Ocean acoustic tomography depends on a suitable reference ocean environment with which to set the basic parameters of the inverse problem. Some inverse problems may require a reference ocean that includes the small-scale variations from internal waves, small mesoscale, or spice. Tomographic inversions that employ data of stable shadow zone arrivals, such as those that have been observed in the North Pacific and Canary Basin, are an example. Estimating temperature from the unique acoustic data that have been obtained in Fram Strait is another example. The addition of small-scale variability to augment a smooth reference ocean is essential to understanding the acoustic forward problem in these cases. Rather than a hindrance, the stochastic influences of the small scale can be exploited to obtain accurate inverse estimates. Inverse solutions are readily obtained, and they give computed arrival patterns that matched the observations. The approach is not *ad hoc*, but universal, and it has allowed inverse estimates for ocean temperature variations in Fram Strait to be readily computed on several acoustic paths for which tomographic data were obtained.

## I. INTRODUCTION

Ocean acoustic tomography consists of four basic steps (Munk *et al.*, 1995). First, the signals from acoustic propagation over some distance through the ocean are recorded. Second, the equivalent acoustic propagation is computed using a “reference ocean,” or a suitably realistic representation for the acoustic environment. This computation is the acoustic forward problem, which is an analysis to understand the essential factors governing the acoustic propagation and a computation of an acoustic arrival pattern that matches the recorded arrival patterns reasonably well. The discrepancies between the computed and measured arrival patterns indicate the essential information provided by tomography. Third, a representation of the ocean variability is constructed such that the variability can be modeled by a set of parameters, and the inverse components and inverse itself are computed. Last, the inverse solution is tested by computing acoustic propagation through the “corrected” acoustic environment to verify that the new, computed arrival pattern agrees with that recorded. Analyses of the model variances and inverse uncertainties are required and integral aspects of the inverse problem.

The reference ocean employed for the forward problem is most often a smooth climatology such as the World Ocean Atlas (Antonov *et al.*, 2010; Locarnini *et al.*, 2010; Dushaw, 1999), a smoothed hydrographic section (Worcester *et al.*, 1999; Sagen *et al.*, 2017), or an ocean model or state estimate (Dushaw *et al.*, 2009; Dushaw *et al.*, 2013). There are several examples of instances where the effects of small-scale variability (internal waves, spice variations, small mesoscale) are essential to understanding the forward problem, however (Van Uffelen *et al.*, 2010; Colosi, 2016; Dushaw *et al.*, 2016b; Dushaw *et al.*, 2017). The influences of small-scale variability on long-range acoustic propagation were perhaps first highlighted with the observations of the SLICE'89 experiment at 1-Mm range in the central North Pacific (Duda *et al.*, 1992). Shadow-zone arrivals, which are stable, ray-like arrivals that occur at times and depths outside the arrival pattern predicted using a smooth sound speed section (Dushaw *et al.*, 1999; Dushaw *et al.*, 2009; Van Uffelen *et al.*, 2010; Dzieciuch *et al.*, 2013), are obvious examples of the influence of small-scale variability. How can such data be exploited for ocean estimation by an inverse when they cannot be represented by a conventional forward problem? The addition of simulated, yet realistic, small-scale variability to a forward-problem environment can be essential to computing an inverse estimate.

For our inverse computations, we employed geometric rays. Rays provide a convenient and accurate way to compute the components of the inverse. Ray tracing has been shown to provide accurate acoustic predictions, even in challenging environments with significant small-scale variability. Dushaw *et al.* (2013) provide a brief review; see also Jensen *et al.* (2011). In recent years, full-wave physics has been employed to derive travel-time sensitivity kernels (TSKs) for tomography that show the precise spatial sensitivity of particular travel times in range and depth (Skarsoulis *et al.*, 2009; Skarsoulis and Cornuelle, 2004; Dzieciuch *et al.*, 2013). Such analyses have supported the ray approximation as a valid and reasonably accurate representation for acoustic sampling. It appears readily apparent, however, that although the geometric ray model can provide accurate acoustic predictions, the notion of associating an individual ray with a particular arrival peak is not as clear cut as originally anticipated for tomography. Rather, the ubiquitous small-scale variability of the ocean generally causes both a spread in the arrival peaks of rays and the appearance of many similar rays associated with a peak rather than a single ray. Rather than a hindrance, this property can be exploited to obtain accurate inverse estimates from acoustic data. One aim of this paper is to describe inverse estimates obtained using the tomography data that have been obtained in Fram Strait, where small-scale variability is ubiquitous, if not dominant.

The necessary relation between signal processing and inverse schemes, particularly when small-scale contributions are part of the tomography problem, is described in Sec. II. The roles of the small-scale variability in interpreting and employing “shadow zone arrivals,” or the extensions of the branches of time fronts caused by small-scale scattering, for the purposes of tomography are described in Sec. III. In Sec. IV, the importance of the small-scale variability in interpreting data obtained in Fram Strait is described, and examples of inversions of data there exploiting small-scale variability are given. The inversions include mesoscale effects with the reference ocean in obtaining estimates for average temperature. Finally, the results and implications of this analysis are summarized and discussed in Sec. V.

## II. SMALL-SCALE VARIABILITY AND DATA PROCESSING

When small-scale variability is employed to create a more realistic reference ocean, a data-processing filter consistent with both that variability and the inverse approach (e.g., rays or TSKs) must be employed. The small-scale variability affects the details of an acoustic arrival pattern, contributing to such things as wander, spread, and bias to an individual ray arrival (Colosi, 2016). Individual arrivals can be split into many, a result of a single path being split into multipaths. Determinations of such details in the observations are influenced by whatever signal processing or filtering has been applied to the data; thus, small-scale contributions and signal processing are inextricably linked.

Ideally, implementation of ocean acoustic tomography would begin with an acoustic source that could create a strong impulse of perhaps 10 ms duration. Such an impulse, which would necessarily be broadband in frequency, would generate acoustic waves that would then propagate to a suitable receiver (we set aside such physical constraints as non-linear acoustics or cavitation as would likely occur by such an impulse). In ordinary mid-latitude, long-range propagation, the received coda at the depth of the sound channel would begin with a small set of sharp pulses, corresponding to geometric ray arrivals, followed by an increasingly stochastic rumble terminating with a loud pulse of the finale. In many regions of the world's oceans, the finale corresponds to the slowest acoustic energy carried by acoustic modes that have traveled along the sound channel axis.

Such impulsive acoustic sources are, in practice, impossible, and, further, the sound strength of such pulses would be prohibitive from an environmental standpoint. Explosive sources do not offer such time resolution. Mechanical sources cannot be built with sufficient bandwidth and intensity to produce the ideal signal. Rather, sophisticated, lengthy coded signals are employed. After transmission of such a signal, typically extending several minutes in duration, signal processing techniques are applied to compress the signal to obtain an equivalent “impulse response.” Computation of the true response is impossible, however, since environmental factors, Doppler effects, etc., affect the lengthy signal in various ways. The practicing oceanographer thus approaches the necessary signal processing in different ways, depending on the purpose of the analysis, with the different approaches providing different realizations of the idealized impulse response. Sagen *et al.* (2017) provide a case study of such processing for acoustic propagation in Fram Strait, showing the effects of assuming different signal bandwidth and coherence times on estimated time series. Such processing can affect the signatures of small-scale variability in the data, either to highlight or to obscure its properties, hence, the data processing scheme employed must be matched to the intended approach to travel time inversion.

## III. TOMOGRAPHY WITH SHADOW-ZONE ARRIVALS

The obvious need for a reference ocean enhanced with small-scale variability is demonstrated in the problem of computing inversions using shadow-zone arrival data (Dushaw *et al.*, 1999; Dushaw *et al.*, 2009; Van Uffelen *et al.*, 2010; Dzieciuch *et al.*, 2013). Such arrivals give stable time series of travel times that are readily identified with extensions of branches of a time front computed from a smoothed ocean realization. Indeed, shadow-zone arrivals were a significant fraction of data obtained during the decade-long (1996–2006) Acoustic Thermometry of Ocean Climate (ATOC) project. They represent valuable data, but a ray path associated with them cannot be computed using a smooth reference ocean. An inverse employing these data with such a reference ocean is therefore technically not possible. Such arrivals can be computed, and a ray path determined for them, by augmenting a smooth reference ocean with small-scale variability. Researchers, e.g., Dushaw *et al.* (2009) and Dzieciuch *et al.* (2013), have referred to these arrivals as “nongeometric,” but that description is incorrect. Similarly, the label “shadow zone arrivals” is also a misnomer. These arrivals can be readily identified with geometric rays computed using a more complete, or realistic, reference ocean.

The nature of shadow-zone arrivals can be illustrated using a notional 2000-km-long acoustic path along 25 °N in the North Pacific. A good representation for the sound speed section can be computed from the 2009 World Ocean Atlas (WOA09; Antonov *et al.*, 2010; Locarnini *et al.*, 2010; Dushaw *et al.*, 2013). The internal wave model of Colosi and Brown (1998) can be used to approximate the effects of internal waves on acoustic propagation. The parameters used for the model were a thermocline depth *B* = 1000 m, spectrum indices *j*_{max} = 100 and *j*_{∗} = 3, and $\zeta 02=32$, appropriate for a normal Garrett–Munk internal wave spectrum. *N*_{0}, the reference buoyancy, was computed as the depth integrated buoyancy divided by a 1000-m thermocline depth giving a value 0.0091 rad s^{−1} (4.7 cy h^{−1}). Time fronts computed for such an acoustic path, with and without the effects of internal waves, are shown in Fig. 1. The characteristic extensions of the lower cusps, or caustics, of the time front as a consequence of internal wave scattering are evident; this extension can be several hundred meters (Van Uffelen *et al.*, 2010; Fig. 1). A receiver at 4000-m depth, for example, would detect several more ray arrivals than could be predicted using just the smooth climatology.

Figure 1 suggests such arrivals could have a travel time bias relative to the smooth ocean prediction, but such arrivals are not universally biased. The computed arrival pulse is obviously no longer a single ray arrival, but has a width of perhaps 50 ms. Dzieciuch *et al.* (2013) suggested that one approach to reconciling this spread and the associated dependency of a computed TSK on internal wave realization would be to compute averages of many realizations. The implication appears to have been that a travel time estimate would also be computed from a corresponding average of the acoustic data. (Many of the TSKs of Dzieciuch *et al.*, 2013, appear to have been computed using surface-reflected rays, which are more stable than purely refracted rays. Surface-reflected rays do not have upper turning points within upper ocean variability.) The potential consequences of including internal wave effects in a reference ocean, from a bias to multiple arrivals that have to be taken into account, have yet to be fully assessed. This task is beyond the scope of this paper, and, indeed, is likely dependent on the local sound speed and ocean characteristics.

The specific rays associated with the predicted arrivals give the spatial sampling for the inverse problem. The cusp of the ray arrival at 1346.6 s was considered as an example. Three sets of rays from the lower panel of Fig. 1 may be of interest (Fig. 2). For the smooth ocean reference ocean, only four eigenrays were obtained to a notional receiver at the maximum depth of the time front cusp at 3590 m. For the same reference and receiver depth, but including internal wave variations, 210 rays were obtained. The cusp of the time front is a caustic that corresponds to a high ray density, or equivalently, a high acoustic amplitude. One effect of the internal wave variations is that the high-amplitude reception becomes split into many ray paths with varying lower and upper turning depths. Last, at a depth of 4000 m, or 410 m below the cusp of the smooth-ocean time front, ten rays are obtained, with similar variations in turning depth.

One way to assess the impact of small-scale fluctuations on the ray paths, hence, the inverse, is to compute the ray weighting as a function of depth. With a separation of sound speed into the reference sound speed, *c*_{0}(*z*, *r*), plus a perturbation, *δc*(*z*, *r*), the acoustic travel time for a ray becomes after a perturbative expansion,

with the ray's reference travel time computed from the reference ocean,

(Munk *et al.*, 1995). In these equations, a fixed-ray approximation is assumed; that is, the sound speed perturbations are assumed not to significantly disturb the ray path geometry, Γ. The small internal-wave scales included in the reference ocean have horizontal scales of *O*(100–1000) m, while the perturbative sound speed that is of oceanographic interest has horizontal scales of *O*(10–1000) km. The second term of the first equation has a weighting along the ray path $ds/c02(z,r)$; the ray path and its weighting are known as the ray kernel (Dzieciuch *et al.*, 2013). The ray kernel reflects the weight that is applied to the perturbation *δc*(*z*, *r*) when the path integral is computed. The ray kernel quantifies the impact particular travel times can have on the spatial resolution and uncertainty of an inverse solution.

The ray kernels for the rays of Fig. 2 are shown in the right-hand panels. In this case, the kernel has been summed horizontally, giving the weighting as a function of depth. In the topmost panel, the ray from the smoothed reference ocean gives ray weighting with sharp increases in value at the ray turning depths. The aspect ratio of the ray figures can be misleading; most of the ray arc length occurs at the ray turning depths. One effect of the small-scale variability is to induce variations in the upper and lower turning depths of the rays. As a consequence of the irregular turning depths, the ray weighting is blunted and broadened in depth. The ray kernels for the rays to receivers at 3590 or 4000 m depths, are similar, although there are many more rays to the shallower depth where the acoustic intensity is greater.

While the primary aim of this computation was to show the degree of similarity between direct and shadow-zone ray sampling, an obvious implication of this analysis is that the ray sampling derived from a smoothed ocean reference has characteristics that are artificial compared to ray sampling from a more realistic ocean. One immediate conclusion evident from the scattering from small-scale sound speed variations is that the ray-loop resonance described by Cornuelle and Howe (1987) is not likely realistic, given the horizontal instability of the ray paths.

Last, we note that “shadow-zone arrivals” are not limited to the lower (or upper) cusps of the time front. The ray arrivals recorded in the Canary Basin during 1997–1998 for the purpose of measuring the properties of Meddies included a clear, unpredicted arrival at the end of the arrival pattern (Fig. 3; Dushaw *et al.*, 2017). This arrival was recorded by a receiver near the sound channel axis. It arose from the extension of a branch of the time front to later times as a result of small-scale scattering. Its existence is entirely a consequence of such scattering, and any inverse employing this arrival would necessarily require a reference ocean incorporating small-scale variability.

## IV. THE SMALL SCALES OF FRAM STRAIT

Over the past decade, tomography has been employed to measure the temperature variability within Fram Strait. The extraordinary small-scale variability of the region has made it challenging to understand the regional oceanography by hydrographic sections or moorings. A pilot tomography measurement in 2008–2009, the “Developing Arctic Modelling and Observing Capabilities for Long-term Studies” (DAMOCLES) program, was described by Sagen *et al.* (2016), while the subsequent “Acoustic Technology for Observing the Interior of the Arctic Ocean” (ACOBAR) progam (2011–2012) was described by Sagen *et al.* (2017; Fig. 4). The DAMOCLES tomography data consist of a 314-day time series of travel times during 2008–09, while the tomography data time series from ACOBAR consist of about 250-day record lengths on three paths during 2010–2011 and during 2011–2012. Since one of the ACOBAR paths (mooring A to mooring B) consisted of reciprocal transmissions, the ACOBAR tomography data set consists of eight time series of about 250-day record length. Over the two time intervals of ACOBAR deployments, the array position was unchanged, so the tomography measurements on each path combine to form time series of about 600-day duration.

The small-scale variability has a large impact on the acoustic propagation in the region. Sagen *et al.* (2017) examined the acoustic predictability of the ACOBAR data by employing smoothed hydrographic sections and comparing acoustic predictions to suitably filtered observations. They sought a deterministic solution to the ray sampling problem, requiring a suppression of small-scale effects by filtering methods. With the unique stratification in Fram Strait, the precise acoustic propagation is dependent on the mesoscale state, which is constantly changing (Dushaw *et al.*, 2016b; Sagen *et al.*, 2017).

Acoustic predictions using a smoothed climatology or hydrographic section give only a small set of rays, but a typical arrival pattern for the region consists of a single, broad arrival of perhaps 100 ms width, sometimes together with a few distinct pulses from bottom-reflected rays. With a Rossby radius of deformation of 4–6 km, the small mesoscale variability of the region is essential to understanding the nature of the acoustic propagation (Dushaw *et al.*, 2016b). The contributions from internal-wave variability are minor, as a consequence of the stratification in this region.

Temperatures, averaged along the acoustic path, have been estimated from the DAMOCLES tomography data using a numerical ocean model as a reference ocean (Sagen *et al.*, 2016). The inverse method for these acoustic data was described by Dushaw and Sagen (2016), while to good approximation sound speed is a proxy variable to temperature in Fram Strait (Dushaw *et al.*, 2016a). These publications, together with Dushaw *et al.* (2016b), which discusses the importance of the small-scale variability in understanding the acoustic forward problem, are prerequisites to the present analysis. Dushaw (2017) provides a report drawing together the relevant components of those publications used for computing time series of temperature in Fram Strait according to the approach described by this paper. Inverse methods, and objective mapping approaches, in particular, have been associated with tomography for over 30 yr; Worcester (2001) provides a review.

One aim of the previous analysis was to demonstrate that this data type can be used in conjunction with numerical ocean models. While the use of the ocean model as a reference ocean was effective, two technical problems were apparent. First, the ray predictions using the model were overly dispersive, giving an arrival pattern that did not match the observations. Similarly, a limited number of rays were obtained using the ocean model sound speed environment. Both of these properties presented technical difficulties for the inverse estimate, limiting the number of data that could be used in the inverse and requiring an overly large prior data uncertainty to obtain inverse solutions that agreed with the observations. Even with these inverse strategies, the inverse estimates gave predicted travel times that were slightly biased compared to the data. Corrected travel times were slightly too large (the inverse slightly overcorrected the travel times), corresponding to a slight cold bias in the estimates for temperature. Last, since an obvious use of these observations is as a test of several available ocean models for Fram Strait, we seek an estimate for temperature that is independent of any particular model.

### A. A reference ocean with small-scale variability

To alleviate the several issues just described, new inverse estimates were obtained by employing a reference ocean formed by the World Ocean Atlas as the background sound speed plus sound speed variations caused by small mesoscale variability (Fig. 5). The mesoscale model was required by (Dushaw *et al.*, 2016b) to describe the acoustic forward problem for DAMOCLES. The inverse strategy employed here was suggested as a possible alternative by Sagen *et al.* (2016). The dispersal of the arrival patterns predicted using only the World Ocean Atlas was similar to observations in Fram Strait, but only a few rays were obtained (Fig. 6). When realizations for small mesoscale variations were added to the World Ocean Atlas, many more rays were obtained and the predicted patterns were similar to observations, both DAMOCLES (Dushaw *et al.*, 2016b) and ACOBAR. In this case, rather than just a few rays, several hundred rays were obtained; six of these rays are shown in Fig. 7. Despite the large number of rays, their spatial sampling was similar, normally cycling between the ocean's surface and 1000–1500 -m depths. The arrival pattern produced by the large number of rays was in agreement with the arrival patterns computed using full-wave propagation methods such as the parabolic equation (Dushaw *et al.*, 2016b). It seems apparent that although individual rays are likely problematic representations for the acoustic sampling, considered in total, ray predictions do work for representing acoustic propagation in Fram Strait.

### B. Ray kernels

The ray kernels associated with rays from the WOA09 with and without the small-scale contributions are distinctly different (Figs. 6 and 7, middle panels). The differences result from the greatly varying ray turning depths that occur when the rays encounter the small-scale variations in Fram Strait and the tendency of the smoothed sound speed of WOA09 to produce surface reflecting rays. The sensitivity of the ray paths to small-scale variability in Fram Strait is a product of the nature of the background sound speed gradients of the region (Dushaw *et al.*, 2016a; Sagen *et al.*, 2017). Although the ray kernels appear quite different, suggesting inverse results from the two sets of rays may be different, the actual forward problem operator, e.g.,

depends on the product of ocean model functions, *V _{j}*(

*z*), and ray weighting. This combination is relatively independent of precise ray kernel, as can be illustrated using first and third vertical functions (empirical orthogonal functions, EOFs) employed for the inverse (Figs. 6 and 7, right panels).

### C. Components of the inverse

With the exception of the reference ocean just described, all of the components of the inverse of Fram Strait tomography data to obtain estimates for temperature have been described in recent publications. In highlighting the differences between tomography and moored observations of Fram Strait, Dushaw and Sagen (2016) also describe the mechanics of the inverse and the vertical and horizontal functions employed to model ocean variability. This model was grounded on hydrographic and moored observations of Fram Strait (Dushaw *et al.*, 2016b). The computation of travel time data employed for the inverse, that is, the difference between the predicted and measured travel times, was described by Sagen *et al.* (2016). Essentially, these travel time differences were derived by randomly associating peaks of the recorded arrival pattern with rays computed using the reference ocean. A data uncertainty of 100 ms was assumed to account for the imprecision of this association. A key aspect of this strategy was to rely on the large quantity of data to obtain accurate inverse estimates and reduce the uncertainties. The scatter of the data does not allow particularly accurate inverse estimates from an individual acoustic transmission, but accuracy can be achieved by averaging inverse estimates from many transmissions in a daily to weekly average. Last, Sagen *et al.* (2016) detail how absolute temperature can be computed from the inverse estimate for sound speed and the reference ocean.

In this case, we desire as clear a representation of the impulse response as possible, particularly the small-scale effects, because we want to exploit the characteristics of the small-scale variability. The data were processed with a matched filter that assumed infinite bandwidth (10 000 Hz) and coherence time (10 000 s; Dzieciuch, 2014), resulting in highly detailed arrival patterns. The essential strategy for data processing in acoustic tomography has been centered on deriving travel times of peaks in the arrival pattern. This strategy stems from the need to associate a measurement kernel (a ray or TSK) with a particular arrival as a prerequisite to computing the inverse. A legacy of this strategy is that the acoustic arrival data are usually presented as “dot plots” that represent continuous arrival patterns as discrete sets of peaks (Figs. 8 and 9). In the arrivals for Fram Strait, this basic strategy is likely not physical; we doubt that the various peaks within the broad, 100-ms arrival pulse correspond to particular rays. As described by Dushaw *et al.* (2016b), the detailed structure of the broad arrival pulse is not stable. Rather, a continuous acoustic field, perhaps represented by a large number of acoustic rays, contributes to the broad arrival pulse. It is expedient to use several peaks as an approximation to the underlying continuous pulse, however, with those peaks assigned to specific ray paths. A more rigorous inversion scheme may eventually be devised that employs the continuous broad pulse arrival in its entirety, but such a scheme is not yet apparent to us.

To be as realistic as possible, a new reference ocean and set of ray paths should be computed for each inverse or transmission. The repeated computation of the ray paths is challenging, however. While any particular ray computation takes less than a minute, such ray computations would be required for each of the *O*(1000) available transmissions associated with each 250-day time series. Since only *O*(10) peak travel time data are available from each hydrophone record and transmission, while *O*(500) rays are available from a single ray computation using the reference ocean, an expedient is apparent. The forward problem operator, *G*, required for the inverse, was computed using the complete set of *O*(500) rays from a single reference ocean. The values for this matrix together with the ray travel times then formed a pool with population *O*(500) from which the small number of values required for the data used each inverse were selected at random. After the reference ocean was corrected by the inverse, ray computations were used to verify the solutions agreed with data, however (Figs. 8 and 9). Even relying on the single, initial ray trace, the computations of the inverse estimates and associated ray travel times for a 250-day time series took about two days on a high-end personal computer.

### D. Assessing the temperature estimates and bias

Estimates of absolute temperature can be readily computed by combining the reference ocean with the estimated sound speed perturbation, as described by Sagen *et al.* (2016). The best-resolved quantity is the temperature averaged over the path length and the upper 1000 m of ocean. The small-scale variability added to the reference ocean has negligible effect on this average. Estimates for the DAMOCLES time series from its 130-km path in 2008 and for an ACOBAR time series from a 301-km path across Fram Strait from 2010 to 2012 are shown in Figs. 10 and 11. Similar estimates were readily computed for all ACOBAR paths (Dushaw, 2017). Consistent with the basic nature of the data, the estimates from individual acoustic transmissions show considerable scatter. The rapid variations in temperature evident in the two time series, weaker in the observations from the longer path, reflect the extraordinary mesoscale environment of Fram Strait (Dushaw *et al.*, 2016b; Sagen *et al.*, 2016).

Formal inverse uncertainties for the individual DAMOCLES and ACOBAR estimates were 30 m °C and 75 m °C, respectively, but such values are superseded by the reality of the scatter of the individual estimates. Sagen *et al.* (2016) computed a smoothed time series of temperature from the DAMOCLES time series by applying a two-day running mean. Given the irregular nature of the sampling, a better smoother is desirable. We computed a smoothed estimate for time series of temperature by employing a least-squares cubic spline fit, allowing for both misfit and data error (de Boor, 2001; blue lines in Figs. 10 and 11). The distribution of the misfit of individual estimates from the smoothed estimates indicates the individual estimates have about an 80 m °C root-mean-square (RMS) scatter, in both DAMOCLES and ACOBAR time series. The uncertainties associated with the smoothed estimates are smaller. To obtain an estimate for uncertainties of the smoothed temperature, we assumed that the data from two days of transmissions are combined to form a value of the smoothed estimate and that Gaussian statistics apply. There are 16 total transmissions during 2 transmission days. A nominal value for the uncertainty, associated with an estimated time series of average temperature with two-day resolution, therefore, may be $80\u2009\u2009m\xb0C/16$, or 20 m °C. Such a value is fairly small by oceanographic standards [an expendible bathythermography (XBT) measurement has an accuracy an order of magnitude larger; a conductivity-temperature-depth (CTD) measurement has an accuracy an order of magnitude smaller], and comparable to the error introduced into the estimate by the effects of salinity (Dushaw *et al.*, 2016a). Indeed, an uncertainty in average temperature of 20 m °C is challenging in the context of data assimilation. Most present-day data assimilation schemes would have to assume a far larger uncertainty for these estimates to avoid artifacts from “over-assimilation,” given the myriad biases, errors, and complexities of ocean models.

As noted above, one motivation for adopting the revised inverse strategy was to correct a small apparent bias in previous estimates. A comparison of the data-estimate misfit in the estimates employing a numerical ocean model (Fig. 9 of Sagen *et al.*, 2016) to the misfit employing the present reference ocean (Fig. 8), subjectively indicated the bias of the revised inverse is reduced. With the various complications of these data, it proved difficult to define an objective measure of the bias between data and corrected travel times. Computing the difference between the means of data employed for the inverse and travel times computed using the inverse solution was an insensitive approach. Such differences are 10–30 ms for either DAMOCLES or ACOBAR time series, irrespective of inverse approach.

The bias can be assessed by comparing the distributions of travel times, however. The data from the first 50 days of the DAMOCLES transmissions illustrate the observed and computed distributions (Fig. 12). With a numerical ocean model as a reference, the initial computed travel times had a dispersal much larger than the data. As a consequence, travel times computed from the inverse solution were slightly too large (the inverse over corrected) and exhibited a skewed distribution compared to data (Fig. 12, lower panel). By using the WOA09 plus a stochastic contribution, the initial computed travel times matched the nature of the data better, resulting in a better inverse estimate. As shown in Fig. 12 (upper panel), the final distribution of computed travel times agreed well with the distribution of data. The distribution of data shown in Fig. 12 shows an extended tail into later travel times, attributed to noise or outliers. To make a quantitative comparison, the mean and standard deviation of travel times between 88.90 and 89.05 s were computed, values chosen to mostly eliminate the effects of the tails of the distributions. The predictions using inverse solutions based on the numerical ocean model reference ocean required a larger travel time limit of 88.15 s to encompass that distribution. The data had mean and standard deviation 88.972 ± 0.043 s, the previous inverse solution gave 89.026 ± 0.042 s, while the new inverse gave 88.968 ± 0.043 s. The new inverse solution has travel times that are in better agreement with the observations.

While the bias is noticeable when comparing computed travel times to data, its equivalent in temperature is rather small. The previously estimated DAMOCLES time series had mean and RMS 1.11 ± 0.33 °C, while the new estimate has a mean and RMS 1.19 ± 0.31 °C. The new estimate has therefore corrected a bias of −0.08 °C in the previous estimate.

## V. SUMMARY AND DISCUSSION

In measurements of the arrival patterns of long-range acoustic propagation for the purposes of tomography, there are several instances where the influences of small-scale variability are essential to understanding the acoustic forward problem. That is, to be able to compute an arrival pattern that is reasonably similar to the observed pattern, the effects of scattering from small-scale variability must be included. Examples of arrivals that require the effects of small-scale acoustic scattering include deep arrivals in basin-scale transmissions in the North Pacific during the 1996–2006 ATOC program, a late, unexpected arrival recorded in 1997–1998 near the sound channel axis in the Eastern Atlantic, and the single, broad arrival that has been observed on acoustic paths within Fram Strait over the past decade. Some of these arrivals have been called “shadow-zone” or “nongeometric” arrivals, but these are misnomers. The arrivals are readily associated with geometric rays in a realistic, rather than smoothed, ocean environment. Since small-scale effects are required to understand the forward problem in these cases, it is natural that small-scale effects would be required to compute the inverse for tomography. Stochastic models for the small-scale variability can be readily computed for this purpose. The use of such models as an element of the reference ocean employed for the inverse has no obvious adverse or negative consequences, so long as small-scale variability is modeled in a way that is at least consistent with the small-scale processes.

An inverse scheme employing small-scale contributions was applied acoustic data obtained in Fram Strait to obtain estimates for the variability of average temperature. The ocean estimates obtained by the inverse were verified to give acoustic predications that agreed with the observations both in absolute travel time and in the distribution of the arrival pattern. Such inverse estimates were readily obtained on four distinct acoustic paths of 130–300 km ranges. Estimates for temperature, averaged over a 300-km acoustic path length and the upper 1000-m of ocean, and smoothed to about two-day temporal resolution, had uncertainties of about 20 m °C. Such a small uncertainty ultimately results from the inherent averaging of the acoustic sampling over long range and the averaging of the data from many acoustic transmissions.

The use of small-scale contributions with a reference ocean requires new approaches to both analysis and interpretation. The scattering by small-scale features causes a multitude of rays to be associated with an individual “ray arrival,” rather than the individual ray obtained in a smoothed ocean. The scattered rays have varying lower and upper turning depths and varying horizontal sampling. Despite these variations, the ray kernels of these multiple rays represent similar ray sampling. Indeed, it is the ray sampling derived from a smoothed reference ocean that is unrealistic, with excessive weight obtained at the well-behaved upper and lower turning depths. The ray loop resonance described by Cornuelle and Howe (1987), wherein the ray loop periodicity leads to resolution of particular matching wavelengths of ocean variability, does not likely occur in a realistic ocean. Ray loop periodicity is disrupted by the effects of the small-scales. Similarly, the original notion for tomography of identifying a particular ray with a particular arrival pulse is at best an approximation. On the one hand, it appears that individual stable rays do not seem to exist, while on the other hand the scattering by small-scale features makes an arrival pulse more like a narrow distribution of arrivals than a single pulse. The latter property requires that any particular inverse approach be consistent with the inevitably necessary signal filtering scheme used to derive the travel times of “peaks” in the arrival pattern. Despite these complications, it has been demonstrated time and again by comparisons to other approaches that geometric rays can be used to accurately model acoustic arrival patterns, even in complicated environments. It seems apparent that although we must abandon the notion of individual rays (or accept them as approximations), we must also accept the complete set of rays as a valid model for the acoustic propagation.

While ray paths are mathematically of zero width, hence, rays sample the ocean in an artificial, if not unphysical, way, two practical aspects of the inverse computation ameliorate the sampling issue. First, to compute the inverse, the ray path itself is integrated by convoluting it with the quite large-scale functions that comprise the ocean model for the inverse. Attempts have never been made to exploit the fine sampling of the acoustic rays to resolve small-scale structures of the ocean. Second, the inverse solution is nearly always averaged to form well-resolved quantities such as a range- and depth-average sound speed (Dushaw and Sagen, 2016). Indeed, a determination of even a range-averaged profile with depth is, more often than not, problematic, relying on the existence of a suitable set of stable, resolved rays with different turning depths. Inversions with rays become questionable only if the ray sampling so poorly represents the true acoustic sampling that the integrals or averages become erroneous. Such poor representation by rays is unlikely; indeed, Dzieciuch *et al.* (2013) found the opposite was true.

Ultimately, any inverse of acoustic data for an ocean estimate has two basic goals: (1) to obtain a realization for the ocean (by any means) for which computed arrival coda agree with the observations, and (2) an estimate for the uncertainty of that ocean realization. Aside from the traditional “resolved rays and objective mapping” approach, inversions have employed other approaches including: brute force search of model space, the matched-peak and approach of Skarsoulis *et al.* (2010) employing two-dimensional EOFs as basis functions, the use of acoustic mode travel data (Sutton *et al.*, 1994), or the suggested use of TSKs (Skarsoulis *et al.*, 2010; Sagen *et al.*, 2016; Sagen *et al.*, 2017) as a basis for inversion. The use of a reference ocean that is time dependent (Morawitz *et al.*, 1996; Sagen *et al.*, 2016) can have advantages both to linearize the inverse by reducing the differences between reference and actual oceans and reduce the formal uncertainties associated with the inverse. The present result adds another trick to the tomographer's toolbox, the sometimes essential, but innocuous, addition of small-scale variability to the reference ocean, together with a modest relaxation of the requirement for stable, well-resolved ray paths.

## ACKNOWLEDGMENTS

B.D.D. was supported by Office of Naval Research Grant No. N00014-15-1-2186. The ACOBAR project was funded under the 7th EU Framework Programme (Grant No. 212887). Analysis of the ACOBAR data has been carried out under funding from the Research Council of Norway through the ACOBAR II (Grant No. 226997) and UNDER ICE (Grant No. 226373) projects at the Nansen Environmental and Remote Sensing Center. ENGIE E&P Norway provided additional support. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the Office of Naval Research.