Change of Support: Post I

Let's take a break from Variography. We could dive deeper into variography, but for now, let's save that for later as we cover other topics first. This next topic is a very important topic so I'm going to give an example that is hopefully relatable to most people since I believe we all at one point had to take a cooking class or Home Economics class or our parents made us cook, or we chose to try to cook ourselves (and almost burned the house down) and maybe it didn't turn out.

Scenario...

Imagine you have a recipe to make a single cupcake because well... you don't want a recipe that makes a whole cake nor a recipe that makes multiple cupcakes, just sometimes after work you just want one cupcake and not to save the rest to try to eat or you are watching your figure :)

Cupcake Recipe (Single Serving):

• 1/4 cup flour
• 1/4 cup sugar
• 1/8 cup butter
• 1/8 cup milk
• 1/4 tsp baking powder
• 1/4 tsp vanilla extract
• 1 egg
• Temperature: 350°F (175°C)
• Cooking Time: 15-20 minutes

Let's say now you are going to throw a party. You can't just make one single cupcake and you love the recipe you have. You want to share the taste of the cupcake with everyone in the form of a cake. Now, the common thought would be to determine how big of cake you want compared to the cupcake and multiply all the ingredients by a factor...lets say you determine that factor to be 8:

• 2 cups flour
• 2 cups sugar
• 1 cup butter
• 1 cup milk
• 2 tsp baking powder
• 2 tsp vanilla extract
• 8 eggs
• Temperature: 350°F (175°C)
• Cooking Time: 15-20 minutes

Have you ever seen a cake recipe with 8 eggs???? At the closest, it was pound cake. I mean the quiche I make has 4 eggs. If I make two quiches...then its 8. But mostly for normal cake, you won't see 8 eggs. The texture would be super dense and heavy. Eggs provide structure, moisture and help with rising (or leavening). The cake would ultimately come out rubbery and that wouldn't be nice would it?

Also, if you cook the cake for 15-20 minutes like that small cupcake, you will probably have an undercooked center, uneven cooking, it will collapse so it won't be pretty and it probably won't rise all the way. The cake needs more time to reach the center.

Ok so how do I make a cake that tastes like that perfect cupcake? How do I upscale that cupcake (point data) to the bigger cake for my party (block scale)? You need to adjust the proportions (especially for the eggs and cook time) for the scale up from a tiny cupcake to a cake.

So we might do something like this:

Cake Recipe (8-inch Round Cake):

• 2 cups flour
• 2 cups sugar
• 1 cup butter
• 1 cup milk
• 2 tsp baking powder
• 2 tsp vanilla extract
• 2 eggs
• Temperature: 350°F (175°C)
• Cooking Time: 30-35 minutes

So as you can see all ingredients, excluding the eggs, were multiplied by 8. Eggs you just need to add another one. Cooking time is a bit longer to allow the heat to reach the center so that when you stick in the toothpick, it comes out clean and it won't be a sunken gooey mess.

In Geostatistics, we take samples at a small scale (drill hole data which is like the cupcake in this example) but we want to estimate at a much larger scale such as a (I always want to put an before SMU for some reason--maybe because S sounds like ess so an ess makes sense) SMU (selective mining unit --- small volume of rock that can be safely extracted or separated - are you mining it with a bucket wheel excavator or the opposite end would be a pickaxe or shovel). We need to take that small, point based data and transform it into larger spatially averaged data. But averages won't work. The transformation we will use accounts for how measurements at small scales might behave differently when aggregated into larger volumes.

When we go from points to blocks, the statistical properties (i.e., variance) need to be properly adjusted. We have change of support corrections to help transform point data into what theoretically we should expect to see out of a model. This deserves a whole post in itself to discuss which corrections exist, why they were created, what is the difference between Diffuse and Mosaic. This will hopefully help drive it all home. Maybe Diffuse vs. Mosaic will be its own post since we need to talk about deposit styles and variables. I've got one more example for next time that helps explain volume variance correction as well.

Do you use change of support corrections? Which one?

#WisdomWednesdayWithCW #Geostatistics #ResourceGeology #Mining #Geology #Metallurgy #ChangeOfSupport