Porch: Estimating velocity and diffusion from tagging data 



695 



pected velocity) and diffusion of a population. As such, 

 they should provide useful alternatives for estimat- 

 ing the movement parameters of the advection-dif- 

 fusion equation or individual-based simulations. 

 They are applicable to box models only to the extent 

 that such box models are analogous to finite differ- 

 ence approximations to the advection-diffusion equa- 

 tion. (In that case the transfer coefficients can be 

 written as functions of the local velocity and diffu- 

 sion rates.) 



This article is divided into five sections. The first 

 section highlights the conceptual differences between 

 trajectory-based and abundance-based estimators of 

 velocity. The mathematical details of the proposed 

 trajectory models are developed next. Section two 

 focuses on the process of advection and section three 

 on the process of diffusion. The performance of the 

 models and their sensitivity to violations of the as- 

 sumptions are evaluated by using stochastic simu- 

 lations in the fourth section. Finally, some practical 

 considerations and extensions to the methodology are 

 discussed. 



General concepts 



Velocity, advection, and diffusion defined 



The term 'velocity' refers to the speed and direction 

 in which an object (tag) moves. Mathematically, the 

 velocity U at position x and time t is defined by the 

 differential equation 



at 



(1) 



The map of U that assigns a velocity to each point in 

 space and time is called the velocity field. 



In theory, the position of any given tag at any given 

 time can be predicted from its velocity history by in- 

 tegrating Equation 1. In application, however, it is 

 usually more practical to describe the collective ve- 

 locity histories of a group of tags in terms of a com- 

 mon expected component u(x,t) and unique random 

 components u'(x,t). The expected component is known 

 as the advection field, and the random component 

 gives rise to the diffusion field. By using this descrip- 

 tion, the position x iT of the i'th tag after T units of 

 time can be expressed in the general form 



<,o+r 



tio+T 



x lT = x l0 + \u(x t ,t)dt+ \u'(x t ,t)dt, (2) 



',0 



tto 



where x w and t i0 are the initial position and time, 

 respectively. 



The first integral in Equation 2 determines the 

 expected position of the tag, and the second integral 

 determines the displacement of the tag relative to 

 its expectation. In terms of a group of tags with com- 

 mon starting points, the first integral describes the 

 advection of the group as a whole, and the second 

 integral determines how the group spreads about its 

 expected center of mass. 



Approaches to estimating velocity 



The purpose of estimating movement rates and other 

 parameters from tagging data is usually to elucidate 

 the behavior of a larger population — often within the 

 context of managing that population. Any such ap- 

 plication implicitly assumes that the tagged and 

 untagged members of the population move the same 

 way. This basic tenet is accepted throughout the re- 

 mainder of this article. The discussion in this sec- 

 tion focuses on the ancillary assumptions that dif- 

 ferent estimation approaches must make. 



The classical formula for estimating the advection 

 of a tag is 



" = 772/ 



iT 



WO 



(3) 



i=i 



where n is the number of observations. Jones ( 1959, 

 1976) developed an alternative formula, which, in 

 the present notation, is 



l iT ~ -SO 



i=l 



(4) 





The practicality of the estimators (Eqs. 3 and 4) is 

 limited because for these the advection field is as- 

 sumed to be relatively constant, which is unlikely 

 over the temporal and spatial scales relevant to popu- 

 lation management. Therefore, it will often be prof- 

 itable to enlist a more dynamic model of advection. 

 The parameters of the dynamic model can be esti- 

 mated by minimizing an appropriate objective func- 

 tion of some measure of the effect of advection on 

 the population. To date, the measure of choice for 

 tagging data has been the local abundance of recov- 

 ered tags. The measure used in this article is the 

 trajectory of each individual tag. 



The accuracy of any estimator depends on the as- 

 sumptions behind the construction of the objective 

 function. These involve assumptions regarding the 

 probability density of the measure and the structure 



