Random Walk or Grid Pattern search?
Let’s consider two approaches to searching a field with a metal detector and assume that the size of the field is such that in one hour your coil can pass over 10% of the area of the field. One approach we will consider is to walk around the field at random without overlapping swings. The second approach is to walk a grid pattern and overlap swings by 50%.
In the random walk you start out covering the ground at a rate of 10% per hour. But as you randomly cross over ground that you have previously covered, the rate at which you cover new ground decreases. The mathematical expression for the fraction of ground covered is [1-e-rt].
Using a grid pattern with 50% overlap you cover 5% of the ground per hour.
But the reason for overlapping swings is that the detection pattern under the coil does not go straight down from the edges of the coil. The pattern is more bowl shaped, so in a single swing you do not cover all the volume beneath the coil. A single swing only covers about 67% of the volume. On the other hand, a 50% overlap covers about 92 % of the volume. So just looking at the area covered is not fair. It will be better to look at the volume of ground covered.
The volume of ground covered by the random walk is roughly [1-e-.67rt]
The volume of ground covered by the 50% overlapped grid is [.92 x .50 x .10 x t]
Notice that the random walk is covering the ground faster until more than half the field has been searched.
So if you have enough time to search the whole field the overlapped grid search will pay off. But if you only have a couple of hours, a random walk can be just as effective.
Another strategy, if you only have a few hours, is to use a grid pattern with no overlap. You will probably never hear anyone recommend this approach because everyone knows that it is important to overlap sweeps.
For the first 10 hours you cover the volume at a rate of 6.7 % an hour. Then after 10 hours you have covered all the area and you start over, sweeping at right angles to the first pass. During the second ten hours you cover new volume at a rate of 1.7 % an hour. After about 17 hours the 50 % overlap strategy wins.