the previous post I used some geometry to show that the January UAH anomaly of 0.72˚ is not supported by the data on channel 5, footprint 15. The idea that the area of a temperature graph can determine the anomaly may not be immediately obvious, so I wanted to show that this must be so with an informal proof. I leave formalizing the proof as an exercise to the reader.
It's important this be done because in the raw data we're working with temperatures, not anomalies. But the published temperature changes are in anomalies, not temperatures. We need to be able to switch between the two and compare the two with confidence.
What Is An Anomaly?
By definition, an anomaly is a deviation from some standard. A temperature anomaly is a deviation from some standard temperature. The value of the standard temperature doesn't matter too much, so long as it's well defined. We'll say the standard temperature has some value we'll call ⣿. Like any other temperature, we can graph ⣿ on a chart. We'll say the temperature ⣿ is graphed by the rectangle a in the diagram at the beginning of this post. When we say the temperature anomaly is 0˚, that's equivalent to saying the temperature is ⣿.
What Is The Temperature Area?
The temperature area is simply the area of the graph below the curve showing the anomaly. So if the anomaly is zero, the temperature area is the area of the bar that maps ⣿.
The X axis of such a graph represents time. When can measure time in different ways. For example the X axis of a month with 31 days can be measured with 333,450 scan lines along the X axis. Or it can be measured as 31 days. Or it can be measure as 1 month. It doesn't really mater which unit we choose anymore than it matters if we call 36 inches 3 feet or a yard.
For convenience, we'll say the X axis is 1, for 1 month. (Note that I used 31 as the width in my previous post. That works just fine, but I'll probably use 1 from here on out to make it easy on myself.) This means when we want to calculate a temperature area, we need only be concerned with it's height. This is because the area of a rectangle equals width times height and width is always 1.
So, for example, the temperature area of ⣿ is always ⣿.
We can graph anomalies individually, or we can take the best fit to the anomalies. When we take the best fit we get a triangle above or below ⣿. If we instead graph anomalies we get lots of little triangles along the graph, but in the end the area of the anomalies is the same. Again, for convenience, we'll take the best fit and work with a single triangle.
If the anomaly at the beginning of the graph is zero and at the end of the graph is some positive number, ❢, then a triangle is formed with an area equal to ½❢ and the temperature area now equals ⣿ + ½❢. This is shown in rectangle b in the diagram at the beginning of this post.
If the anomaly at the beginning and end of the graph is some positive number ❢, then two triangles are formed both with an area equal to ½❢ and the temperature area now equals ⣿ + ❢. This is shown in rectangle c in the diagram at the beginning of this post.
If the anomaly at the beginning of the graph is zero and at the end of the graph is some negative number, ❧, then a triangle is formed with an area equal to ½❧ and the temperature area now equals ⣿ - ½❧. This is shown in rectangle d in the diagram at the beginning of this post.
We can see from these examples that as the anomaly changes, the temperature area also changes in direct proportion. Or, equivalently, we could say that as the temperature area changes the anomaly changes in direct proportion.
So if we ever see a change in temperature area and a change in anomaly not in sync, we know something is wrong.
Previous Posts In This Series:
Trying To Find The UAH January Anomaly In The Raw Data, Part 1 Of 2
Overview Of The Aqua Satellite Project, Update 1 Features
Aqua Satellite Project, Update 1 Released
Spot Checking The Spot Check
NASA, UAH Notified Of QA Spot Check Findings
About The Aqua Satellite Project
UAH January Raw Data Spot Check
So, About That January UAH Anomaly
A Note On UAH's High January Temperature