GPS Measurement Error Gives Rise to Spurious 180° Turning Angles and Strong Directional Biases in Animal Movement Data

Background Movement data are frequently collected using Global Positioning System (GPS) receivers, but recorded GPS locations are subject to errors. While past studies have suggested methods to improve location accuracy, mechanistic movement models utilize distributions of turning angles and directional biases and these data present a new challenge in recognizing and reducing the effect of measurement error. Methods I collected locations from a stationary GPS collar, analyzed a probabilistic model and used Monte Carlo simulations to understand how measurement error affects measured turning angles and directional biases. Results Results from each of the three methods were in complete agreement: measurement error gives rise to a systematic bias where a stationary animal is most likely to be measured as turning 180° or moving towards a fixed point in space. These spurious effects occur in GPS data when the measured distance between locations is <20 meters. Conclusions Measurement error must be considered as a possible cause of 180° turning angles in GPS data. Consequences of failing to account for measurement error are predicting overly tortuous movement, numerous returns to previously visited locations, inaccurately predicting species range, core areas, and the frequency of crossing linear features. By understanding the effect of GPS measurement error, ecologists are able to disregard false signals to more accurately design conservation plans for endangered wildlife.

Using the standard trigonometric relationships, since the measured step length (L t ) forms the hypotenuse of a right angled triangle with adjacent side of lengthx t+1 −x t and an opposite side of lengthŷ t+1 −ŷ t . The absolute value of the determinant of the Jacobian matrix for this change of variables isL tLt+1 . With these change of variables Eq. S.4 is, The last change of variables is to determine the measured turning angleτ t =θ t+1 −θ t . Noting that −τ t =θ t −θ t+1 and cosτ t = cos(−τ t ). Integrating outL t ,L t+1 andθ t , The probability densities of u t and θ t have previously been described in more detail in Solow (1990) where they are referred to as A i and D i respectively.

Solving the turning angle integral for turning angles
In this section I solve the triple integral Eq. S.6.
Integration on the plane (L t ,L t+1 ) is performed over the first quadrant. Changing to polar coordinates, Using another change of variables, Taking the antiderivative, (S.14) Let,

Probabilistic model for directional bias
The derivation and analysis of the probabilistic model for directional bias is analogous to the probabilistic turning angle model. I consider the special case where the animal is stationary and is located at the bias point where (x t , y t ) = (x t+1 , y t+1 ) = (Ψ, χ) = (0, 0). The distribution of measurement error is assumed bivariate Normal distributed (Eq. S.1). The joint probability of any pair of locations (x t ,ŷ t ) and (x t+1 ,ŷ t+1 ) is, (S.18) I used the change of variables, whereŵ t andẑ t are the x and y distances to the bias point. With the change of variables, (S.20) I used another change of variables, where absolute value of the determinant of the Jacobian matrix isL tMt . The directional

Solving the triple integral for directional bias
In this section I solve Eq. S.21, where x = 2 cos ζ t and, where d/dx can be taken out of the integral because the last integral is convergent for all x between −1 and 1.