Geolocation of Icelandic Cod using a modified Particle Filter Method David Brickman Vilhjamur Thorsteinsson

What does one do when …

Note that the T simulation is good, but the recapture estimate is way off Note that track goes into deep water – not considered likely for Icelandic Cod Varying parameters improves results but not by that much. The best that a “standard” particle filter can do is DST recap position model recap position DST tag position model simulation 600m 200m

Why does this occur?? T field around Iceland is ~ parabolic so that particles drifting from tag location, and trying to follow T data, have 2 possible directions to choose. Temperature field ~ parabolic Climatological September T at 100m Aside: T field for this study comes from a state-of-the-art circulation model for the Iceland region developed by Kai Logemann (Logemann and Harms Ocean Sci., 2, 291–304, 2006)

OPTIONS 1. Accept that this is the best that the PF method can do and Do nothing Hide these results (~5 out of 27) 2. See whether modifications to the PF method can produce better simulations

The Data: 27 useable DSTs Example of tag being inserted into cod fish (from Star-Oddi website)

Example of DST data

Movement model: Where x n = (lon,lat) position at time n = the “state” V = (max) swim velocity = model parameter U = random # from uniform distribution dx = change in (lon,lat) position dt = timestep “Standard” Particle Filter (PF-1) (Andersen et al. 2007, CJFAS 64: ) dxdx Vdt xnxn x n+1 particle z-level = DST z-level Particles start at the initial tagging position, and evolve according to a

Observation model: Where y n = observation at time n (i.e. temperature) from the DST NB: last time includes the “recapture” observation  = error g(x) is the model temperature field at x derived from a numerical circulation model

Error Model -- Particle Filter Weight: Standard assumptions for a SIR filter yield: The probability of the observation given the state is and following Andersen et al. (and others):

Particle Filter: “PF-1” At t=0 NP particles are seeded at the (known) DST tagging position Each particle evolves according to the movement model (A) At each timestep evaluate P  particle filter weights w (B) Sample with replacement NP particles from w, preferentially choosing those with higher probability (i.e lower error). Use the standard SIR cumulative distribution method. (C) Propagate these particles to the next step Repeat A-C Continue to end of series, at which time the recapture position is an (important) observation to be incorporated into P. NB: no backward smoothing procedure coded.

Example of a Good Result Standard PF PF-1 However, note that offshelf drift is not considered biologically realistic

Modifications to standard PF Two modifications added: “Attractor” function: To increase the influence of the final (recapture -- R) position, a time-dependent term was added to the error model: distance from recapture position factor that increases as final time is approached Allows a future observation to influence present state Adds 2 parameters: time0 and  a

Interpretation of Attractor term Consider 2 particles returning the same T (i.e. T-error) – late in the simulation The estimate reported by particle 2 is considered more likely because it is closer to the recapture position. 1 2 recap position

Demersal error term: Intended to correct the tendancy for particles to follow increasing temperatures by drifting southward Observed in many simulations but considered biophysically unlikely. where z n i is the depth of the i-th particle at time n (= DST depth) and z btm (x n i ) is the model bottom depth at location x n i  d is a vertical scale parameter

Interpretation of Demersal term Assumes that the school of fish are clustered within  d of the bottom and penalizes those fish that do not fit into this “demersal” vertical distribution. Consider 2 particles at the same depth, reporting the same T the estimate reported by particle 2 considered more likely as that fish is exhibiting a more demersal behavior. Action ~ negative diffusion 1 2

New terms incorporated in an error distribution (at every timestep, for each particle): E =  {T-error + attractor-term + demersal-term} i.e. additive error distribution of un-normalized error terms, sampled using a SIR-type procedure; New Error Model Preferentially choose particles with lowest error

How to think about this -- Heuristically For this type of problem (i.e. DST) the backward smoothing procedure is essential as it is the way that the recapture observation influences the solution: Up to recap obs, PF yields optimal “local” solution Use of backward smoothing produces optimal “global” solution. E-distribution “attempts” to solve the global problem in one pass through the data.

Regarding E -- Consider minimizing a likelihood function over all observations: BTW: solved L(y|  ) for optimal parameters using a Direct Search algorithm. DS algorithm: see Kolda et al. 2003, SIAM V.46, no.3, pp

Note that the Demersal term could be incorporated into the Movement Model by including bathymetry: shallow deep shallow deep dxdx Vdt xnxn x n+1 Present model Model using bathymetry dxdx Vdt xnxn x n+1

Results

Addition of Attractor function only (PF-2) Cf: no attractor

Addition of Attractor function plus Demersal term (PF-4) Cf: attractor only

Comparison of PF-1 versus PF-4 ( NB: Different DST)

Summary / Conclusion Standard PF seen to perform poorly on a number of DSTs PF method modified by adding: Attractor term that “sucked” particles toward the recapture position allows future data to influence present result Demersal term that favoured particles that adhered to a “gadoid-type” behavior keeps particles onshelf Attractor + demersal terms can be considered to be rules or behaviors imposed on the particles. Result likely depends on Temperature field: demersal term may not be necessary

Forcing better biological behavior (addition of demersal term) resulted in poorer simulation of temperature timeseries. i.e. quantitatively WORSE results “Best” result is subjective OR Modified PFs performed better than standard PF, especially on difficult DSTs. However, When signal processing theory meets fisheries biology adjustments may have to be made