTC on 15 July 2006, shown in Figure 3. The arrival has a travel time of 1173 s, a trace velocity of 430ms−1, and a bearing deviation of 1.8°. The slowness parameters are estimated with a standard time-domain beamforming technique, using Fisher statistics [Melton and Bailey, 1957].
 An ensemble of meridional wind models is generated by varying orthogonal function coefficients between −0.1 and 0.1. This range is chosen after analyzing the statistics of a year of these coefficients obtained in the SVD decomposition. In this paper, we consider variations in the first four meridional EOF coefficients. The other coefficients are kept fixed to the a priori value; the full set of coefficients is used to compute the meridional wind field, following equation (4). Throughout the rest of the paper, we will refer to these models as the two-parameter, three-parameter, and four-parameter models. We further regularize the model space by a uniform pdf: only models within 50ms−1 of the a priori G2S model are considered for further analysis. Distributions of travel time, trace velocity, and bearing deviation are computed for the remaining ensemble of models using ray theory for a stratified medium with background flow [Brekhovskikh and Godin, 1999].
 Raypaths and travel time can be computed using the following ray equations:
 Here r denotes the position along the raypath, and ν is the normal to the wavefront, defined to be the surface of constant phase through the position r. In equation (6), the Einstein summation convention is used. The variable s denotes the ray length from the origin to position r. The wave normal ν is not parallel to the tangent to the ray, which is described by , except for a windless atmosphere. is the group velocity, which describes the propagation velocity along the ray, c and v0 are the temperature-dependent (adiabatic) sound speed and the background horizontal wind, respectively. The following initial conditions (s=0 m) are used:
 Here θ is the angle with respect to the horizontal, and φ is the azimuthal angle, clockwise from north. ν can be obtained from the Eikonal equation:
 To compute travel times, we evaluate the following integral along the raypath:
 Eigenrays are determined using a Levenberg-Marquardt search algorithm. The source location is assumed to be known exactly; eigenrays that fall within 0.1km of the center of the array are considered. This is justified, given the aperture of the array (150 m).
 Histograms showing distributions of travel time, trace velocity, and bearing deviation for a four-parameter meridional wind ensemble are shown in Figure 4. Clearly, a wide but limited distribution of values exist that could possibly be simulated at LITE, given the parameterization and the theory. Travel times are confined to the range 1140 to 1190s, trace velocities to lie between 420 and 510ms−1, and bearing deviations to lie between 1.5 and 2.0°. Note that the trace velocity distribution deviates from a Gaussian-like shape. This is related to the relatively large geometrical spreading of the rays that are associated with large trace velocities. Travel time, trace velocity, and bearing deviation distributions that we consider for the 0601 UTC eruption are also shown in Figure 4. All models that fall within the joint data distributions explain the data. Note that travel time puts a much narrower constraint on the model space compared to trace velocity. Comparatively, the parameter is much better constrained. Also note that the variation in bearing deviation falls completely in the noise that is assumed for this quantity. This can be explained by the invariance of the zonal wind field, which influences bearing deviation for meridional paths.
 A posteriori model distributions are computed using equation (1) and are shown in Figures 5, 6, and 7 for the two-, three-, and four-parameter models, respectively. The model with the maximum likelihood is represented by the black diamond. The probability is computed for the various uncertainties in the data space considered. Clearly, for less constrained data, the posterior distribution broadens. For the two-parameter meridional wind model a clear maximum can be found; the posterior distribution has a Gaussian-like shape. Increasing the dimensionality of the model space leads to a broadening of the posterior distribution. In addition, secondary maxima appear in the posterior distribution and the distribution departs from a Gaussian-like shape, indicating the need for an intrinsically nonlinear inversion procedure. Nonlinear inverse problems are complicated to solve when a local-search algorithm is used, such as a conjugate gradient method [Lalande et al., 2012]. Typically, the various maxima need to be visited and considered separately [Tarantola, 2005]. In addition, the initial guess is often crucial in arriving at the global minimum of the objective function. Therefore, nonlinear inverse problems are more typically tackled by direct search methods [Sambridge, 1999].
 The posterior model broadening can be explained as follows. The following two key results are inevitable when additional model parameters are added to a stochastic inversion problem when a component of the model m, as in our case, is formed as a linear combination (e.g., m=Aβ) of parameters β where A is a matrix. Firstly, the fitting residual for the optimal value of the parameters must either decrease or remain the same. This is self-evident from the linear formulation. A choice of zero for the additional parameters simply results in the same problem as for the case of fewer parameters. Thus, the residual can only decrease or remain the same. Secondly, since the parameter β is a stochastic variable, the linear combination m is also a stochastic variable. It is straightforward to show that the variance of a linear combination of stochastic variables can only increase or remain the same as additional stochastic variables are added. Therefore, the a posterior probability density will broaden. This is consistent with our result.
 Figure 8 shows the same information as Figures 5, 6, and 7. Here the basis function coefficients have been multiplied by their respective basis functions and plotted as a function of height and posterior probability. The model with the maximum probability is shown with the red line; the blue line shows the original G2S profile. As noted before, the uncertainty in the inverse solution grows with the dimensionality of the parameterization and the uncertainty in the data space. However, all the models indicate that the magnitude of the original meridional tidal oscillation has been underestimated by 20 to 40ms−1, depending on the chosen parameterization. The two-parameter meridional wind model predicts the highest deviation from the a priori model, whereas the four-parameter model predicts the smallest deviation from the a priori model.
 The choice for optimizing up to four parameters originates from the analysis of the eigenvalue spectrum of Figure 1 and the comparison of the original atmospheric specifications with the EOF parameterization as shown in Figure 2. From this, we have concluded that HWM's thermospheric meridional wind structure, dominated by the semidiurnal and the diurnal tide, can be well represented by a limited number of EOF parameters. Figure 9a shows the vertical distribution of the first six meridional EOFs. Indeed, the thermospheric meridional wind structure is well parameterized by a limited number of EOFs; the vertical structure of the sixth EOF in the thermosphere is very small compared to the first five EOFs. Figure 9b shows that the inclusion of a fifth parameter in the inversion procedure only changes the maximum-likelihood result in the lower thermosphere slightly. An overview of maximum likelihood is provided in Table 1. The maximum likelihood increases from 99.8% to 100.0% when the fifth inversion parameter is considered. For a five-parameter search, the posterior distribution is somewhat widened (Figure 9c). We have found that grid searches are still manageable for four parameters. More sophisticated search methods that sample the model space more efficiently (i.e., the Neighborhood Algorithm, see Sambridge ) are advised for larger model spaces; this is a topic of future work.
Table 1. Maximum Likelihood Associated with the Various Inversion Runs
Number of Parameters Varied
 The analysis developed in this section shows that this acoustic remote sensing problem is generally nonlinear and that the inversion result is strongly dependent on the choice of the parameterization and the uncertainties in the acoustic data. Whereas the solution is fairly well constrained for a parameterization of the upper atmosphere with a fairly low resolution, secondary maxima appear when the level of detail in the model or the uncertainty in the data is increased. More sophisticated global search methods are then required to avoid local minima in the objective function.
 In addition to reducing uncertainty in the knowledge of upper atmospheric wind speed models, wind model updates are beneficial in the context of propagation studies. To further illustrate this, we apply the wind model updates to simulate nonlinear acoustical propagation effects for a thermospheric arrival in the next section.
5 Application of Updated Profiles to Nonlinear Propagation Studies
 In this section, we make use of the updated meridional wind profile to simulate the propagation from Tungurahua to LITE, including the nonlinear stretching effects that are observed [Rogers and Gardner, 1980]. It is known that the thermospheric waveform shape is produced primarily by the vertical structure and the intrinsically nonlinear nature of sound propagation in a rare gas, such as in the MLT.
 The context of this application is that the authors are involved in an effort to determine signal source strength from the period lengthening characteristic of the strong (due to decreased ambient density) nonlinear propagation in the thermosphere [Waxler et al., 2012], following the suggestion by Kulichkov . It has been found that while the amplitude of thermospheric returns is quite sensitive to atmospheric attenuation, the signal duration (for sufficiently large explosions) is not [Waxler et al., 2012]. It seemed natural to apply the newly developed yield determination method as a consistency check of both methods. The logic is as follows: first, updates to the meridional thermospheric wind speed are estimated using signal travel times and waveform parameters following the method described in the previous section. Second, the source yield is estimated using the observed signal duration. Once the source yield has been estimated, the method provides a predicted waveform for the observed thermospheric return.
 Figures 10a and 10b show the adiabatic sound speed and the wind profiles. The updated meridional wind model is plotted with a solid line, and the original profile is plotted with a dashed line. The wind model used is selected from the ensemble of four-parameter models as the one with the maximum likelihood, computed with the posterior distribution analysis discussed in section 4. A deviation of about 25ms−1 from the a priori meridional wind model is estimated. Figures 10c and 10d show simulations using a wide-angle high Mach number modal propagation model [Assink et al., 2011b] for f=0.1 Hz. The full-wave simulation is used as a check for diffraction around 0.1Hz. In all the simulations, the nominal values of the Sutherland and Bass  attenuation coefficients have been used. The simulations using the original profiles are shown in Figure 10c; the updated specifications are used in the simulation of Figure 10d. Figure 10d also shows the eigenray between Tungurahua and LITE.
 Nonlinear ray theory [Robinson, 1991; Gainville, 2008; Lonzaga et al., 2012] is used to model the nonlinear propagation effects due to the propagation in the rarified regions of the atmosphere. The nonlinear ray theory used is the so-called weak shock theory approximation. Raypaths and travel times for weak shock theory are the same as for linear models; only amplitudes differ. Thus, linear ray theory is used to compute propagation paths and travel times. The nonlinear propagation effects include the stretching and steepening of an incoming waveform which deforms the shape of the pulse [Pierce, 1981; Rogers and Gardner, 1980]. Due to the stretching, low-frequency content is created that is not part of the source spectrum. Because of the steepening, high-frequency harmonics are generated. The higher frequencies are absorbed while propagating in the thermosphere, whereas the lower frequency content is not much affected by the absorption. Therefore, the dominant frequency of a thermospheric arrival is determined by the low-frequency part of the spectrum. Varying the source amplitude influences the amount of nonlinearity along a raypath, as nonlinear propagation effects are more significant for larger pressure disturbances. The peak pressure at 1km is used as an input parameter in the nonlinear raytracing model. The raypaths are shown in Figure 11a; the eigenrays and caustic surfaces are shown in blue and red, respectively. The raypath of the simulated arrival is depicted by the thick blue line; the arrival that is associated with the other eigenray has a much larger transmission loss (+ 60 dB) due to geometrical spreading and absorption effects and is most likely not observed.
 The peak-to-peak pressure is varied to obtain a match in the dominant period between the measured and simulated thermospheric waveforms at LITE for the 0601 UTC event. The resulting thermospheric waveform simulation is shown in Figure 11b. The simulation output is shown in blue and matches up very well with the measurements, shown in red. A 180° phaseshift has been applied, as the raypath has passed through a caustic point of the second order [Červený, 2001]. Note that the arrival around 1100s is interpreted to be a reflection off a wind shear layer, which is not captured by ray theory. In recent work by Chunchuzov et al. [2011b], a theory has been developed that discusses the correspondence of mesospheric fine-scale structure and the observed infrasound signal. We believe that the inversion of such structure using such arrivals and a more complete full-wave theory could be carried out in a sequential step.
 Comparing the simulated versus the estimated waveform at 1km from the source, we find a good match (Figure 11c). The estimated waveform at 1km has been obtained by assuming spherical spreading from Tungurahua to RIOE and correcting for the loss due to geometrical spreading. The simulated waveform is slightly larger than the waveform measured at RIOE, corrected for spherical spreading. Although the match is good in this case, it is important to mention that we have assumed a simple geometry and have neglected the influence of local meteorology on the near source field, both of which can have a significant effect [Waxler et al., 2011]. Many sensors are typically required in the near vicinity of the source to study the radiation pattern in the near field [Bonner et al., 2013]. Although the Ecuadorian infrasound arrays have provided useful data for propagation studies, the arrays were deployed as part of the Acoustic Surveillance for Hazardous Eruptions (ASHE) [Garcés et al., 2007] project to monitor volcano activity, not to study near-field acoustic radiation in detail.
 The propagation modeling presented in this section, using updated wind profiles obtained with acoustic data, shows that the simulated waveforms are consistent with the waveforms measured at RIOE and LITE. This also suggests that dominant frequency is another observable that can be used in combination with travel time, bearing, and trace velocity to reduce uncertainty in the knowledge of upper atmospheric specifications. However, the dependency on source amplitude and the associated intense calculations make inclusion of dominant frequency in the procedure less attractive for now. Therefore, we recommend a consistency check as demonstrated in this section instead. Since the signal duration, and thus the yield estimation, is insensitive to the attenuation, we feel that the fairly good correspondence of the source yield determination with the near-field signal observation can be taken as an indication of the validity of the method.
 In this paper, we have introduced a Bayesian method to estimate upper atmospheric wind speed updates, using an a priori wind model database and infrasound observations of explosive eruptions of the Tungurahua volcano in Ecuador on two arrays. A grid search method in combination with Bayesian statistics is used to search for potential solutions in the model space; similar methods have been used before in other geophysical studies [Sambridge, 1999; Tarantola, 2005]. We believe that this is the first study in which such methods are applied to real infrasound data, allowing for a rigorous analysis of the inverse problem.
 We seek to minimize travel time, trace velocity, and bearing misfits that have been determined in an earlier study [Assink et al., 2012]. The propagation is essentially northward, which means that travel time and trace velocity are determined to leading order by the adiabatic sound speed and the meridional winds, while bearing deviations are determined by the zonal winds. The Tungurahua data set is characterized by accurate travel time measurements, whereas the trace velocity and bearing measurements are more likely biased due to the array geometry and degrading wind filters. Hence, we choose to invert for meridional winds only.
 The theory operator is a matter of discussion. Ray theory is computationally fast and works well away from shadow zones, caustic regions and small-scale structure. Due to the high-frequency nature of the geometrical acoustics solution, the characteristic space scale of the medium parameters variation is required to be much larger than the acoustic wavelength [Brekhovskikh and Godin, 1999]. Before solving the inverse problem, it is important to understand what part of the data set is likely to be explained by the theory. Full-wave methods should be used when diffractive effects are considered, such as interaction with small-scale structure in the atmosphere [Chunchuzov et al., 2011a] or reflections from sharp wind gradients [Kulichkov, 2010]. We believe that the inversion of such structure using such arrivals and a more complete full-wave theory could be carried out in a sequential step. Here the use of ray theory is justified because we are interested in variations of the tidal winds, which are large features in the upper atmosphere.
 Uncertainties in the priori model and data space are used to make statistical inferences about the posterior model space. The ensemble is generated through realizations from the database of empirical orthogonal functions that are obtained through a singular value decomposition of a year of G2S specifications. This is justified, as our main interest is in the region of the atmosphere where the specifications are described by semiempirical models (MSIS-E 00, HWM07), for which most of the variability is captured in a year. In this paper, we assume that the adiabatic sound speed and zonal wind can be kept fixed while varying the meridional wind field. In reality, wind and temperature profiles are dynamically coupled [Andrewes et al., 1987]. We plan on future inversion studies in which the self-consistency of these quantities is maintained. We consider this a topic of future work that is to be carried out in collaboration with the upper atmospheric research community.
 In contrast to the solid earth, where little information is known about velocity models, a global database of temperature and wind data throughout the atmosphere is available to build the model space. Empirical orthogonal functions are an intelligent choice of parameterization here, as these basis functions are based on the statistical variability within the available atmospheric data set. The most constrained functions are the most robust features in the model space [Snieder and Trampert, 1999]. In this case, the atmospheric tides can be well parameterized by a limited number of basis functions. However, as the parameterization is based on the atmospheric database used, this also implies that no data can be explained that falls outside the predicted range. Alternatively, one could impose a different parameterization, such as the Gaussian correction factors used by Le Pichon et al. [2005b]. However, such a parameterization would not be based on a existing atmospheric database and would be more subjective.
 The parameterization of the atmospheric state that has been used in this work does not include range-dependent atmospheric states, such as caused by gravity wave forcing. We justify the use of a range-independent atmospheric state, since the coherent variability of the thermospheric travel time fluctuations, with periods similar to the atmospheric tides [Assink et al., 2012], indicate that large-scale spatial features are dominant in this particular case. Furthermore, gravity waves of smaller scales dissipate or coalesce into larger gravity wave features in the region of interest, i.e., the lower thermosphere. Moreover, the horizontal variation in medium properties is much weaker compared to the vertical, and in the region of the atmosphere where we are attempting to provide updates, only about 50km is traversed horizontally (see Figure 11a), which appears to be the smallest wavelength of small-scale gravity waves in this region [Drob et al., 2013]. While not excluding the possibility of range dependence, we believe that the good agreement between the predicted and the observed waveforms does support a range-independent state. However, as the interaction of gravity waves on infrasound is significant in the higher regions of the atmosphere [Hedlin et al., 2012; Drob et al., 2013], we plan on future studies in which the effect of gravity wave forcing on updated wind model estimates is further studied.
 The results show that for an increased level of detail in the model and uncertainty in the model and data space, the posterior model distribution assumes a complicated form. The a posteriori distribution broadens, and secondary maxima appear. Secondary maxima are a feature of nonlinear objective functions with more than two parameters. Therefore, we favor the use of a global grid search over local-search methods, as the grid search allows us to identify the global maximum. We note that the broadening and the appearance of multiple local minima for the misfit function is a general feature of inversion and not at all specific to the Bayesian framework. The Bayesian framework is useful in the description of the uncertainties associated with the prior and a posteriori distributions. As was discussed in section 4, the posterior model broadening is inevitable as the variance of a linear combination of stochastic variables can only increase or remain the same as additional stochastic variables are added. Therefore, the posterior probability density will broaden.
 In the case of strong nonlinearity, the problem cannot be solved with linearized methods [Lalande et al., 2012] and a more rigorous analysis is required. For strongly nonlinear inverse problems, direct search methods are often the only feasible method to solve the inversion problem, as gradient-based methods fail for (strongly) nonlinear inverse problems [Tarantola, 2005]. In this paper, we make use of an exhaustive search in the HWM model space to solve the inverse problem. It is estimated by our analysis that the southward flow has been underestimated by about 20ms−1. We note that the direct search approach would be computationally inefficient for large dimensional model spaces. Many global optimization methods are available that are designed to efficiently search multidimensional model parameter spaces, such as Simulated Annealing, Genetic Algorithms, and the Neighborhood Algorithm [Sambridge, 1999]. Future research could focus on the application of these algorithms.
 The results obtained in this paper are for the source-receiver pair Tungurahua-LITE, as it was the objective to estimate wind speeds from the Tungurahua data set. Although this data set provides a large database of stratospheric, mesospheric, and thermospheric arrivals from strong and repetitive explosions, the arrays were not optimally designed to measure trace velocities of thermospheric arrivals with infrasound frequencies around 0.05Hz. Studies like these would certainly benefit from infrasound arrays that are designed to measure trace velocity and bearing with very small uncertainties. In addition, the posterior distributions would be more constrained in the case of more than one far-field array. We plan on future inversion studies using multireceiver data sets, such as obtained during the Utah Testing and Training Range [Talmadge et al., 2010] and the Sayarim 2009/2011 [Fee et al., 2013] measurement campaigns. The arrays that have been deployed during these campaigns were designed to measure trace velocity and bearing at frequencies around 0.05Hz with much greater precision than the ASHE arrays. The use of these data sets, in combination with more sophisticated search methods, would allow one to invert for more atmospheric parameters. For example, one could simultaneously invert for meridional and zonal wind parameters.
 Finally, the description of data and model uncertainties is important in solving the inverse problem, as these regularize the inversion problem by weighting the misfit function. The more objective this information is, the more objective the inversion result is. In this study, we have regularized the model space by a piecewise probability density function in which the perturbed profile was required to be within 50ms−1 of the a priori model, which is a reasonable zeroth order constraint given the HWM07 model. The observables have been characterized by a Gaussian probability density, with uncertainties assumed to be statistically independent of each other. In the future, we plan on considering covariances between the various infrasound observables in the description of the a priori data [Tarantola, 2005].
 This paper discusses the inverse problem of estimating meridional wind perturbations, using a priori wind models and infrasound data. Such wind model updates may be useful in reducing uncertainty in the knowledge of wind speed models in the region above the stratopause and are beneficial in the context of propagation studies, as the atmospheric specifications are based on spatially and temporally averaged measurements that do not (fully) explain infrasound observations. We make use of empirical orthogonal functions to parameterize the Horizontal Wind Model. A grid search method is used, in combination with Bayesian statistics, to find a distribution of most likely models that fit the data. We believe that this is the first study in which such methods are applied to real infrasound data, allowing for a rigorous analysis of the inverse problem. The influence of the parameterization and the data uncertainty on the inversion result is discussed. It is found that the posterior distribution broadens and global maxima appear when the parameterization becomes more detailed and the uncertainties in the data grow.
 We present a case study for which an meridional wind model update is obtained given information on the source and receiver locations, travel time, and trace velocity. We apply a nonlinear ray tracing technique to check the physical self-consistency of the proposed solution, as nonlinear propagation effects further constrain the propagation path. We find an excellent agreement between the duration of the simulated and observed waveforms, with the source peak pressure constrained by acoustical measurements near the source.
 This document was prepared under award NA08NWS4680044 from the National Oceanic and Atmospheric Administration, U.S. Department of Commerce. The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the views of the National Oceanic and Atmospheric Administration or the U.S. Department of Commerce. The GEOS5 analysis fields utilized in conjunction with other data sources within the Naval Research Lab (NRL) G2S atmospheric specifications were provided by the Global Modeling and Assimilation (GMAO) at the NASA Goddard Space Flight Center (GSFC) through the online data portal in the NASA Center for Climate Simulation. The NOAA GFS analysis fields also utilized in the G2S specifications were obtained from NOAA's National Operational Model Archive and Distribution System (NOMADS), which is maintained at NOAA's National Climatic Data Center (NCDC). Discussions with Läslo Evers are acknowledged. The authors are grateful to Jean-Marie Lalande for a thorough review of the initial manuscript. The manuscript was substantially improved by reviews from Doug Drob and two anonymous reviewers.