Impact of unequal distances among acoustic sensors on cross-correlation based fisheries stock assessment technique

Cross-correlation based fisheries stock assessment technique utilized array of multiple acoustic sensors which were equidistant pair. However, at practical implementation of this technique, equal distances among acoustic sensors is sometimes challenging due to different practical phenomenon. Therefore, in this study, we work on this issue and investigated the impact of unequal distances among the acoustic sensors. We found that cross-correlation based technique proved its effectiveness even for the unequal spacing among acoustic sensors. We considered chirp generating species of fish and mammals, i.e., damselfish (Dascyllus aruanus), humpback whales (Megaptera novaeangliae), dugongs (Dugong dugong), etc., species, and three acoustic sensors array for simulation purposes. Some limitations including negligence of multipath interference, assuming the delays to be integer were compromised during simulations.

www.nature.com/scientificreports/ sensors makes this technique difficult to implement practically, which is recognized as a limitation of this technique. With an aim to overcome this limitation, we started our investigation. In this paper, we have considered three acoustic sensors, where the sensors maintain unequal distances. Three acoustic sensors can be organized by two types of topologies, i.e., acoustic sensors in line (ASL) scheme and acoustic sensors in a triangle (AST) scheme 9 . We have worked with both schemes to investigate the impact of unequal distances among the acoustic sensors. From diverse types of fish sounds, we have considered chirp sound and use its frequency for simulations. This type of sound is very common in damselfish (Dascyllus aruanus), humpback whales (Megaptera novaeangliae), dugongs (Dugong dugon), etc., species. Firstly, we have worked to establish a theoretical impact and then we have verified the theory by simulation. In this research, MATLAB R2010a was used as our simulation tool.

Background
In this section, the methodology of cross-correlation based population estimation technique 9,10,15 is deliberated. We have considered three acoustic sensors array in this study. This type of array can be either ASL form or AST form as in Ref. 10 . In this section, first, we will describe the methodology of ASL scheme and then AST scheme. As we have considered chirp sound generating fish and mammals in this study, this section will describe the estimation process with respect to this chirp signal. Let us consider a three-dimensional spherical area containing N fish and mammals and three acoustic sensors. In this area, the fish and mammals are evenly distributed over the whole region. The fish and mammals are the sources of chirp signals and the acoustic sensors are the receivers.
In the ASL scheme, the three equally separated acoustic sensors (H 1 , H 2 and H 3 ) stay in a straight line at the center of the sphere as shown in Fig. 1. The distances among the acoustic sensors are such that, d DBS12 (distance between H 1 and H 2 ) = d DBS23 (distance between H 2 and H 3 ) = d DBS (distance between the equidistant pair of acoustic sensors). Now, if chirp signal is produced by a fish or mammal, it will be recorded by three sensors with the corresponding time delays and attenuations. Henceforth, we can express the CCFs of the chirp signals received at each pair of sensors individually by a delta function 16,17 . For each fish or mammal, we will find a delta function after cross-correlation. These delta functions inhabit any position in the sample space of the corresponding pair of acoustic sensors. Now, we can consider each delta function as a ball and the samples between the corresponding pair of acoustic sensors as the bins into which the balls may fall 16,17 .
We will define the number of bins, b (as shown in Fig. 2) as twice the number of samples between the acoustic sensors, minus one as 16,17 :  www.nature.com/scientificreports/ Here, S R is the sampling rate and S P is the speed of propagation. R of CCF is defined as the ratio of standard deviation (σ) to the mean (µ) is chosen as the estimation parameter of this process, as it requires no prior knowledge of the signal strength 16 . Two estimation parameters, R 12 and R 23 are derived from two CCFs, C 12 (τ) and C 23 (τ), respectively (where, τ is the time delay), to calculate the final estimation parameter of ASL scheme 10 . Then, the final estimation parameter, R 2CCF Equal:ASL of ASL scheme is obtained by taking the average of R 12 and R 23 , and can be expressed as 10 : where, σ 12 and σ 23 represent standard deviation and µ 12 and µ 23 represent the mean of the CCFs, i.e., C 12 (τ) and C 23 (τ), respectively.
It is very complex to calculate the R of CCF using mathematical expressions. So, the cross-correlation related problem is reframed into a probability problem using the well-known occupancy problem 16 . After reframing, R of CCF can be written as 16 : Hence, Eq. (2) can be expressed as 10 : Here, b 12 and b 23 denote the number of bins of the CCFs, C 12 (τ) and C 23 (τ), respectively. In this case, b 12 = b 23 = b according to Eq. (1); as the values of S R and S P are constant during the estimation process and d DBS12 = d DBS23 = d DBS . Hence, Eq. (4) becomes Using Eq. (5), we can estimate N, as b is known from Eq. (1) and R 2CCF Equal:ASL can be calculated from the CCFs. In AST scheme, the three sensors (H 1 , H 2 and H 3 ) form an equilateral triangle for estimation purpose, where the centroid of that triangle stays at the center of the sphere as shown in Fig. 3. So, the distances among the acoustic sensors are such that, d DBS12 (between H 1 and H 2 ) = d DBS23 (between H 2 and H 3 ) = d DBS31 (between H 3 and H 1 ) = d DBS (between the equidistant pair of acoustic sensors).
In the AST scheme, three CCFs, C 12 (τ), C 23 (τ), and C 31 (τ) are used for estimation 10 . The estimation parameters, R 12 , R 23 , and R 31 are derived from three CCFs, i.e., C 12 (τ), C 23 (τ), and C 31 (τ), respectively, to calculate the ultimate estimation parameter of the AST scheme. Then, the estimation parameter R 3CCF Equal:AST of the AST scheme is obtained by averaging the R 12 , R 23 , and R 31 , and can be expressed as 10 :

impact of unequal distances among acoustic sensors
The previous section showed the estimation process with equal distances among the acoustic sensors. This section will investigate a theoretical and simulated impact of unequal distance among the acoustic sensors on population estimation. At first, we will investigate the theoretical approach and then simulated approach to justify the theory.
Here, from diverse types of fish sounds, we have considered chirp sound which is usually generated by damselfish (Dascyllus aruanus) 18 , humpback whales (Megaptera novaeangliae) 19 , dugongs (Dugong dugon) 20 , etc., species. A sound analysis of Plectroglyphidodon lacrymatus and Dascyllus aruanus species of damselfish (family pomacentridae) demonstrated that their generated chirp sounds consisted of trains of 12-42 short pulses of three to six cycles, with a durations from 0·6 to 1·27 ms; and the peak frequency varied from 3400 to 4100 Hz illustrated in Ref. 21 . The expression of this signal is found in Refs. 9,10,15 as: where, f 1 is the starting frequency in Hz, f 2 is the ending frequency in Hz, d is the duration in second, P is the starting phase, and A is the amplitude. Figure 4 shows simulated form of chirp signal which represents a simple form of chirp with duration of 1 s.

Theoretical impact.
In this subsection, we will describe the theoretical impact of unequal distances among the acoustic sensors on cross-correlation based population estimation technique. Firstly, we will investigate the theory for the ASL and AST schemes. Three unequally spaced sensors denoted by H 1 , H 2 , and H 3 are considered along a line for estimation. Here, the middle sensor H 2 is placed at the center of the estimation area. Unequal distances signify that d DBS12 ≠ d DBS23 ≠ d DBS31 . However, the process is analogous to the estimation process described in "Background". Therefore, we can write Now, b 12 and b 23 can be expressed using Eq. (1) as: And,  , equals 1, we find After reframing Eq. (14), we can find This is the condition between the equal and unequal sensor separation cases among the bins for three acoustic sensors in the ASL scheme.
However, there is another way to establish a special condition for this scheme. If we consider C 31 , i.e., CCF due to cross-correlation between acoustic signals received at H 3 and H 1 , another condition will be found. This CCF is not considered in equal sensor separation case, as only the CCFs due to the equidistant pair of sensors are used for that case. Considering the additional CCF, C 31 (τ), the ultimate estimation parameter of the ASL scheme with unequally separated sensors can be achieved by averaging the estimation parameters, R 12 , R 23 , and R 31 , derived from the CCFs, C 12 (τ), C 23 (τ), and C 31 (τ), respectively, can be expressed as: Here, b 31 is the number of bins of C 31 (τ), which can be written as In the ASL scheme, d DBS31 = d DBS12 + d DBS23 and b 12 ≠ b 23 ≠ b 31 . Hence, from Eq. (17), we can write as Therefore, from Eq. (16), we can write After reframing Eq. (19), we can write which implies a special condition for unequal distances among acoustic sensors using three sensors in ASL scheme. Now, we will establish a condition for unequal distances among acoustic sensors using three sensors in AST scheme. Analogous to special condition of ASL, AST scheme also has the similar property, i.e., If we take the ratio R 3CCF

Equal:AST R 3CCF
Unequal:AST equals 1, we will find After reframing Eq. (21), we can write This is the condition for the equal and unequal sensor separation cases among the bins for three acoustic sensors of the AST scheme.
Simulated impact. In this subsection, we will verify the theory by using simulated results. A MATLAB simulation environment was considered to obtain the simulations of proposed scheme. To acquire simulations, www.nature.com/scientificreports/ a uniform random distribution of fish and mammals was considered. The entire simulations were obtained with respect to chirp generating fish and mammals. No signal parameters were changed in the simulation. We have generated fish sounds considering different real-time parameters, i.e., frequency, time duration, bandwidth, etc. 9 . Similarly, fish signals can be represented as swift frequency wave 9 . The toolbox of MATLAB provides functions to generate swept-frequency waveforms such as the chirp function.
The following parameters were used in the MATLAB simulation as stated in Table 1. The motivation of this parameter setting is to give a clear idea about the total simulation, i.e., procedure and results, of this research.
A negligible amount of power difference among the acoustic pulses transmitted by each fish or mammal was considered to ease the simulations. We have used 500 iterations to achieve these simulated results, which mean the simulations run for 500 times and then average to achieve the plots. A SNR of 30 (26.02059 dB SNR) was considered to accomplish the simulation 11 Figure 5 shows the comparison of simulated and theoretical results. Figure 5a corresponds to the ASL scheme of three acoustic sensors and Fig. 5b   We can see from Figs. 5, 6, and 7, the simulated results closely agreed with the theoretical results. Therefore, these results demonstrate the robustness of estimation even with unequally sensors separation. At the same time, the effectiveness of the conditions derived in Eqs. (15) and (22) is proved.
However, this research has certain bounds, i.e., negligence of the effect of multipath interference, consideration of uniform random distributions, assumption of a negligible amount of power difference among acoustic pulses, assuming the delays to be integer, and consideration of acoustic sensors to be stayed at the middle of the estimation area. Similarly, as a passive acoustic method, this technique can be only applicable to the soniferous species.
During practical implementation of this technique, several matters should be taken in to account, i.e., dispersion factor, limited bandwidth problem, proper signal to noise ratio (SNR), etc. Dispersion coefficient is the main factor of the distance dependent attenuation It was found that with the increase of dispersion coefficient, i.e., dispersion loss in the medium, a significant amount of deviation occurred from actual fishery quantity to the estimated 12 . Limited bandwidth of the underwater channel poses a barrier during acquisition of fish signals, which has infinite bandwidth. To overcome this problem, a proper scaling is a mandatory task 11 . Similarly, a low www.nature.com/scientificreports/ signal to noise ratio (SNR) is also an impediment to obtain an accurate estimation. It was found that estimation with minimum 26.02 dB SNR can perform like the noiseless estimation 11 . Researches are underway to overcome the barrier of center placement of acoustic sensors. Many things can affect the speed of sounds, e.g., nature of the medium, (gas, liquid or solid), temperature, additive substances, i.e., such as salt in water, etc. Sounds travel faster through denser and hotter materials. At normal temperature, sound speed is 1493 ms −1 in fresh water, which is 1533 ms −1 in sea water 22 . However, in our research, we consider the propagation speed of fish sound is 1500 ms −1 during simulations conclusion The researchers of cross-correlation based passive monitoring technique considered equidistant acoustic sensors during their investigations previously. This consideration was a limitation of this estimation process. In this research, we have worked to eliminate this limitation of equidistant sensors. We also have worked with three acoustic sensors and derived two conditions, i.e., one regular and one special condition, for ASL scheme and one condition for AST scheme. We found that the acoustic sensors can be unequally spaced but in a way that verifies these two conditions. The conditions for both ASL and AST schemes are verified by simulations. We also have proved the robustness of this estimation process in the case of unequal sensor separation. Finally, it was our key goal to remove the barrier of the assumption of equal distance among the acoustic sensors and we have removed that in this study which satisfies our goals properly.