A substantial proportion of the transmissions was recorded at two or more receivers. For these multiple recordings, the detection range between the transmitting fish tag and each individual receiver detecting the signal was approximated as the distance between the mean position of the receivers detecting the signal and the position (Xri,Yri) of each recording receiver. This was based on the assumption that a likely location of a transmitter whose signal was detected by several receivers and was in the middle of the receivers (cf. Hedger et al. 2008). The largest of the calculated detection ranges for each transmission was considered as a minimum estimate of the detection range for that particular transmission and termed minimum detection range Dmin (eqn 1).
- (eqn 1)
The minimum detection ranges as defined here are likely underestimates of the true detection ranges, but allow for an exploration of the relative probability of signal detection as a function of range. This was achieved by taking the proportion of hourly transmissions N recorded at an increasing number of multiple receivers (i = 2–7) to the expected number of recorded transmissions Ne. These proportions served as an estimate of the probability of detection Pd at the associated minimum detection range Dmin(i) (eqn 2).
- (eqn 2)
The probabilities of detection Pd were calculated for detection data grouped by wind speeds W in 0·5-m·s−1 bins ranging from 0 to 5·5 m·s−1. In order to avoid depth dependency of the detection probability, this analysis was confined to transmissions in the 0–6-m depth interval. The expected number of recorded transmissions was calculated as the number of expected transmissions per hour (Nt) adjusted for the mean code detection efficiency Ce and the proportion Sz of the total recorded transmissions transmitted within the depth interval z (eqn 3).
- (eqn 3)
To calculate Sz for any given hour, the number of depth recordings within each one metre interval was divided by the number of depth recordings integrated over all depths for that hour. The sound pressure I (μPa) at distance D can be modelled according to (eqn 4), where I0 is the transmitter power in dB (re 1 μPa at 1 m) and αe is the sound attenuation coefficient in Nepers m−1 (Medwin & Clay 1998).
- (eqn 4)
The first part (the power function) of (eqn 4) accounts for losses due to spherical spreading, and the latter part (the exponential function) for the attenuation losses (primarily absorption and scattering). Wind mixing of air bubbles into the water should theoretically increase the attenuation by increasing both absorption and scattering. It was therefore assumed that the probability of detection was a constant P0 (P0 = 1) until a sound pressure threshold It at range D0 (see also Melnychuk & Walters 2010), beyond which the probability of detection could be described by a scaled sound profile. In the present study, we chose to use P0 = 1. Multiple recordings associated with distance Dmin and probability of detection Pd (see (eqn 2)) were used as data input to the model. An estimator of the probability of detection was then obtained by scaling the sound profile by a sound pressure threshold parameter It (eqn 5).
- (eqn 5)
The attenuation coefficient α [dB m−1] relates to αe [Nepers m−1] by a factor, such that α = αe∙8·686 (Medwin & Clay 1998). As it is the more common representation, the attenuation coefficient was chosen to be presented in dB m−1 in this paper. The dependence of α on wind was modelled by a power function (eqn 6), where W denotes mean wind speed (m s−1) for the data interval, α0 denotes the attenuation coefficient under no wind influence and β the power dependence of α on wind speed.
- (eqn 6)
An acoustic transmission model that can be used as an estimator of the probability of detection as a function of range and wind can be created by combining (eqn 5) with (eqn 6) as follows (eqn 7):
- (eqn 7)
In this sound transmission model, the parameters It, α0 and β were estimated by nonlinear least squares regression on the Pd (Dmin,W) obtained from (eqn 2). After the parameter fitting, the estimated It was used with (eqn 5) to estimate attenuation coefficients for each wind interval. The wind speed interval 0–0·5 m s−1 was considered to have a positively biased attenuation coefficient since the wind speed interval in the preceding hour could only be equal or higher. For all other wind speed intervals, the wind speed during the preceding hour could be either lower, equal or higher, and these intervals were thus considered unbiased. The 0–0·5-m s−1 wind speed interval was therefore excluded from the nonlinear regression parameter estimation.
A similar procedure as for the wind dependence was used to model the depth dependence. The detection data were divided into 1-m-depth intervals, and Pd (Dmin,z) was calculated for each depth interval and Dmin combination using eqns 2 and 3. Only one wind speed interval was used [0–1 m s−1]. Within each depth interval, the attenuation coefficient was then estimated by nonlinear regression with (eqn 5) and detection probability modelled for 0–1500 m range.