Modeling sound attenuation in heterogeneous environments for improved bioacoustic sampling of wildlife populations

Acoustic sampling methods are becoming increasingly important in biological monitoring. Sound attenuation is one of the most important dynamics affecting the utility of bioacoustic data as it directly affects the probability of detection of individuals from bioacoustic arrays and especially the localization of acoustic signals necessary in telemetry studies. Therefore, models of sound attenuation are necessary to make efficient use of bioacoustic data in ecological monitoring and assessment applications. Models of attenuation in widespread use are based on Euclidean distance between source and sensor, which is justified under spherical attenuation of sound waves in homogeneous environments. In some applications there are efforts to evaluate the detection range of sensors in response to local environmental characteristics at the sensor or at sentinel source locations with known environmental characteristics. However, attenuation is a function of the total environment between source and sensor, not just their locations. In this paper I develop a model of signal attenuation based on a non-Euclidean cost-weighted distance metric which contains resistance parameters that relate to environmental heterogeneity in the vicinity of an array. Importantly, these parameters can be estimated by maximum likelihood using experimental data from an array of fixed sources, thus allowing investigators who use bioacoustic methods to devise explicit models of sound attenuation in situ. In addition, drawing on analogy with classes of models known as spatial capture-recapture, I show that parameters of the non-Euclidean model of attenuation can be estimated when source locations are unknown. Thus, the models can be applied to real field studies which require localization of signals in heterogeneous environments.

signal and the source and therefore more general models of attenuation are needed.
Ideally, the end use of bioacoustic data in monitoring and assessment of biological pop-68 ulations should integrate explicit models of sound attenuation with parameters that are 69 themselves estimated in situ along with biological parameters of interest such as density, 70 position of sources, occupancy, or other ecological state variables. In this paper I sug-71 gest flexible classes of models for modeling attenuation in heterogeneous environments using 72 cost-weighted distance in which effective distance is defined by a cost function that involves 73 spatially explicit structure describing a heterogeneous landscape. This non-Euclidean dis-  The data from a field experiment of this sort are power or signal strength measurements, S ij , at each sensor having location x j , from each source i = 1, 2, . . . , n (n = 16 in this case) having location s i . The model for these observations can be formulated in terms of other declines with distance d from the source, and assume the transformation produces a normally distributed variable such that attenuation is well approximated by the model An intuitively appealing model for sound attenuation in heterogeneous environments is the cost-weighted distance (CWD) model in which attenuation is governed not by Euclidean distance but by a cost-weighted distance metric which depends on the habitat structure in the vicinity of the sensor. The cost-weighted distance can be computed for a path P = )} consisting of m segments between any two points v 1 and v m+1 on the landscape and it is defined by where cost(v g , v g+1 ) is a parametric function describing the cost of movement between pixels 112 v g and v g+1 , which must be prescribed (see below) and dist(v g , v g+1 ) is the Euclidean distance 113 between pixels. The cost-weighted distance then is the sum over all pixels along a given path  The relevance of this distance metric to inference about sound attenuation arises when the cost function is parameterized in terms of the landscape structure. For example, if a covariate z(v) exists then one sensible function describing the cost of passing from pixel v g to pixel v g+1 is The parameter α 2 represents the resistance of the covariate z(v) (higher values incur higher cost of transmission and vice versa), and it should be estimated from observed data on signal strength or time of arrival. I provide an estimation framework based on maximum likelihood below. To acknowledge this new distance metric in the model for sound attenuation, and that it depends on an unknown parameter α 2 , express the model as In general attenuation is frequency dependent (Wiley and Richards 1982) and thus param- the least-cost path between any two points can be computed using the function costDistance.

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To see the effect of cost weighted distance on "effective distance" Fig. 2 shows contours 148 Figure 2: Effective distance to a sensor (shown by +) placed at (3,3) under the leastcost path model with parameter α 2 = 0.4 (left), α 2 = 1.4 (center) and α 2 = 2.4 (right). As resistance increases, effective distance contours get closer together in response to dense structure (green).

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The cost-weighted distance metric described above is amenable to direct likelihood analysis from data on observed signal strength at fixed locations and with fixed sources (e.g., as in Fig. 1). The observed data from an experiment are the detection/signal strength pairs (y ij , S ij ) for each source and each sensor. Recall that signal strength is truncated at some value c chosen to reflect a reasonable threshold below which signals cannot be distinguished from ambient noise. Conditional on the J known source locations x j , the likelihood for the data from source location s i is These probability distributions depend on the model parameters as in the likelihood given above but I omit that dependence to be concise. A standard assumption in spatial capturerecapture is to assume no a priori information about the location of a source so that Pr generally, source density gradients can be accommodated by modeling explicit covariate effects in Pr(s). For example, suppose the sound sources are birds and they are likely to be using habitat preferentially, even the same habitat which is affecting sound attenuation, then we might assume where z(v) is the measured habitat structure for any location v and θ is a parameter to be 172 estimated.

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R code for computing the posterior distribution of detected sources is given in Appendix