This post aims to educate geoscientists without or with limited prior knowledge about estimation about the fundamentals of geostatistics. We are aware a whole swath of other methods exist for estimations. However, since geostatistics is now a dominant force in estimation, we will focus on that and gently show how it produces its estimates. If interested in alternatives we recommend checking “Local models for spatial analysis by C. D. Lloyd” to get a complete overview.

NOTE: do not be afraid to read further if you are scared of maths: this post does not contain any maths (well, only one formula, to summarize)! For the more adventurous readers we wrote the mathematics behind Kriging, RBFs and Machine Learning in a separate post. They are just modifications to what we will describe below.

With this out of the way, we can start.

Think briefly: how would you estimate how tall somebody will be when they are grown up?

A typical way would be to measure the length of existing adults and average the results. The expected length will be close to this average. Most of you will know how this is done, but we’ll calculate it here to build on for later.

A simple average to produce a first estimate:

Let us assume we measured the length of 6 adult:

The average length is then a simple matter of adding all their lengths together and dividing by the number of people:

So, for every new child, we estimate they will be around 176 cm tall when grown up.

Local estimate: Netherlands

If we know a baby was born in the Netherlands we would possibly get a better estimate if we did not include any Turkish babies in our average. We then get:

Local estimate: Türkiye

Similarly, if a new baby is born in Türkiye, we would omit the dutch measurements:

And elsewhere?

Based on our prior knowledge we think can get a better estimate by only using the values from that country (thus throwing half our measurements away…). But what happens if the baby was born in Austria, or Belgium? We can’t just ignore all measurements just because we don’t have measurements from those countries. How can we still use the knowledge about where the baby was born?

If we slightly adjust our average calculations for a baby born in the Netherlands we can gently introduce how weights can be used and adjusted to get better estimates. To this end we do not just ignore the measurement lengths for Turkish people, we just set their weights to zero.

In a more general form we multiply each length with a weighting factor (WF) and then divide by the sum of all WF’s. To illustrate, let’s again calculate the average for NL first before adapting it for a Belgian baby.

The baby was in NL, but this time we estimate using WF’s to produce the total length like so:

This gives our total height of 561 cm.

Likewise, we now not just divide by the total number of measurements, but the sum of the WFs:

1 + 0 + 0 + 1 + 1 + 0 = 3

This again gives an Average 561 / 3 = 187

We cannot overstate how important this part is to realize, as it forms the basis for most geostatistics methods!

With this definition of our WF we create our estimate for a Belgian baby. This means we multiply each length with its own WF, which is 1 / distance, like so:

Adding these together gives us a total height x WF’s of 3.0022. Similarly, the the total of all weights is 0.0162.

This gives our weighted Average 3.0022 / 0.0162 = 185.32 cm

As you can see, for a Belgian baby we estimate it to be a little smaller than a Dutch baby, but taller than a Turkish baby.

WARNING: formula coming up!

We can summarize what we explained with many words in the following little formula:

\[e = \frac{\sum \omega * v}{\sum \omega}\]

Well, actually, since the sum of the weights ( $\frac{1}{\sum \omega}$ ) could be seen as a constant that could be included in the original weights. We can then simply things even more to:

\[e = \sum \omega * v\]

With this simplified method, it means for estimations we multiply each value (v) with a WF (w) and add them all together, where each WF is normalized (just means that when added all together, they add up to 1)

One reason we write the formula here is to highlight how flexible this fundamental method actually is. Until now we have used very simple weights, but even in Inverse Distance Weighting there normally is the option to not just use the distance itself, but either the squared distance or even cubed distance. The overall formula doesn’t change when we change the way we establish the weights.

To establish the weights we can be as creative as we want and could use a sine, cosine, fancy polynomials… whatever we think works for the problem at hand.

The thing to remember here (your take home message if you want) is:

If the samples remain the same for different estimations methods, the estimate only depends on the choice of the weights!