Movement-assisted localization from acoustic telemetry data

Background Acoustic telemetry technologies are being rapidly deployed to study a variety of aquatic taxa including fishes, reptiles, and marine mammals. Large cooperative telemetry networks produce vast quantities of data useful in the study of movement, resource selection and species distribution. Efficient use of acoustic telemetry data requires estimation of acoustic source locations from detections at sensors (i.e. localization). Multiple processes provide information for localization estimation including detection/non-detection data at sensors, information on signal rate, and an underlying movement model describing how individuals move and utilize space. Frequently, however, localization methods only integrate a subset of these processes and do not utilize the full spatial encounter history information available from sensor arrays. Methods In this paper we draw analogies between the challenges of acoustic telemetry localization and newly developed methods of spatial capture-recapture (SCR). We develop a framework for localization that integrates explicit sub-models for movement, signal (or cue) rate, and detection probability, based on acoustic telemetry spatial encounter history data. This method, which we call movement-assisted localization, makes efficient use of the full encounter history data available from acoustic sensor arrays, provides localizations with fewer than three detections, and even allows for predictions to be made of the position of an individual when it was not detected at all. We demonstrate these concepts by developing generalizable Bayesian formulations of the SCR movement-assisted localization model to address study-specific challenges common in acoustic telemetry studies. Results Simulation studies show that movement-assisted localization models improve point-wise RMSE of localization estimates by > 50% and greatly increased the precision of estimated trajectories compared to localization using only the detection history of a given signal. Additionally, integrating a signal rate sub-model reduced biases in the estimation of movement, signal rate, and detection parameters observed in independent localization models. Conclusions Movement-assisted localization provides a flexible framework to maximize the use of acoustic telemetry data. Conceptualizing localization within an SCR framework allows extensions to a variety of data collection protocols, improves the efficiency of studies interested in movement, resource selection, and space-use, and provides a unifying framework for modeling acoustic data.

A key objective of acoustic telemetry studies is localization of sources using data from 45 arrays of receivers, i.e., estimation of the location of an individual source from detections at 46 one or more sensors of an array. When regarded formally as an estimation problem, localization is essentially statistical triangulation, and it can be done when signals are obtained Let u t be the unknown location at the time the t th signal was produced. At times t = 1, . . . , T 93 the transmitter produces a signal which may be detected by one or more sensors in an array. 94 Let ∆ t be the interval between signal transmissions (hereafter "signals" or "transmissions"). 95 In some cases ∆ t is constant for all tags and prescribed by design but, in practice, when 96 many tags are deployed at the same frequency the interval is often set to be random in 97 order to introduce an offset in detection times of individuals and avoid interference among 98 tags. For example, an individual tag might be set to emit a signal on a random schedule 99 with a minimum of 50 seconds and a maximum of 100 seconds between signals. Thus 100 ∆ t ∼ Uniform(50, 100). 101 For demonstration purposes, we assume a Markovian movement process conditional on 102 ∆ t according to (e.g., Brownian motion) although we note that a number of alternative movement models are possible [27]. The 104 observed data for each time t are the locations of the sensors that detected the individual 105 (including possibly none). To be consistent with SCR terminology we call this the spatial 106 encounter history and denote it by the vector y t where elements y j,t = 1 if the individual tag 107 was detected by receiver j at transmission t. Receivers have coordinates x j which are fixed 108 by design. In some cases we might have time-difference-of-arrival (TDOA) information but, 109 in practice, such information may not be available and, instead, only a coarse summary of (2) We also require the probability distribution of the spatial encounter history 120 1 In keeping with the terminology in spatial capture-recapture, we refer to this planar region as the statespace of the point process which defines the potential locations of individuals during the study. In practice, the state-space should be chosen to be much larger than the region containing the sensor array because individual locations may be detectable some distance from the sensors on the boundary. Pr(y|u) which, for binary detection encounter data, is determined by a set of detection used in distance sampling [30] and spatial capture-recapture applications: which has parameters p 0 and σ 2 det . In acoustic telemetry applications a logistic model is often produces fewer detections, and fewer and more imprecise localizations. 148 We propose to resolve this trade-off formally by extending the localization model de-149 scribed above through the integration of an explicit movement model to simultaneously 150 estimate the movement and detection processes. For example, instead of assuming Pr(u t ) is 151 uniform we replace this assumption with the Markovian movement assumption given above, 152 that is Then, in effect, the data from the previous (and subsequent) interval provide information about u t via the prior distribution Pr(u t |u t−1 ).

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Use of this movement model based prior distribution is effectively an informative prior 156 in the sense that it restricts potential states of u t to be in the vicinity of the previous state 157 where the extent of this vicinity is determined by the parameter σ 2 u as well as the sampling 158 interval ∆ t . Thus, note that as ∆ t increases, the information provided by previous states 159 diminishes rapidly and the prior tends to a uniform (non-informative) prior defined by the Pr(y t |u t , σ det , p 0 ) Pr(u t |u t−1 , σ 2 u |∆ t |) The inference objective is to jointly estimate the model parameters p 0 , σ det , and σ u as 2.3.1 Inference for the movement-assisted localization model In general, it is challenging to express the likelihood of the observed data (the spatial en- integration of a movement model with a spatial encounter model.

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Here, we seek to estimate the model parameters, which include the detection range param-187 eter σ det , the detection probability parameter p 0 and also the movement variance parameter a Metropolis-within-Gibbs algorithm is that the full-conditional distribution for u t can be constructed by noting that, This does not have a convenient simplified form (to the best of our knowledge) but it does 195 emphasize the point that information about the state of u t derives not only from the data y t 196 but also from the previous (u t−1 ) and subsequent (u t+1 ) states. In turn, those previous and 197 subsequent states are informed by detection data from t − 1 and t + 1, respectively. Thus 198 our generalized localization model is using "all the data" in a manner that is prescribed by 199 the specific movement and detection model imposed upon the system. In many acoustic telemetry applications the time interval between signals will not be known.

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Rather, devices are programmed to emit a signal on a random schedule, e.g., ∆ t ∼ Uniform(a, b), gap between detections of a given tag, then there are "missed signals" which have to be ac- Panel 1: JAGS specification of the localization model using a Brownian motion prior distribution on animal locations. Input data are the detection history data y, a J × T matrix, the time interval information T where ∆ t is the interval between signal t − 1 and t, and the sensor locations X.
commodated to achieve unbiased estimates of the detection process. Clearly it is uncertain 212 whether the 10 minute interval contained 3, 4, 5 or 6 signals that were missed. In this case, 213 one option for including the data obtained at the 10 minute interval is to set ∆ t = 10 and 214 then eq 4 is correct for the observed interval. The effects of missed signals in this approach, 215 however, is to decrease the information about the current state u t provided by previous and 216 subsequent states and biased estimates of the true detection parameters as all missed signals 217 are ignored. 218 Alternatively, we develop an MCMC algorithm that treats the number of missed signals 219 and the interval duration between each successive missed signal as random variables that are 220 estimated as part of the model. To do this, we integrate an additional sub-model to describe 221 the signal rate and associated interval durations. The specific form of the signal rate sub-222 model can be tailored to any study (e.g., programmed acoustic tags versus passive cetacean 223 cues), but generally requires two stochastic components to account for: (i) the number of 224 missed signals and (ii) the interval length between missed signals. The R code is provided 225 in Appendix A, while the signal rate sub-model is briefly described below.

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The signal rate sub-model can be tailored to different protocols and formulated based 227 on inter-signal rates (∆ t ; e.g., defined transmitter settings in telemetry studies) or number 228 of cues per unit time (n; e.g., animal calls). In most applications, assuming a distribution 229 for interval duration also induces a distribution for the total number of missed signals and 230 vice versa. For demonstration purposes, we develop a signal rate sub-model for acoustic tags 231 programmed with signal intervals ∆ t ∼ Uniform(a, b). The constraints for estimating the 232 number of missed signals (n) and their associated intervals (∆ t,1:n ) quickly becomes complex 233 as for any observed interval, (∆ obs , i.e. the length of the gap), there is a known minimum and 234 maximum number of missed signals, a fixed minimum and maximum true interval duration, 235 and a requirement that the intervals must sum to ∆ obs . In our example, we use a normal 236 approximation for the sum of uniform random variables to link ∆ t , and n. Specifically, we 237 assume: where ∆ obs is the observed gap length and n is the latent number of missed signals during 239 ∆ obs . In this example, n is an estimated parameter, while a, b, and ∆ obs are given as data.

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This approach greatly improved MCMC efficiency and provided reasonable estimates for our 241 study system. However, a variety of approaches are possible depending on the study system. We simulated 100 data sets for a system that involved 25 acoustically tagged individuals, 260 subjected to sampling at 100 sensors on a 10 x 10 grid with unit spacing (Fig. 1). The R The standard deviation of the Brownian motion movement process, σ u , was set at 0.25. We 271 selected these particular parameter settings so that the probability of detecting an individual 272 within the array was > 0.90, but quickly decreased as distance from the array increased ( Fig.   273 2). As such, individual locations near the center of the array were detected at 0 -6 sensors, 274 while locations on the periphery were detected at 0 -2 sensors (Fig. 2). The true movement biased toward the interior of the sensor array where sampling is more intensive. Therefore, 281 localizations using classical methods will also necessarily be biased toward areas of higher 282 sampling intensity. This sampling bias must be accounted for in studies of movement and 283 resource selection unless sensor placement itself is random with respect to habitat structure.

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In an SCR framework, however, signals that produce zero detections provide information on 285 the detection process but ignoring all-zero occasions may bias parameter estimates [18]. 286 We analyzed each simulated dataset using four modelling approaches, one approach that

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Both the movement-assisted localization known-interval and unknown-interval models re-330 turned relatively unbiased estimates of the movement and detection parameters (Table 1).

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The independent localization and movement-assisted localization detection-only models dis-332 played positive bias in p 0 due to the exclusion of all-zero occasions (Table 1). In the detection-333 only model, σ det and σ u displayed -1% and -9% relative bias, respectively ( Table 1). The 334 unknown-interval model reduced the relative bias in σ det and σ u to 0% and -8.0%, respec-335 tively (Table 1), while reducing bias in p 0 from 16% to -1% and providing additional inference 336 to locations with zero detections. 337 We observed large benefits in our primary objective of improving localization by integrat-

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ing an underlying movement model. First, the three movement-assisted localization modeling   Table 2: Root mean squared error of posterior means (RMSE), precision of full posterior (Prec; smaller is better), and credible interval coverage (Cov) for localization of u t from 100 simulated data sets analyzed using the independent localization model (Ind loc) and three forms of movement-assisted localization: (Mvmt DO) detection occasions only, (Mvmt KI) assuming the signal interval is known, or (Mvmt UI) modeling the unknown signal interval as a random variable. Values are presented as a function of the number of sensors where the signal was detected. Only the known-interval model localizes to a specific u t when 0 detections. Due to their rarity, results for signals detected at 7 or 8 sensors were excluded from results. Inference to locations with zero detections is possible in both the movement-assisted lo-  processes from acoustic data even when the numbers of missed signals are unknown.

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Integrating the movement process dramatically improved precision of posterior localiza-360 tions while still maintaining nominal or close to nominal credible interval coverage (Table   361 2, Fig. 3). Increased localization precision between the independent localization model and 362 the movement-assisted localization detection-only model are due directly to the integration 363 of the movement model (Fig. 3 a vs b). The known-interval model further extends inference 364 to locations with zero detections, thus the full posterior trajectory is slightly larger than 365 the detection-only model (Fig. 3 c vs b). Finally, the unknown-interval model relaxes the 366 requirement of a known number of missed signals and the interval between those signals 367 (Fig. 3 d). Interestingly, localization at the level of the full trajectory was only minimally 368 influenced by an unknown signal interval in these examples (Fig. 3 c vs d).  with explicit movement and signal rate sub-models. This approach improves localization 509 estimates and can be adapted to a variety of species-and study-specific settings such as 510 unknown signal (or cue) rates, imperfect detection, and the integration of habitat data to 511 inform individual-and population-level movement and space-use.

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Additional files 513 Appendix A R code for simulating movement, signal rate, and the detection process 514 describe in this paper, as well as for fitting all the models and some post-processing.