Introduction

Solar altitude is a function of time, longitude and latitude, and so it can be possible to infer location based on measuring altitude as a function of time. This form of solar navigation can be based on sunrise and sunset times, at least on non-equinox days.

I have explored this for a school-based project I call “SkyView” [1] involving light sensors and Arduino microcontrollers, and I suspect that readers could imagine other applications as well.

I will illustrate the ideas and the accuracy of the method based on the example of sunrise and sunset on Remembrance Day in Halifax, Nova Scotia, a city where respect for the fallen is very strong.

Methods

According to various websites [e.g. 2], sunrise on the Halifax Remembrance Day sunrise is at 7:06AM (11:06 UTC), with sunset at 4:51PM (20:51 UTC).

Rather than devising an inverse formula to infer location from time and solar altitude, the R function optim may be used to find the longitude and latitude that minimize angle the sun makes to the horizon. That angle is given by the altitude component of the list returned by oce::solarAngle().

Thus, a method for inferring the location of Halifax is as follows. The code should be self-explanatory to anyone who can read R.

1

library(oce)

## Loading required package: methods
## Loading required package: mapproj
## Loading required package: maps
## Loading required package: proj4

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misfit <- function(lonlat)
{
    riseAlt <- sunAngle(rise, longitude=lonlat[1], latitude=lonlat[2], useRefraction=TRUE)$altitude
    setAlt <- sunAngle(set, longitude=lonlat[1], latitude=lonlat[2], useRefraction=TRUE)$altitude
    0.5 * (abs(riseAlt) + abs(setAlt))
}
start <- c(-50, 50) # set this vaguely near the expected location
rise <- as.POSIXct("2014-11-11 11:06:00", tz="UTC")
set <- as.POSIXct("2014-11-11 20:51:00", tz="UTC")
bestfit <- optim(start, misfit)

# Plot coastline
data(coastlineWorldFine, package="ocedata")
plot(coastlineWorldFine, clon=-64, clat=45, span=500)
grid()

# Plot a series of points calculated by perturbing the 
# suggested times by about the rounding interval of 1 minute.
n <- 25
rises <- rise + rnorm(n, sd=30)
sets <- set + rnorm(n, sd=30)
set.seed(20141111) # this lets readers reproduce this exactly
for (i in 1:n) {
    rise <- rises[i]
    set <- sets[i]
    fit <- optim(start, misfit)
    points(fit$par[1], fit$par[2], pch=21, bg="pink", cex=1.4)
}
# Show the unperturbed fit
points(bestfit$par[1], bestfit$par[2], pch=21, cex=2, bg="red")
# Show the actual location of Halifax
points(-63.571, 44.649, pch=22, cex=2, bg='yellow')
# A legend summarizes all this work
legend("bottomright", bg="white", 
       pch=c(21, 21, 22), pt.bg=c("red", "pink", "yellow"),
       legend=c("Best", "Range", "True"))

Results and conclusions

The diagram above indicates that varying times by half a minute (corresponding to the rounding interval in public forecasts of sunrise and sunset times) yields approximately 25km of uncertainty in geographical position, at this particular time of year. (Note that a degree of latitude is about 111km.)

Readers interested in exploring the uncertainty through the year should find the R code simple to modify. It is also easy to express the uncertainty statistically.

Further discusion of matters relating to solar angles can be found in my upcoming book [3].

Resources

1. A website for the SkyView project is http://emit.phys.ocean.dal.ca/~kelley/skyview.

2. A website providing the requisite sunrise and sunset times is http://www.timeanddate.com/astronomy/canada/halifax.

3. Dan Kelley, in preparation. Oceanographic Analysis with R. Springer-Verlag.

4. Source code: 2014-11-10-solar-navigation.R.