Porch: Estimating velocity and diffusion from tagging data 



697 



If, for example, faster tags were more likely than 

 slower tags to move into a region where recovery 

 rates are high, then the speed of the advection field 

 would be overestimated. 



At this point it seems convenient to examine the 

 proposed trajectory models in closer detail, noting 

 that recovery factors other than advection do not need 

 to be considered when Equation 7 is satisfied. The more 

 complicated matter of accounting for variations in re- 

 covery rates when Equation 7 is not met is deferred to 

 the Discussion section at the end of this article. 



Trajectory-based predictors for 

 estimating advection 



This section focuses on developing the predictor for 

 the directed (advective) component of motion from 

 the trajectories of recovered tags. The remaining 

 portion of the objective function, which quantifies the 

 differences between the observed and predicted val- 

 ues of the measure, depends on the nature of the 

 probability density and is discussed in detail in the 

 subsequent section entitled "Diffusion and the ob- 

 jective function." 



The general form of the trajectory predictor may 

 be written 



t, n + T, 



\dx = \u(x t ,t)dt , 



(8) 



where x iT = predicted position of tag i after time at 

 liberty T\ 

 u = estimated advection field; 

 x iQ = initial position of tag i; and 

 t i0 = initial date of tag i. 



In order to use this prediction equation, one must be 

 able to evaluate the integral on the right. There are 

 two ways to address this problem. One way is to break 

 the temporal and spatial domain into small strata 

 where the advection rates are approximately con- 

 stant and then to assemble a picture of the large-scale 

 advection field in piece-wise fashion. The other way is 

 to define explicitly a dynamic model of the advection 

 field and to evaluate the integral directly. Each ap- 

 proach is developed in a separate subsection below. 



Piece-wise models 



This approach seeks to assemble a picture of the over- 

 all advection field from estimates of the advection 

 fields in smaller strata. An independent estimate of 

 the average advection in each space and time strata 



can be obtained from any tag that has remained in 

 that strata the entire time between its release and 

 recovery (or between any two position updates) by 

 using the formula 



HT 



Wo 



If observations are available for most of the strata 

 of interest, the entire advection field can be para- 

 meterized quite nicely by using a two-way analysis 

 of variance (AN OVA) model: 



: U + A„ + S Q + /„„ + £, 



(9) 



where u 

 u 



\ 

 I.. 



e, = 



the observed velocity of tag i; 



the overall mean velocity; 



the main effect of area a on u\ 



the main effect of season s on u; 



the area/season interaction effect; and 



the error associated with tag i. 



In two spatial dimensions a separate AN OVA would 

 apply to each velocity component: 



u t = u + A au + S su + I asu + e lu , 

 v i =v+A av +S sv +I asv +e iv , 



where w ( = the observed velocity in the direction 



of the first dimension; 

 v- = the observed velocity in the direction 



of the second dimension; 

 A a = the main effect of area a on u or v; 

 S s = the main effect of season s on u or v; 

 I = area/season interaction effect on u or 



v; and 

 £. = the errors associated with tag i. 



Interpolation routines other than ANOVA may also 

 be used to describe the overall advection field, but 

 ANOVA provides a convenient framework for test- 

 ing whether the advection rates vary among strata. 

 Such tests are valid if the velocity variances are the 

 same in all strata; otherwise one must employ an 

 equivalent nonparametric approach. 



ANOVA or other interpolation algorithms are well 

 suited to situations where the positions of tags can 

 be updated frequently. They are especially promis- 

 ing for programs that employ remote tracking de- 

 vices such as radio or ultrasonic tags (e.g. Quinn, 

 1988; Hines and Wolcott, 1990; Schulz and Berg, 

 1992). In some cases the biology of the organism may 

 even permit effective visual tracking (e.g. Stacho- 

 witsch, 1979). The ANOVA approach is less suited to 



