Is there any recommended method for finding the optimum number and location of rain gauges over a region based on the data from existing rain gauges. There is one Isohyetal method that does not give the exact location.
The answer, as is often the case for questions about optimums, depends on objectives and constraints.
How accurately do you want to measure rainfall? The more gauges the more precision, but obviously at greater cost.
How important is the spatial element, which in turn might depend on the objectives of monitoring? Is it more important to get accurate total volumes, or to identify variability and extremes?
Imagine two cases in the same region, with a hilly area, a hydroelectric dam and an agricultural lowland. The dam operators would want to accurately characterize inflow to their dam, while the farmers might want to measure dry season rainfall to schedule irrigation. A third agency might be satisfied with a single gauge that most closely represented a regional average. Their definitions of optimum would be rather different.
With objectives determined the next question is where do you get data on existing rainfall and variability? Often it's the case that these questions of optimization are about rationalizing or refining existing networks. In this case you will have records from the existing network.
While the detailed approach may vary, the basic principal is to compare the contribution, in terms of information, of each rain gauge to your quantitative objective. You can then eliminate the gauge that contributes the least information. This process can be repeated until a minimum network that meets the project objective is arrived at. Computationally refining this method to look at all possible combinations of subsamples is usual. I.e. for a 10 gauge network you would look at all combinations of 9 gauges, then of 8 etc. until the objectives were met.
A similar approach can be used to add gauges to a network, by looking at the correlations between adjacent rain gauges and siting gauges in areas of poor spatial correlation.
«A New Approach for Optimizing Rain Gauge Networks: A Case Study in the Jinjiang Basin» by Wu et al. is a good worked example of this process using geostatistics. Its reference list provides links to many other case studies.
Wu, H.; Chen, Y.; Chen, X.; Liu, M.; Gao, L.; Deng, H. Water 2020, 12, 2252; (https://doi.org/10.3390/w12082252, open access).
For simple calculation, use this: https://www.calculatoratoz.com/en/optimum-number-of-rain-gauge-stations-calculator/Calc-5418
You need to know the CV and degree of error (example: 10%).
CV can be calculated from average and stdev of monthly rainfall in certain area. The equation is CV = 100*stdev/avg.
Then number of optimum rain gauge will be N = (CV/error)^2.
If you have more data (watershed, lands, or slope and elevation), you can also add this data and make different scenario (flat, mountain, forest
Rainfall varies according to a lot of different factors, which include the location of mountains and hills, bodies of water, and vegetation.
One approach is to collect data interactively with the environment. To do that keep placing rain gauges. If they are reading the same, separate them. So, by trial and error you determine a good distribution.