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Abstract. Modeling animal movements with Brownian motion (or more generally by a Gaussian process) has a long tradition in ecological studies. The recent Brownian bridge movement model (BBMM), which incorporates measurement errors, has been quickly adopted by ecologists because of its simplicity and tractability. We discuss some nontrivial properties of the discrete-time stochastic process that results from observing a Brownian motion with added normal noise at discrete times. In particular, we demonstrate that the observed sequence of random variables is not Markov. Consequently the expected occupation time between two successively observed locations does not depend on just those two observations; the whole path must be taken into account. Nonetheless, the exact likelihood function of the observed time series remains tractable; it requires only sparse matrix computations. The likelihood-based estimation procedure is described in detail and compared to the BBMM estimation.

The periodic sways of a group of ten Pinus contorta var. latifolia (lodgepole pine) trees with slender stems from the Two Creeks site (TW) and ten stout trees from the Chickadee site (CH) both in Alberta, Canada were quantified. Tree displacement at TW was measured during periods of consistent wind direction with three mean wind speeds (1.9, 4.6, and 5.4 m/s) and for two mean wind speeds at CH (5.0 and 7.9 m/s). Spectral analysis of sway displacement data showed a decrease in the frequency with wind speed for trees at TW, but remained unchanged for trees at CH. Significant correlations between tree sway frequency and amplitude during high winds at TW indicate a loss of sway energy concomitant with the occurrence of high collision intensity. These observations support the hypothesis that inter-crown collisions have an important influence on the sway frequency of trees and should be incorporated into efforts to model their sway dynamics. We also present a theoretical collision-damped sway model which supports our empirical findings.

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Kriging is a widely employed method for interpolating and estimating elevations from digital elevation data. Its place of prominence is due to its elegant theoretical foundation and its convenient practical implementation. From an interpolation point of view, kriging is equivalent to a thin-plate spline and is one species among the many in the genus of weighted inverse distance methods, albeit with attractive properties. However, from a statistical point of view, kriging is a best linear unbiased estimator and, consequently, has a place of distinction among all spatial estimators because any other linear estimator that performs as well as kriging (in the least squares sense) must be equivalent to kriging, assuming that the parameters of the semivariogram are known. Therefore, kriging is often held to be the gold standard of digital terrain model elevation estimation. However, I prove that, when used with local support, kriging creates discontinuous digital terrain models, which is to say, surfaces with "rips" and "tears" throughout them. This result is general; it is true for ordinary kriging, kriging with a trend, and other forms. A U.S. Geological Survey (USGS) digital elevation model was analyzed to characterize the distribution of the discontinuities. I show that the magnitude of the discontinuity does not depend on surface gradient but is strongly dependent on the size of the kriging neighborhood.