Porch: Estimating velocity and diffusion from tagging data 



699 



tion of the difference between the observed and pre- 

 dicted positions of all the tags. 



In practice, models capable of effectively captur- 

 ing the dynamics of the advection field may not be 

 simple enough to admit an analytical solution. The 

 obvious alternative is to evaluate the right-hand in- 

 tegral in Equation 8 numerically. Press et al. (1986) 

 contributed an excellent review of many of the more 

 powerful numerical integration algorithms. 



It may also happen that the system is not under- 

 stood well enough to construct a detailed model of 

 tag motion. In that case there is no equation to inte- 

 grate, analytically or otherwise. One option is to re- 

 sort to the piece-wise methods described earlier. 

 There is, however, a second option available provided 

 the effects of space and time on advection are sepa- 

 rable, i.e. provided Equation 1 may be recast as 



resent G and H. Although the functions must be flex- 

 ible enough to accommodate a wide range of move- 

 ment possibilities, they must also be compatible with 

 the search routine used to find the best estimates of 

 the parameters. When polynomials are used, for ex- 

 ample, the search tends to converge on the trivial 

 solution G = H = 0. Depending on the search algo- 

 rithm, it may be possible to impose constraints that 

 circumvent such problems. In any case, it may help 

 to use a theoretical model for either the spatial or 

 temporal aspect of the problem. For example, if there 

 is evidence to suggest that the pattern of motion is 

 periodic, one might write 



x T t +T 



f dx= { sinlcit - d)]dt , 



J s\x\ J 



*o 



dx 



h[t]dt, 



where the frequency (c) and offset parameter {d) are 

 (14) known. The corresponding least-squares function to 



be minimized is 



where g and h are functions of space and time, re- 

 spectively. In this case the right and left sides of 

 Equation 14 may be integrated separately: 



G[x lT }-G[x l0 }-cos[c(t M + T,-d)\+ y 

 cos[c(f,o -d)\ 



1 



f— -dx = G[x T ]-G[x ] (15) 



J s \x\ 



and 



t +T 



jh[t]dt = H[t Q +T]-H[t ]. (16) 



It may not always be possible to derive the func- 

 tions G and H from g and h , but g and h can always 

 be derived from G and H ( provided they are continu- 

 ous on the interval). To the extent that Equations 15 

 and 16 are equivalent, the suggestion is to pose flex- 

 ible functions for G or H and to fit them to the data 

 by minimizing their differences. The least-squares 

 fit, for example, would minimize the quantity 



'£{G[x iT ]-G[x i0 ]-H[t i0 + 'r i -]+H[t i0 ]) 2 . 



i 



The corresponding advection field would be obtained 

 by the formula 



u[x,t] = 



dH 



dt 



dG 



dx 



v-i 



(a proof is given in the Appendix). 



The efficacy of the separability procedure will de- 

 pend on the behavior of the functions chosen to rep- 



The function G would be some arbitrary, but flex- 

 ible, function. 



Diffusion and the objective function 



This section examines the random (diffusive) com- 

 ponent of tag motion. There are two perspectives from 

 which to do this: the absolute sense (the displace- 

 ment of individuals from their expected positions) 

 and the relative sense (the displacement of individu- 

 als in a patch relative to one another). The distinc- 

 tion is important because relative diffusion includes 

 only those physical processes acting within the patch 

 and implicitly excludes processes that might cause 

 the patch itself to vary from its expected path. Abso- 

 lute diffusion, on the other hand, includes random 

 processes operating on all scales. The trajectory ap- 

 proaches espoused in this article model each tag with- 

 out regard to its proximity to other tags; therefore 

 the statistics they produce are not relevant to the 

 diffusion of a patch. Accordingly, the remainder of 

 this discussion will focus on diffusion in the abso- 

 lute sense. 



Measures of diffusion 



Two common measures of absolute diffusion are 

 mean-square dispersion and absolute diffusivity. 



