Perpendicular distance from a declination angle

Converts a declination angle and an aircraft altitude into the perpendicular (right-angle) distance from the track-line to a sighting.

Usage

perp_distance(angle, altitude, units = c("m", "km"))

Arguments

angle: Declination angle below the horizon, in degrees. Values outside (0, 90] cannot describe a sighting below the horizon and return NA.
altitude: Aircraft altitude in metres, typically the ALT column. Non-positive or missing altitudes return NA.
units: "m" (default) or "km".

Value

A numeric vector of perpendicular distances.

The geometry

ANGLEL and ANGLER are, per handbook 8.A.2, "the declination angles, in degrees, below the horizon of a sighting (to the left or right, respectively) when it is perpendicular to the track-line". Because the angle is taken at the moment the sighting is abeam, the aircraft, the point on the sea surface directly below it, and the animal form a right triangle in the vertical plane perpendicular to the track. The perpendicular distance is therefore

$$x = h / \tan(\theta)$$

for altitude $h$ and declination $\theta$. An angle of 90 degrees is straight down and gives a distance of zero; as the angle approaches the horizon the distance grows without bound.

These replaced STRIP in 2022. Handbook 8.A.2 gives the reason: survey altitudes had to rise once offshore wind turbines were in place, and the fixed distance intervals STRIP encodes shift with altitude, whereas an angle carries the altitude dependence explicitly.

Units

ALT is in metres (8.A.1), so distances are returned in metres by default. Note that segment effort in this package is in kilometres — if you pass both to Distance::ds() you must either use units = "km" here or supply convert_units. Getting this wrong scales density estimates by 1000.

References

Kenney, R.D. (2023) The North Atlantic Right Whale Consortium Database: A Guide for Users and Contributors, Version 8, section 8.A.2. NARWC Reference Document 2023-01.

Buckland, S.T., Anderson, D.R., Burnham, K.P., Laake, J.L., Borchers, D.L. and Thomas, L. (2001) Introduction to Distance Sampling. Oxford University Press.

Examples

# Straight down
perp_distance(90, altitude = 229)
#> [1] 0

# 45 degrees below the horizon: distance equals altitude
perp_distance(45, altitude = 229)
#> [1] 229

# Shallow angles put the animal a long way off the track
perp_distance(c(60, 30, 10), altitude = 229)
#> [1]  132.2132  396.6396 1298.7235

perp_distance(30, altitude = 229, units = "km")
#> [1] 0.3966396