Availability: the probability an animal was at the surface to be seen

Computes the probability that an animal was available to be detected while it was within view, from its surfacing and diving intervals and the time the survey platform kept it in view. This is the availability component of g(0), and it is computed from external dive data, not estimated from the survey.

Usage

availability(
  surface,
  dive,
  window,
  se_surface = NULL,
  se_dive = NULL,
  cov_surface_dive = 0,
  key = NULL
)

Arguments

surface: Mean surfacing interval, in seconds. A vector gives one result per element, which is how a per-month table is built.
dive: Mean diving interval, in seconds. Recycled against surface.
window: Time the animal is within view, in seconds. See view_window() to derive it from platform geometry. Recycled against surface.
se_surface, se_dive: Standard errors of surface and dive. Both are needed for a standard error on the result; either alone is an error, since propagating one source of variance and silently dropping the other understates the total.
cov_surface_dive: Covariance between the two interval means. Zero by default, which is right when they are estimated from separate follows and optimistic when they are not.
key: Optional labels — months, platforms, years — carried through to the result so the rows stay attached to what they apply to.

Value

A tibble with one row per element of surface: key, component (always "availability"), value, and se. That is the shape a g(0) correction is assembled from, so availability and perception rows stack.

Why this is a calculation and not an estimator

An animal submerged while the aircraft passes is missed at every perpendicular distance equally. To first order availability is a pure scale factor on $g(x)$: it does not dent the near-zero end of the distance distribution, does not change its shape, and leaves no signature for a likelihood to find. That is exactly why a mis-specified g(0) shifts every candidate in a selection_table() by the same factor and leaves the ranking looking untouched.

So there is nothing in the distances to estimate it from, and none of the inputs here come from the survey being corrected. surface and dive come from focal follows, tagging, or drone observation; window comes from the platform's geometry and speed. Both arrive with their own uncertainty, which is the point: it is visible rather than absorbed into a constant.

The formula

Following Laake et al. (1997), for a mean surfacing interval $E(s)$, a mean diving interval $E(d)$, and a window $w$ during which the animal is in view:

$$a = \frac{E(s) + E(d)\left(1 - e^{-w/E(d)}\right)}{E(s) + E(d)}$$

The two limits are worth holding onto as a check on any number this returns. As $w \to 0$ the window is instantaneous and $a$ becomes $E(s)/(E(s) + E(d))$, the plain proportion of time spent at the surface. As $w \to \infty$ the platform watches forever, every animal surfaces eventually, and $a \to 1$. The middle term is the extra chance that an animal which was down when the window opened comes up before it closes.

There is no single availability, and that is the finding

Ganley et al. (2019) measured this for right whales in Cape Cod Bay and found availability varying by month from 0.27 in January to 0.91 in April, tracking the depth of the copepod layer the whales were feeding on. Those measurements ship as ganley_availability.

A plausible-looking single figure is therefore the most dangerous input this package accepts. A value near 0.83 is a real number for some month, and wrong by a factor of three for others — and because it scales every model equally, no amount of model selection reveals it. Compute one per key and pass a vector, rather than picking a representative value.

Note that this is availability alone. Perception is a separate component and a separate measurement, which Ganley et al. did not make — estimating it needs a second observer team. g0() will say so if you leave it out.

How the standard error is obtained

Availability is deterministic given surface, dive and window, so the uncertainty comes from them. Given se_surface and se_dive, the Jacobian is taken numerically and combined with their covariance matrix — the same delta-method treatment effect_estimates_ddf() uses.

Where the raw focal follows are in hand, resampling them is better than this: Ganley et al. (2019) and others bootstrap the follows and take the standard deviation of the resulting availabilities, which does not assume the interval means are jointly normal. Do that and pass the result on as se_surface and se_dive, or bypass this and build the rows directly.

Without se_surface and se_dive the standard error is NA, deliberately: this function will not invent a precision for a number whose CV routinely dominates the CV of abundance.

References

Laake, J.L., Calambokidis, J., Osmek, S.D. and Rugh, D.J. (1997) Probability of detecting harbor porpoise from aerial surveys: estimating g(0). The Journal of Wildlife Management 61:63-75. doi:10.2307/3802415 The formula implemented here.

Laake, J.L. and Borchers, D.L. (2004) Methods for incomplete detection at distance zero. In Advanced Distance Sampling, pp. 108-189. Oxford University Press. Why perception needs a second observer team, and availability needs something outside the survey entirely.

Ganley, L.C., Brault, S. and Mayo, C.A. (2019) What we see is not what there is: estimating North Atlantic right whale Eubalaena glacialis local abundance. Endangered Species Research 38:101-113. doi:10.3354/esr00938 Focal follows and aircraft field of view applied to right whales, and the monthly variation quoted above.

Roberts, J.J., Yack, T.M., Fujioka, E., Halpin, P.N., Baumgartner, M.F. and others (2024) North Atlantic right whale density surface model for the US Atlantic evaluated with passive acoustic monitoring. Marine Ecology Progress Series 732:167-192. doi:10.3354/meps14547 Corrects perception and availability per platform, team and conditions across 11 institutions, which is the scale at which these corrections actually vary.

Examples

# An instantaneous window is the proportion of time at the surface
availability(surface = 60, dive = 240, window = 0)
#> # A tibble: 1 × 4
#>   key   component    value    se
#>   <chr> <chr>        <dbl> <dbl>
#> 1 NA    availability   0.2    NA

# A real window lifts it: some animals that were down come up in time
availability(surface = 60, dive = 240, window = 30)
#> # A tibble: 1 × 4
#>   key   component    value    se
#>   <chr> <chr>        <dbl> <dbl>
#> 1 NA    availability 0.294    NA

# One row per month, which is how it is actually used
availability(
  surface = c(45, 60, 90),
  dive    = c(300, 240, 150),
  window  = 24,
  key     = c("Feb", "Mar", "Apr")
)
#> # A tibble: 3 × 4
#>   key   component    value    se
#>   <chr> <chr>        <dbl> <dbl>
#> 1 Feb   availability 0.197    NA
#> 2 Mar   availability 0.276    NA
#> 3 Apr   availability 0.467    NA

# With uncertainty on the intervals
availability(surface = 60, dive = 240, window = 24,
             se_surface = 8, se_dive = 25)
#> # A tibble: 1 × 4
#>   key   component    value     se
#>   <chr> <chr>        <dbl>  <dbl>
#> 1 NA    availability 0.276 0.0297