Perpendicular distance from a track to an exact sighting position

Given a point on the track-line, the bearing the track runs on there, and the position of a sighting, returns the perpendicular (cross-track) distance from the track to the sighting, and how far along the track the foot of that perpendicular lies.

Usage

cross_track_distance(
  lat,
  lon,
  bearing,
  target_lat,
  target_lon,
  units = c("m", "km", "nmi")
)

Arguments

lat, lon: Position on the track-line, in decimal degrees — normally the event position of the sighting record.
bearing: Bearing of the track at that position, degrees clockwise from true north. See track_bearing().
target_lat, target_lon: Position of the sighting, in decimal degrees.
units: "m" (default), "km", or "nmi".

Value

A tibble with one row per input:

distance: Unsigned perpendicular distance from the track.
along: Signed along-track distance from (lat, lon) to the point where the sighting is abeam; positive ahead, negative behind.
side: "left", "right", "on-track", or NA.

The geometry

The track through (lat, lon) on bearing defines a great circle. For a sighting at angular distance $\delta_{13}$ and initial bearing $\theta_{13}$ from that point, and a track bearing $\theta_{12}$, the cross-track angular distance is

$$\delta_{xt} = \arcsin(\sin \delta_{13} \sin(\theta_{13} - \theta_{12}))$$

and the along-track distance is $\arccos(\cos \delta_{13} / \cos \delta_{xt})$, signed by $\cos(\theta_{13} - \theta_{12})$. Multiplying by the Earth's radius gives distances. The sign of $\delta_{xt}$ gives the side.

Why not just the distance between the two positions

A sighting is not always logged at the moment it is abeam. If it is recorded 300 m before the aircraft draws level, the straight distance from the event position to the animal is $\sqrt{x^2 + 300^2}$ where $x$ is the perpendicular distance the detection function needs — for a whale 130 m off the track, that is 327 m rather than 130 m, an error of a factor of two and always in the same direction. Distance sampling assumes perpendicular distances, and inflating them biases the detection function towards a wider effective strip and so density downwards.

along is returned so that this can be checked rather than assumed: it is the along-track offset between the logged position and the point where the sighting was abeam. Values near zero mean the two measures would have agreed.

References

Veness, C. (2019) Calculating distance, bearing and more between latitude/longitude points. Movable Type Scripts. The cross-track and along-track formulae are given there in the form used here.

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. Perpendicular distance is the quantity line-transect estimators are defined on.

Examples

# A track running due north; a sighting a little to the east is on the right
cross_track_distance(43, -69, bearing = 0, target_lat = 43, target_lon = -68.99)
#> # A tibble: 1 × 3
#>   distance along side 
#>      <dbl> <dbl> <chr>
#> 1     813.     0 right

# The same animal logged before it was abeam: the perpendicular distance is
# unchanged, and `along` records how far ahead it was
cross_track_distance(43, -69, bearing = 0, target_lat = 43.01, target_lon = -68.99)
#> # A tibble: 1 × 3
#>   distance along side 
#>      <dbl> <dbl> <chr>
#> 1     813. 1111. right