Monte Carlo Method

Monte Carlo methods use repeated random sampling to obtain numerical results; many simulations suggest the distribution of an unknown probabilistic result.  The method uses a probability distribution for the model variables, with a mean and standard deviation.  A random number generator picks the value for each run of the model, or each step of the run, and each run leads to differernt final position.

Three Monte Carlo simulations with the same inputs, except for the number of model runs.
500, 5000, and 25000 runs on the same map. 5000 runs, density of results
500 runs, density of results 25000 runs, density of results

Increasing the number of runs improves the resulting map, but also the time required.

The size of area increases with the number of runs, but the central region also becomes much more clearly defined.

You can then compute probabilities for each location.  The points are stored in a grid with a particular size, and you know the percentage of the points that fall within that area.

For the 5000 runs, the maximum number of runs ending a 500x500 m square was 14.  This translate to 14/5000 or 0.28% chance of that being the correct location.  This of course assumes that the model is correct, and your estimate of the error in the model is also reasonabl.e

The results will have a "bulls eye" pattern, with a most likely location and the probabilities dropping off in all directions.  The pattern may be circular or spread out, depending on the size of the standard deviations applied to the different variables.  In this case larger relative uncertainty in the speed relative to the direction leading to a longer axis for the probability distribution in the east-west direction, which corresponds to the heading.

Last revision 1/30/2017