# A Bayesian method to estimate the depth and the range of phonating sperm whales using a single hydrophone

TL;DR: The author describes a Bayesian method to combine the information contained in those acoustic data plus visual observations that could be used to study the behavior of sperm whales using a single hydrophone in any location no matter what the depth, the relief, or the constitution of the seafloor might be.

Abstract: Some bioacousticians have used a single hydrophone to calculate the depth/range of phonating diving animals. The standard one-hydrophone localization method uses multipath transmissions (direct path, sea surface, and seafloor reflections) of the animal phonations as a substitute for a vertical hydrophone array. The standard method requires three multipath transmissions per phonation. Bioacousticians who study foraging sperm whales usually do not have the required amount of multipath transmissions. However, they usually detect accurately (using shallow hydrophones towed by research vessels) direct path transmissions and sea surface reflections of sperm whale phonations (clicks). Sperm whales emit a few thousand clicks per foraging dive, therefore researchers have this number of direct path transmissions and this number of sea surface reflections per dive. The author describes a Bayesian method to combine the information contained in those acoustic data plus visual observations. The author’s tests using synthetic data show that the accurate estimation of the depth/range of sperm whales is possible using a single hydrophone and without using any seafloor reflections. This method could be used to study the behavior of sperm whales using a single hydrophone in any location no matter what the depth, the relief, or the constitution of the seafloor might be.

## Summary (3 min read)

### Introduction

- Submitted on 7 Mar 2013 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not.
- A Bayesian method to estimate the depth and the range of phonating sperm whales using a single hydrophone.
- Journal of the Acoustical Society of America, Acoustical Society of America, 2007, Vol. 121, pp. 1519-1528. pp. 1519-1528. ISSN 0001-4966 Open Archive Toulouse Archive Ouverte is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.

### A Bayesian method to estimate the depth and the range of phonating sperm whales using a single hydrophone

- Christophe Laplanchea Laboratoire Images, Signaux et Systèmes Intelligents, Groupe Ingénierie des Signaux Neuro-Sensoriels, Université Paris 12, Créteil, France Some bioacousticians have used a single hydrophone to calculate the depth/range of phonating diving animals.
- The author’s tests using synthetic data show that the accurate estimation of the depth/range of sperm whales is possible using a single hydrophone and without using any seafloor reflections.
- Therefore, if the unidimensional array is vertical, then the circle is horizontal, and its depth and radius give the depth and the horizontal range of the sound source.
- One can seldom clearly detect the seafloor echoes of clicks emitted by sperm whale using a hydrophone close to the sea surface towed by a research vessel.
- The Bayesian approach has already proven to be efficient to locate sound sources using TOADs Spiesberger, 2005 .

### A. Trajectory model

- First, one would need a representation of the underwater trajectory of the whale using a mathematical model.
- Let t A be the time when the sperm whale flukes-up and starts diving, t B the time when the whale starts clicking, t C the time when the whale stops clicking, and t D the time when the whale resurfaces.
- One can then define the trajectory T using the depth, range, and heading of the whale.

### B. Prior information

- This latter definition of T uses 3n+5 parameters, that is to say redundant information.
- Indeed, the coordinates of the last summit E n+1 can be calculated given the coordinates of the first summit E 1 and using the values of speeds and change of heading in the n segments.
- Sperm whales usually keep a constant vertical speed while descending to reach their prey and while ascending to reach the sea surface Miller et al., 2004a .
- Relative to the research vessel, between the fluking and the resurfacing points.
- Such uncertainties are modeled in the following section using random variables.

### E. Probability distribution of the prior

- The author will give a more accurate description of the prior distributions appearing in Eq. 13 .
- By choosing such a prior, the author indicates that the underwater movement of the whale is not erratic, and that it is likely that the whale tends to swim in a given direction.
- The probability density functions have a maximum at the most likely prior value of the parameters, and its width illustrates the confidence the authors grant to this most likely prior value.
- One could choose for instance gamma probability distributions.
- For practical reasons, the author has chosen truncated normal distributions.

### G. MCMC algorithm

- One can use a Markov Chain Monte Carlo MCMC algorithm to draw samples of the posterior.
- One can use the Metropolis-Hastings algorithm Robert and Casella, 2004 to draw such samples.
- Calculate the acceptance ratio of this sample to determine if the new sample Instead of drawing a whole new trajectory at each iteration i, draw the summits one by one.
- In practice, the second method requires a lower total amount of samples, as it requires less than I /C samples per subchain to lead to a correct estimate T̂. The author has used the subchain method.

### H. Data set

- The author has run simulations using the free software SBPLASH implemented in MATLAB.
- The efficiency of the algorithm is illustrated using synthetic data.
- The depth, range, vertical speed, and horizontal speed of the whale when following this trajectory are given in Figs. 3–6, respectively.
- The initial value of the trajectory is in the author’s simulations the rectilinear, constant speed trajectory linking the prior locations of the points E 1 and E n+1 Figs.
- These parameters here are constants which are empirically chosen.

### A. Convergence

- The algorithm generates trajectories with values close to the data after I0=100 iterations.
- The acceptance orithm left, circles of the depth z of the sperm whale throughout the dive.
- Draws of the first s=1 and last s=n+1 summits are more often accepted than the others 2 s n , given the different constraints and priors they are bound to.

### B. Depth, range, and speeds

- The vertical and horizontal speeds of the whale are estimated using Eq. 19 .
- The estimate of the depth and the range of the whale are calculated using these values.
- Results of vz, vr, z, and r are plotted plus/minus twice their standard deviation.

### IV. DISCUSSION AND CONCLUSION

- The algorithm correctly estimates the depth and the range of the whale throughout the dive Figs. 3 and 4 .
- The author is, however, confident regarding the choice of I1 for the given data set.
- The parameters used in the probability distributions of the priors have been fixed and empirically chosen.
- As stated regarding the standard deviation of the prior speed, an optimal value of could be estimated while sampling and estimating the trajectory parameters.
- Not requiring seafloor echoes, the method could be used to estimate the depth and the range of foraging sperm whales using a single hydrophone in any location no matter what the depth, the relief, or the constitution of the seafloor might be.

Did you find this useful? Give us your feedback

...read more

##### Citations

23 citations

16 citations

### Cites methods from "A Bayesian method to estimate the d..."

...Markov-chain Monte-Carlo localization algorithms, such as that described by Laplanche (2007), or state-space algorithms, such as those employed by Jonsen et al. (2003, 2005) or Tremblay et al. (2009) may be able to take advantage of these additional data, thus such algorithms should be considered…...

[...]

11 citations

7 citations

7 citations

### Cites background or methods from "A Bayesian method to estimate the d..."

...(2012) The Journal of the Acoustical Society of America, vol. 132 (n°5). pp. 3225-3233....

[...]

...Visual methods use photo-identification to differentiate individuals, map their surface movements, and catalogue their clustering preferences (Whitehead, 2003, pp. 206–285)....

[...]

...Electronic mail: laplanche@gmail.com procedure (Davis and Pitre, 1995; Laplanche, 2007; Tiemann et al., 2006)....

[...]

##### References

12,022 citations

### "A Bayesian method to estimate the d..." refers methods in this paper

...It could be interesting to use a convergence diagnosis, like the criterion proposed by Gelman and Rubin 1992 , to make a more rigorous choice of I1 and I2 ....

[...]

271 citations

### "A Bayesian method to estimate the d..." refers background in this paper

...They emit echolocation clicks at a tremendous source level Møhl et al., 2003, 2000 and in series Whitehead and Weilgart, 1990 ....

[...]

249 citations

### "A Bayesian method to estimate the d..." refers background in this paper

...There are exceptions however: Sperm whales may, for instance, horizontally translate during the ascent likely due to the presence of conspecifics Miller et al., 2004b ....

[...]

...Sperm whales usually keep a constant vertical speed while descending to reach their prey and while ascending to reach the sea surface Miller et al., 2004a ....

[...]

...They breathe at the sea surface, fluke-up and swim downwards to reach their prey, hunt at depth, and reascend back to the sea surface Miller et al., 2004a ....

[...]

213 citations

### "A Bayesian method to estimate the d..." refers background in this paper

...They emit echolocation clicks at a tremendous source level Møhl et al., 2003, 2000 and in series Whitehead and Weilgart, 1990 ....

[...]

206 citations

### "A Bayesian method to estimate the d..." refers background in this paper

...There are exceptions however: Sperm whales may, for instance, horizontally translate during the ascent likely due to the presence of conspecifics Miller et al., 2004b ....

[...]

...Sperm whales usually keep a constant vertical speed while descending to reach their prey and while ascending to reach the sea surface Miller et al., 2004a ....

[...]

...They breathe at the sea surface, fluke-up and swim downwards to reach their prey, hunt at depth, and reascend back to the sea surface Miller et al., 2004a ....

[...]