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Table 6 Formulae for calculating mean position of a tag in two dimensions Χ ¯ , Υ ¯ during a time interval Δ t for observation-weighted Simpfendorfer et al.[7] and model-weighted positioning estimators using arithmetic and harmonic approaches

From: Methods for assessment of short-term coral reef fish movements within an acoustic array

Method Arithmetic Harmonic
Observation-weighted* Χ ¯ = i = 1 n R i X i i = 1 n R i Y ¯ = i = 1 n R i Y i i = 1 n Ri X ¯ = i = 1 n R i i = 1 n R i X i Y ¯ = i = 1 n R i i = 1 n R i Y i
Model-weighted X ¯ = i = 1 n ω i X i i = 1 n ω i Y ¯ = i = 1 n ω i Y i i = 1 n ω i X ¯ = i = 1 n ω i i = 1 n ω i X i Y ¯ = i = 1 n ω i i = 1 n ω i Y i
  1. Symbols: n = number of receivers in the array; R i  = the number of receptions at the ith receiver during time interval Δt; X i  = the X-coordinate of the ith receiver; Y i  = the Y-coordinate of the ith receiver; ω i  = weighting factor, ω i  = (max(d i )/d i ), where d i  = model-estimated tag distance, d i =f(R i ), from the ith receiver and max(d i ) = the farthest estimated tag distance from any of the detecting receivers.
  2. * Harmonic means corrected from Simpfendorfer et al. [7].