# Bias correction and uncertainty characterization of Dead-Reckoned paths of marine mammals

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## Abstract

### Background

Biologgers incorporating triaxial magnetometers and accelerometers can record animal movements at infra-second frequencies. Such data allow the Dead-Reckoned (DR) path of an animal to be reconstructed at high resolution. However, poor measures of speed, undocumented movements caused by ocean currents, confounding between movement and gravitational acceleration and measurement error in the sensors, limits the accuracy and precision of DR paths. The conventional method for calculating DR paths attempts to reduce random errors and systematic biases using GPS observations without rigorous statistical justification or quantification of uncertainty in the derived swimming paths.

### Methods

We developed a Bayesian Melding (BM) approach to characterize uncertainty and correct for bias of DR paths. Our method used a Brownian Bridge process to combine the fine-resolution (but seriously biased) DR path and the sparse (but precise and accurate) GPS measurements in a statistically rigorous way. We also exploited the properties of underlying processes and some approximations to the likelihood to dramatically reduce the computational burden of handling large, high-resolution data sets. We implemented this approach in an R package “BayesianAnimalTracker”, and applied it to bio-logging data obtained from northern fur seals (*Callorhinus ursinus*) foraging in the Bering Sea. We also tested the accuracy of our method using cross-validation analysis and compared it to the conventional bias correction of DR and linear interpolation between GPS observations (connecting two consecutive GPS observations by a straight line).

### Results

Our BM approach yielded accurate, high-resolution estimated paths with uncertainty quantified as credible intervals. Cross-validation analysis demonstrated the greater prediction accuracy of the BM method to reconstruct movements versus the conventional and linear interpolation methods. Moreover, the credible intervals covered the true path points albeit with probabilities somewhat higher than 95 %. The GPS corrected high-resolution path also revealed that the total distance traveled by the northern fur seals we tracked was 40–50 % further than that calculated by linear interpolation of the GPS observations.

## Keywords

Biologging Dead-Reckoning High-resolution animal tracking Bayesian Melding Energy expenditure Global Positioning System Uncertainty statement Brownian Bridge## Background

Pseudotracks of animal movements calculated from tri-axial magnetometer and accelerometer tags are derived from infra-second sensor readings providing movement details at a scale that imply a high degree of accuracy and precision. However, error propagates through the pseudotrack calculating algorithm additively so that errors in actual location estimates can be relatively high, and quantifying the various sources of error, such as ocean currents and the confounding between gravitational and animal movement acceleration, is difficult. Current methods for fitting pseudotracks to known locations (such as ARGOS or GPS locations) do so deterministically making no attempt to account for the degree of error between the pseudotracks and known locations of animals or providing any sort of measure of error in the georeferenced pseudotracks. We therefore developed a BM approach to correct the high-resolution path of marine mammals reconstructed from tri-axial magnetometer and accelerometer tags with GPS observations, and quantify the combined uncertainty from all sources in the corrected path.

Bio-logging tags are increasingly being attached to animals to log and relay data about the movements and behaviour of animals that cannot be directly observed [1, 2]. Many of these tags incorporate Global Positioning System (GPS) sensors to directly and accurately determine animal locations. Unfortunately, GPS tags have a limited sampling frequency due to battery life, and often have limited exposure to satellites due to animal behaviour and habitat. This is particularly true for marine mammals that dive frequently and are only on the surface for a relatively small proportion of time. Thus, GPS can only provide a sparse and irregularly spaced record of animal locations.

The gaps in locations between infrequent GPS observations can be filled by concurrently deploying a “Dead-Reckoning” (DR) tag consisting of an accelerometer, a magnetometer, a time-depth recorder (TDR) and other supporting components [1, 3]. Such DR tags can sample at infra-second frequencies (e.g., 16 Hz) and provide a detailed record of an animal’s movement. Data downloaded from the tag can be processed by a Dead-Reckoning Algorithm (DRA) to reconstruct the DR path of the animal [1, 4, 5].

The detailed implementation of a DRA may vary in different applications [1, 4, 6], but the basic idea is as follows. First, the animal’s orientation (direction of velocity) is estimated from the smoothed accelerometer and magnetometer readings via an approximate solution to the Wahba’s problem [7]. Next, the animal’s speed can be estimated by data from other sensors, such as a TDR or speed sensor [8], or it can be derived from acceleration data or assumed to be a constant value. Speed is in turn combined with the orientation and a known starting point to create the DR path.

The DR path provides remarkably detailed information about an animal’s movements, especially fine-scale fluctuations that GPS cannot capture. However, the DR path can be biased because of poor measures of swim speeds, systematic and random error in the accelerometer and magnetometer sensors, undocumented animal movements caused by ocean currents, confounding between movement and gravitational acceleration, and discretization in the integration of the speed all lead to errors in the DR path [1, 9, 10, 11]. These biases and errors can be significant if not corrected using the relatively accurate GPS observations (by as much as 100 km at the end of a seven-day trip in the case study we explored below).

The conventional approach for correcting for DR path bias has been to add a linear bias correction term to the DR path, which directly shifts the DR path to the locations indicated by the GPS observations [1]. However, this approach lacks rigorous statistical justification and does not consider measurement error in the GPS observations. It also does not provide any measure of uncertainty about the correct path taken by the animal, because it is fully deterministic.

Our goal was to develop a statistically rigorous method for track reconstruction that overcomes the limitations of the conventional approach to determine DR paths of moving animals. We thus sought to correct the biased DR paths and quantify the uncertainty in the corrected paths.

## Methods

We collected Dead-Reckoning data from northern fur seals (*Callorhinus ursinus*) tagged in the Bering Sea, and determined their swimming paths using 1) the conventional method for DR path reconstruction [1] and 2) our proposed Bayesian Melding approach.

### Animal tagging and data processing

Two lactating northern fur seal were captured and tagged on Bogoslof Island (Alaska, USA) as part of the Bering Sea Integrated Research Program (BSIERP) [3, 12]. Three tags were attached to the fur of each seal with 5 min epoxy: a DR “Daily Diary” tag and TDR MK 10-F with Fastloc® GPS technology (both produced by Wildlife computers), and a VHF tag, which was used to ensure the success of retrieving the other tags. The accelerometers and magnetometers of the DR tag were set to sample 16 times per second (16 Hz) while the TDR pressure sensor sampled at 1 Hz. The GPS sensor was programmed to make one attempt every 15 min to connect with the satellite.

We produced the DR path for two foraging trips made by the two female seals (denoted as “Trip 1” and “Trip 2”) using the “TrackReconstruction” R package on the 16 Hz data set. This R package was developed based on [1, 4] and its detailed information can be found in [13]. We sub-sampled the DR path to 1 Hz, using only the first of the 16 locations in each second to construct the corrected animal path. We projected the GPS observations of longitude and latitude to Easting and Northing in kilometers (km) in a point-wise fashion as per [1].

### Conventional bias correction method

*T*by \(y_1\) and \(y_T\), respectively; 2) assume \(x_1 = y_1=0\) and calculate the corrected path \(\hat{\eta }_t\) as

### Bayesian Melding

The Bayesian Melding (BM) approach was pioneered to combine direct observations of air-pollutant concentrations from a sparse network of monitoring stations and the computer model outputs at each pixel of a map, based on known pollutant sources and geophysical information [14]. This BM approach was later adapted by different fields to characterize such things as hurricane surface winds [15], ozone levels [16], and wet deposition [17]. All of these applications have demonstrated the remarkable flexibility and effectiveness of the BM approach.

The GPS observations obtained from our fur seals are analogous to the monitoring station measurements in the first BM application [1], and the DR path of our moving seals is similar to the computer model output of drifting air-pollutants. Using the GPS to correct DR paths combines the location information from two independent sources (the GPS and DR path), which is the strength of Bayesian Melding.

#### Model choice

*Animal’s true path*In our BM approach, we assumed the one-dimensional path \(\eta (t)\) of the animal was a Brownian Bridge process, whose mean

*f*(

*t*) at a time

*t*and covariance function \(R(s, t) = \mathrm{Cov}\,(\eta (s), \eta (t))\) for two time points

*s*,

*t*are:

*GPS observations*The GPS observations of the locations were denoted by \(Y(t_k), k=1, 2, \ldots , K, t_1=1, t_K=T, t_k \in \{2, \ldots , T-1\}, k=2, \ldots , K-2\), which are unbiased observations of the true location:

*k*stands for the

*k*-th GPS observation obtained in this trip and the \(t_1, t_2, \ldots , t_K\) are irregularly spaced in (1,

*T*).

*DR path*We used \(X(t), t=1, 2, \ldots , T\) to denote the DR path without any error correction and incorporated the bias of the DR path by assuming:

*h*(

*t*) is a parametric function designed to capture the systematic bias, and \(\xi (t)\) is a Brownian Motion process of mean zero, which can model the unstructured error in the DR path.

In principle, *h*(*t*) could capture the structured error that can be expressed as a deterministic function of time in the DR path, such as those caused by a constant bias in the accelerometer readings, the currents, and other external forces. However, there are still some “random” errors caused by the measurement error in the accelerometer, magnetometer, or speed measurements that cannot be expressed as a deterministic function of time. These random errors may add white noise to the velocity estimates (speed in two dimensions) in the DRA, which will accumulate in the DR path. The sum of white noises over time results in a Brownian Motion process. Thus, we considered \(\xi (t)\) to be a Brownian Motion process, whose covariance function is \(C(s, t) =\sigma ^2_D (\min (s, t)-1)\), where \(\sigma ^2_D\) can be viewed as the variance of the white noise added to the velocity in the DRA.

Various models can be considered for *h*(*t*), such as \(h(t)= \sum _{i=1}^Q \beta _i t^{i-1}\). For simplicity, we chose to illustrate our method using \(h(t) = \beta _0\). We had explored more complicated models [11] and found them to have little impact on the corrected paths and uncertainty in the two fur seal data sets we tested.

This simple model in (2) can cover various factors of biases or errors in the DR path. For example, if there was a constant bias in the speed estimate, the integration of this constant bias over time is a linear function of time, which can be captured by our *h*(*t*) part. On the other hand, the random error in the animal’s speed is absorbed into the \(\xi (t)\). This model choice may not be the best to describe all the bias and error factors, but it worked well for our data sets, as shown below.

*Priors for parameters* The final ingredients in our BM model were the prior distributions of the parameters. We fixed \(\sigma ^2_G\) at 0.0625 based on the extensive tests of the Fastloc® GPS device [24, 25] and the average observed number of satellites present during the two fur seal foraging trips. Non-informative priors were chosen for the remaining parameters, such that \(p(\sigma ^2_H) \propto \frac{1}{\sigma ^2_H}\) and \(p(\sigma ^2_D) \propto \frac{1}{\sigma ^2_D}\), and \(p(\beta _0) \propto 1\).

*Univariate versus bivariate modeling* The two dimensions of the path (i.e., Northing or Easting) were analyzed separately in our BM approach. In theory, our method could be improved if we simultaneously analyzed the two dimensions by considering them as a bivariate Brownian Bridge process. Using a bivariate model would require us to consider a time varying correlation parameter to avoid assuming the animal constantly moves in one direction for the whole trip. However, it is unclear whether the additional parameters needed to represent the time varying correlation function would actually improve its prediction performance. This should be explored in future studies.

#### Computation of the BM model

- 1.
Find a set of reasonable values of \(\sigma ^2_H\) and \(\sigma ^2_G\) based on the GPS observations and the DR path at the corresponding time points. The weight of each pair \(\sigma ^2_H\) and \(\sigma ^2_G\) was decided using the likelihood of the data.

- 2.Conditioning on each \(\sigma ^2_H\) and \(\sigma ^2_G\), calculate the conditional posterior of \(\eta (t)\) in two steps:
- (a)
The posterior of \(\eta (t)\) at the GPS time points and parameters in

*h*(*t*) were decided based on the GPS observations and the DR path at the corresponding time points. - (b)
The rest of \(\eta (t)\) was broken into periods separated by the GPS observations and updated based on the DR path only for the period together with the posterior of \(\eta (t)\) at the two GPS end points.

- (a)
- 3.
Numerically integrate the conditional posterior with the weights found in Step 1.

In the simplified situation where *h*(*t*) (systematic bias) is a constant zero, our algorithm in Step 2-(b) flattened the conventional bias correction by a factor of \(\frac{\sigma ^2_H}{\sigma ^2_D + \sigma ^2_H}\) and added it to the linear interpolation between the two consecutive GPS observations. As the GPS observations were relatively precise, the linear interpolation decided the general direction of the animal’s movement. The DR path nevertheless offered detailed information on the animal’s movement, in sprite of some of these details being just errors of the DR path. Under our model, the animal’s true path, the Brownian Bridge process, contributed \(\sigma ^2_H\) variance to the DR path while the error process \(\xi (t)\) accounted for \(\sigma ^2_D\) variance. Therefore, a proportion \(\frac{\sigma ^2_H}{\sigma ^2_D + \sigma ^2_H}\) of the details from the DR path could be treated as a “signal” from the animal’s true path, which we added to the linear interpolation. In other words, Step 2-(b) can be viewed as a compromise between linear interpolation and conventional bias correction, where the weights on each method in the compromise were decided based on \(\sigma ^2_D\) and \(\sigma ^2_H\). This compromise is illustrated below.

Setting \(h(t)=0\) for all *t* is a simplified situation that helps explain one step of our method. In practice, however, our BM method also considered the bias function *h*(*t*) in Step 2-(b) and marginalized the randomness of the variance parameters \(\sigma ^2_D\) and \(\sigma ^2_H\) in Step 3. As a consequence, the correct path from our BM approach is not just a weighted average of the linear interpolation and the conventional method, and does not have a mathematical formula that can be easily interpreted.

In summary, our BM approach requires the inputs of projected GPS observations (Northing or Easting), the corresponding Northing or Easting of the uncorrected DR path, and the variance of the GPS observation \(\sigma ^2_G\). These inputs return the posterior mean of \(\tilde{\eta }(t)\) as the corrected path, and the posterior standard error \(\tilde{\sigma }(t)\) serves as an uncertainty statement about the corrected path.

### Cross-validation

*m*consecutive GPS observations (

*m*= 1 or 5) were unavailable and carried on correcting the DR path without these

*m*GPS observations. We then compared the predictions from the corrected path to the left-out GPS observations. This procedure was repeated for all the GPS observations except for the start and end points. The difference between the observations \(Y(t_k)\) and the predictions \(\check{\eta }(t_k)\) were summarized as the cross-validation root mean squared error:

## Results and discussion

### Exploratory data analysis about GPS observations

The two foraging trips made by the fur seals were each about 1 week. Trip 1 was 7 days and had 276 valid GPS observations, while Trip 2 lasted about 7.5 days and had 129 GPS observations (Fig. 2). Some of the gaps between the GPS observations were quite large as shown by the histograms of time gaps and (great circle) distances (in km) between any two consecutive GPS observations (Fig. 3). Although a large proportion of the time gaps were within 1 h, 13 % of those in Trip 1 and 30 % of those in Trip 2 were greater than 1 h—with time gaps averaging 36 min for Trip 1 and 82 min for Trip 2. In terms of distances between GPS locations, 7 % of the spatial gaps in Trip 1 and 17 % in Trip 2 were greater than 5 km. The largest between-GPS distance in these two trips was nearly 50 km. Thus, the GPS observations were irregularly spaced in time and space, and the gaps between them were quite large.

### Bayesian Melding results

Applying our Bayesian Melding approach to the Easting and Northing of the two trips to fill in the gaps between GPS observations successfully corrected the bias of the DR path in all four analyses (Fig. 2). Plotting the corrected path together with the CI in the analysis and the DR path of Trip 1 further illustrates our method (Fig. 4). Bias of the DR path dramatically increased with time (and was about 100 km by the end of this trip—Fig. 4), and the DR path contained fluctuations consistent with the fluctuations of the GPS observations. As shown by the black curve (Fig. 4), our BM approach successfully produced a path that passed through the GPS observations. Similar plots were obtained from the analysis of the other three data sets (data not shown).

Zooming in on a portion of the track shown in Fig. 4 reveals how our approach works in the fine scale (see Fig. 5). Figure 5 also provides a way to visually compare the conventional method and linear interpolation (ignoring the DR path and connecting the consecutive GPS observations by straight lines)—and shows that our method differs from linear interpolation between GPS observations by keeping some of the tortuosity exhibited by the DR path (also seen in Fig. 2). The posterior mean from our BM approach appears to be a flattened version of the conventional bias correction, where the bumps in the conventional method are damped to the linear interpolation between two GPS points. A mathematical explanation of this is provided by [11].

The CI for \(\eta (t)\) in Fig. 5 shows a clear “bridge” structure. Namely, the CI was narrow when \(\eta (t)\) was close to the GPS observations and became wider between two GPS points. The posterior SE at the GPS points almost equaled the \(\sigma _G = 0.25\) (km) in the input to our BM, which decided the upper limit of our precision about the corrected path. For the time points between two GPS points, we had less accurate information from the DR path, which increased the posterior SE.

As an overall summary of the accuracy of our corrected paths, we calculated the averaged posterior SE (APSE, \(\frac{1}{T}\sum _{t=1}^T \tilde{\sigma }(t)\)). For Trip 1, the APSE was 0.524 km in the Easting direction and 0.444 km in the Northing. In contrast, the APSE of Trip 2 Easting and Northing were 1.45 and 1.28 km respectively, which were greater than those from Trip 1. This indicates that the BM corrected path for Trip 2 was less accurate than that for Trip 1. It also reflects the fact that fewer GPS observations were obtained for Trip 2, and the gaps between consecutive GPS observations were larger, which means that there was less accurate information to correct the biased DR path.

### Coverage percentage of our CI

Results from the leave-one-out cross-validation analysis of the four data sets

Data set | BM approach | Conventional | Linear interpolation | |
---|---|---|---|---|

Coverage (%) | CV-RMSE | CV-RMSE | CV-RMSE | |

Trip 1 Easting | 99.3 | 0.37 | 0.49 | 0.43 |

Trip 1 Northing | 98.2 | 0.39 | 0.50 | 0.51 |

Trip 2 Easting | 99.2 | 1.03 | 1.02 | 1.47 |

Trip 2 Northing | 100 | 0.85 | 1.15 | 1.06 |

Results from the leave-5-out cross-validation studies of the four data sets

Data set | BM approach | Conventional | Linear interpolation | |
---|---|---|---|---|

Coverage (%) | CV-RMSE | CV-RMSE | CV-RMSE | |

Trip 1 Easting | 97.8 | 0.73 | 1.04 | 1.13 |

Trip 1 Northing | 95.9 | 0.80 | 1.25 | 1.16 |

Trip 2 Easting | 93.7 | 2.52 | 2.67 | 3.84 |

Trip 2 Northing | 92.9 | 3.06 | 3.33 | 4.44 |

### Comparisons with the conventional method and linear interpolation

Our BM approach had a smaller CV-RMSE than linear interpolation in all the four data sets, while the conventional method had a larger CV-RMSE than linear interpolation in some cases (e.g., Trip 1 Easting and Trip 2 Northing in the LOOCV). Heuristically, more data should give better results. The conventional method had more data than the linear interpolation (where the DR path was ignored) and thus it should have had a smaller CV-RMSE than linear interpolation, which was not true in our case study. In this way, the conventional method failed to use the additional information properly to produce better predictions.

In the comparison of our BM to the conventional method, the CV-RMSE of our BM approach was smaller than those from the conventional method in seven out of the eight cross-validations. For Trip 2 Easting in LOOCV, the CV-RMSE of our method was slightly larger than that of the conventional method. This might be caused by \(h(t) = \beta _0\) failing to capture the bias of the DR path sufficiently well. This issue can be addressed by allowing more flexible structure as in *h*(*t*) [11].

Although many of the consecutive GPS observations in our case study were close, there were some huge gaps (e.g., a 4-h time gap and 50 km distance gap) as shown in the large right tail of the histograms in Fig. 3. In these cases, linear interpolation did not work, while our method provided a good estimate/prediction of the animal’s location based on the DR path. This was not as apparent in the tabulated results of our cross-validation analyses because most of the gaps were small. This conclusion is also reflected by the improvement that resulted from our method when going from the leave-one-out CV to leave-5-out CV (Tables 1, 2). In practice, the DR tag and our BM method significantly improve track reconstruction when the gap between GPS observations cannot be controlled.

Computational time in seconds of different bias correction methods

Trip | Dead-Reckoning | BM | Conventional |
---|---|---|---|

1 | 137.7 | 103.4 | <0.5 |

2 | 192.8 | 40.8 | <0.5 |

### Distance traveled by the animal

Total distance (km) traveled in the two fur seal foraging trips

Trip | Linear interpolation | BM | Conventional |
---|---|---|---|

1 | 418.25 | 585.95 | 815.36 |

2 | 443.30 | 662.91 | 1023.88 |

## Conclusions

We found that the GPS observations were too sparse and irregular to sufficiently describe the foraging paths of northern fur seals. The uncorrected DR paths included useful detailed information about the paths, but were severely inaccurate. Our proposed BM approach successfully provides a corrected path along with credible intervals of uncertainty—and can be further improved with more flexible parameterization [11]. Our analysis of the BM method also provides a statistically rigorous foundation for using the DR path to answer many other bio-logging questions.

Our BM approach requires some statistical knowledge and takes more time to compute when compared to linear interpolation and conventional bias correction methods. However, our method can be directly applied using our R package (as shown in additional file 1 of this paper) without understanding all of the formulas, and the computational time is reasonable when compared to the time to download the data and perform the DRA. Moreover, our method achieves greater prediction accuracy than the other two methods as shown by the cross-validation studies—and also quantifies the uncertainty in the corrected path with credible intervals, which cannot be obtained in the linear interpolation nor the conventional method. Further improvement of BM approach is likely to come with the inclusion of multivariate and non-stationary modeling of the animal’s path.

## Notes

### Authors' contributions

YL developed and implemented the proposed Bayesian Melding approach, carried the data analysis and cross-validation studies, and drafted this paper. BB edited the paper, collected the biologging data and provided YL guidance on analyzing the Dead-Reckoning path. JZ first suggested the Bayesian Melding approach and oversaw YL’s research. AT oversaw the animal tagging experiment and edited the paper. All authors read and approved the final manuscript.

### Acknowledgements

Yang Liu and James Zidek thank the National Sciences and Engineering Research Council of Canada for supporting their research, and Prof. Nancy Heckman for additional financial support. All data collection procedures were conducted under the National Oceanic and Atmospheric Administration (NOAA) Permit No. 14329 and abided by the guidelines of the Committee on Animal Care at the University of British Columbia (Permit No. A09–0345). We are indebted to A. Baylis, J. Gibbens, R. Marshall, R. Papish, A. Will, C. Berger, A. Harding, R. Towell, B. Fadley, K. Call, C. Kuhn and N. Liebsch for assistance or advice with animal captures and instrument deployment. The data was collected as part of the BEST–BSIERP “Bering Sea Project” funded jointly by the US National Science Foundation and the North Pacific Research Board.

### Competing interests

The authors declare that they have no competing interests.

## Supplementary material

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