In general, no measurement is perfect and the imperfections give rise to uncertainty in the results. Consequently, the result of a measurement or of a prediction based on measured data is only an estimate of the true value of what is being measured. Therefore, the calculation of the uncertainty is an important part of any measurement and an estimate of the true value of the measurement is only complete when accompanied by a statement of the uncertainty of the estimate [1].
Anyone who has geocached for a while can attest to the uncertainty of the position calculated by a GPS receiver. Sometimes it is spot on but other times it seems to bounce all over. Trees can be a problem as can tall buildings as anyone who has cached in a city can testify. But, most geocachers probably aren’t aware that they are relying on two aspects of Einstein’s Theory of Special Relativity to reduce the uncertainty of the measurement so that they can find plastic containers hidden in the woods. The following comes from the book, An Introduction to Uncertainty in Measurements by Kirkup and Frenkel [2]:
A GPS receiver can determine its position on the Earth with an uncertainty of less than 10 meters. This is made possible by atomic clocks carried on satellites orbiting the Earth with an approximate half-day period and a distance of about 20,000 km. The atomic clocks are stable to about one part in 10^13 (equivalent to gaining or losing one second in about 300,000 years). The receiver contains its own clock (which can be less stable) and by comparing its own clock-time with the transmitted satellite clock-timers, the receiver can calculate its own position. The comparison of clock-times must take into account the first-order Doppler shift, of about one part in 10^5 in the case of the GPS, of the frequency of a clock moving towards or away from a fixed clock. A further requirement of the accuracy of the GPS is the relativity theory of Albert Einstein. Two of the relativistic effects that must be taken into account are the slowing of satellite clocks moving transversely relative to fixed clocks and the speeding up of clocks far from the Earth's surface due to the weaker gravitational field. These two effects act in opposition and have magnitudes of about one part in 10^10 and five parts in 10^10, respectively. So, two major branches of theoretical physics have made possible timekeeping metrology of extremely high accuracy and have revealed subtle properties of times and space.
In addition, the location of the satellites relative to each other and relative to the GPS unit affects the uncertainty of the GPS determination of its position. When the satellites are close together in the sky the determined position is more uncertain than when the satellites are far apart.
The de facto method of calculating certainty has become the method of the ISO Guide to the Expression of Uncertainty in Measurement (GUM) [3, 4] and the method of GUM will be used to calculate the uncertainty for this puzzle. You won’t need to actually calculate the uncertainty of your GPS, but you will need to solve an uncertainty calculation to find the cache location.
According to GUM, if the measured value y is a function of n input quantities x such that 
, then the uncertainties of the input terms propagate into the uncertainty of y according to:
provided that the input terms are mutually uncorrelated [2]. This equation may be rewritten as:
where 
and 
are the uncertainties of y and 
, respectively and 
is the sensitivity coefficient 
.
The term 
is a partial derivative of 
with respect to 
. A partial derivative is not the same as a derivative and if you are not familiar with it, you will need to research partial derivatives to solve this puzzle.
The puzzle involves calculating the uncertainty of two values which together are functions of eight input quantities such that:
The terms a, b, c, d, e, f, g and h are constants. The values of the input quantities, the constants and the uncertainties of the input terms are given in the table below.
You must calculate 
and 
to determine the estimate of the cache location (an estimate due to the uncertainty of course, but it will be as exact as possible!).
You will need to round and to three numbers to the right of the decimal place. Values less than 5 should be rounded down and 5 and above should be rounded up.
The cache can be found at:
You can check your answers by following this link:
[1] 'The Expression of Uncertainty in Testing,' United Kingdom Accreditation Service, Middlesex, UK, Report UKAS Publication ref: LAB 12, 2000.
[2] Kirkup, L. and Frenkel, B., 'Introduction to the GUM,' Cambridge University Press, 2000.
[3] ISO/IEC Guide 98-3:2008, Uncertainty of Measurement -- Part 3: Guide to the Expression of Uncertainty in Measurement (GUM:1995), Geneva, International Organization for Standardization.
[4] Wübbeler, G., Krystek, M., and Elster, C., Evaluation of Measurement Uncertainty and Its Numerical Calculation by a Monte Carlo Method, Measurement Science & Technology, 2008, 19(8)