# DEM of difference - estimate the error - simple method

I have two DEMs of the same area, measured at two different dates. Subtracting one from the other, I can create a DEM of difference (DoD) to find out about the volumetric change in m³.

• DEM 1 has a resolution of 0.05 m with an uncertainty (RMSE error) of +/- 0.03 m.
• DEM 2 has a resolution of 1.00 m with an uncertainty of +/- 0.10 m.

The volume that I calculated between both DEMs is 150.000 m³.

How can I include the uncertainty of both DEMs in the resulting volume? I would like to state that there was a volumetric change of 150.000 m³ +/- xy m³ or a volumetric change of 140.000 - 160.000 m³.

I have already been searching for error propagation methods, but I am not very into the topic and I often do not understand how to do it. Any ideas?

## 3 Answers

The formula for volume difference using prismes (if the area of the two DEMs are exactly the same) and the propagation of errors:

where:

h_i - elevation from first DEM

H_i - elevations from the second DEM

a_1 - resolution of the first DEM

a_2 - resolution of the second DEM

n - number of points in the first DEM

m - number of points in the second DEM

In your case a_1 = 0.05, a_2 = 1.0, m_h = 0.03, m_H = 0.1 and if n = 4 000 000 and m = 10 000 then the mean error of volume difference is ~10 m3. (The effect of the first DEM is very small).

ADDITION TO THE FORMULA

• Does this assume the quoted errors are independent per pixel? Because things will be different if a DEM RMSE is a measure of a systematic uncertainty across all pixels. – Spacedman Dec 15 '18 at 22:39
• Yes, elevations in DEM are supposed independent. – Zoltan Dec 16 '18 at 0:15
• Zoltan, thanks a lot for the answer! Could you name the reference for the formulas, for citing those? – goofy_439 Dec 16 '18 at 9:38
• I've just applied the general rule of propagation of errors to this situation. I've not published them yet. About propagation of error you can find several sources. – Zoltan Dec 16 '18 at 10:16
• I'm afraid this answer, whether or not it might be mathematically correct, does not apply to actual DEMs, because it implicitly makes invalid assumptions about what the RMSE of a DEM actually means. Consequently, it obtains an answer that is very much too small. I have posted an explanation as an alternative answer. – whuber Dec 18 '18 at 18:53

### Summary

The answer depends on many details, but at a minimum you need to account for the area over which the DEMs are compared before you can draw a meaningful conclusion. I provide an overview of what to watch out for and some examples to show the analysis is simple to do when you make some conservative simplifying assumptions.

To get a correct answer, you need to understand what the "RMSE" really means. In most circumstances with DEMs, it is estimated by comparing the DEM's value at a relatively small number of widely scattered points to the "ground truth" at those points. Intuitively, it measures the typical size of deviation between what the DEM says and what the true value is.

You need to understand several things about this.

First, errors are spatially correlated. DEMs tend to present relatively smooth representations of relatively smooth elevation models--they rarely look speckled the way a low-light image does, for instance. This means that if a DEM at one location has a value, say, 0.10 meters high, then most likely its immediate neighbors have similarly high values. Typically, you have to move tens, hundreds, or even thousands of cells away until the error is no longer predictable.

Second, errors are not homogeneous. (The technical term is that they are heteroscedastic.) This means that the typical amount of error can vary according to other factors, especially slope. This is intuitive: where slopes are high, even a slight error in the position of an elevation reading translates into a relatively large error in the elevation itself, whereas where the elevation is horizontal, an error in position contributes no error to the elevation. (I analyzed this in some detail in a post at https://gis.stackexchange.com/a/56573/664.)

Third, errors are not necessarily independent. For instance, if one of the two DEMs being compared was derived from the other--or if both were derived from a common original source--then where one is too high the other is likely to be too high and where one is too low the other is likely to be too low. This is a beautiful thing when comparing DEMs point-by-point, because such errors cancel. But it's also conceivable that there will be no cancellation at all, such as when the DEMs are developed independently using completely different methods.

How, then, are we to exploit information about typical errors in DEMs? Unfortunately, it depends. In this case, with the limited information we have, you should proceed conservatively: make sure you don't underestimate the error. To this end, we must suppose that (a) the regions of the DEMs you have compared exhibit strong spatial autocorrelation; (b) the region of interest is rugged terrain; and (c) the DEMs were independently generated. In such circumstances we have to suppose that all the values in the first DEM could be uniformly off by an amount even greater than suggested by the 0.03 m RMSE; and that all the values in the second DEM could be uniformly and independently off by an amount even greater than suggested by the 0.10 m RMSE.

Under these assumptions, elementary probability theory states that the RMSE of the difference is at least the root sum of squares:

RMSE(difference) >= sqrt(0.03^2 + 0.10^2) = 0.1044.

As you can see, the error is dominated by that of the second DEM. This suggests we not worry too much about the details and hope that the value 0.1044 for the RMSE of difference is reasonable. But this error has to be applied to every cell.

As an example, consider a difference of "150.000" cubic meters. In some countries this means 1.5 E2 (one hundred fifty) cubic meters while in others it means 1.5 E5 (one hundred fifty thousand) cubic meters. I'll adopt the first interpretation -- it doesn't matter for this example. The issue is, how much area does that cover?

• Suppose, for instance, the DEMs are compared over an area of 10,000 square meters. Thus, the difference of 150,000 cubic meters is an average difference of

150,000 cubic meters / 10,000 square meters = 15 meters.

To this we must apply the RMSE of 0.1044 meters to give a standard error of that difference: it is 15 plus or minus 0.1044. Evidently, in this case, the RMSE shows the difference has been estimated with a high relative precision.

The corresponding RMSE of the volume is

10,000 square meters * 0.1044 meters = 1044 cubic meters,

giving a value of 150,000 plus or minus 1044 cubic meters.

• On the other hand, suppose the DEMs really were compared over an area of 1,000,000 square meters. Their average difference in this case would be

150,000 cubic meters / 1,000,000 square meters = 0.15 meters.

Now the standard error of 0.1044 is appreciable. Indeed, using the 68-95-99.7 rule, we ought to conclude that there isn't significant evidence of any real change!

The corresponding RMSE of the volume is

1,000,000 square meters * 0.1044 meters = 104,400 cubic meters,

giving a value of 150,000 plus or minus 104,400 cubic meters.

### Conclusion

Use the conservative estimates described here to obtain an approximate sense of the net error in difference. If that error is small enough for your purposes, you don't have to sweat the details. If that error is large, then look into the possibilities of low spatial correlation and/or positive correlation between the DEMs for opportunities to reduce the error estimate. You will likely have to research the metadata of both DEMs carefully to find the information you need.

• Very interesting analysis on the details of a theory with such relevance in the work we do. Allow me to clarify only that: 10,000 square meters * 0.1044 meters = 1,044 cubic meters. – Gabriel De Luca Dec 19 '18 at 1:03
• this approach sounds useful and logic. For the area I compare I end with a volume of 150 000 m³ +/- 9 200 m³, which is very useful and seems realistic. Do you have sources for the calculations? – the_chimp Dec 19 '18 at 14:56
• I am not aware of any single reference for the analysis I made, referring to spatial correlation, heteroscedasticity, and lack of independence. The calculations themselves, though, are basic: they are taught in any good introductory statistics textbook. I like Freedman, Pisani, & Purves, Statistics (any edition--buy an older used one or check it out of a library). – whuber Dec 19 '18 at 15:28

Find the absolute error of the volume depending on the surface, because the RMSE is an uncertainty in the height.

Suppose that your 150000m3 are comprised in a single square meter of surface (the height difference between both DEM is 150000m), then the total error in the height difference is the sum of both RMSE = 0.13m, then your calculation of volume throws a total error of 0.13m3. But that is not your case.

Suppose that its surface is 150000m2 and the difference in height is 1m. The maximum that you can guarantee (under the worst conditions) is that your error will be less than 0.13m * 150000m2 = 19500m3.

Therefore and apologizing if my logical thinking is faltering at some point, multiply the area covered by 0.13m and you will find the absolute error of volume on that surface. The measured volume does not matter unless you want to express the relative error (dimensionless).

Corrections are appreciated.

EDIT (based on the @Zoltan comment):

This is the conflictive sentence:

the total error in the height difference is the sum of both RMSE = 0.13m.

The propagation on that error is the square root of the sum of the squares of both errors = 0.1044m.

For an area of 10000m2, it is more accurate to estimate that error in the calculation of the difference in volumes will be less than 1044m3. But estimating that the error will be less than 1300m3 may be imprecise but not false.

UPDATE (based on this answer):

Now that I see the edition of @Zoltan to his answer I understand the error that I am committing in my intuitive procedure. Thank you.

• Your idea is interesting but false. Error propagation is not linear. – Zoltan Dec 16 '18 at 14:24
• Your intuition is basically correct except for the fact that it's not the RMSEs that add--it's their squares that add. That would reduce 0.13 = 0.10 + 0.03 to 0.104 = sqrt(0.10^2 + 0.03^2). Nevertheless, you get the right order of magnitude in your answer. The more fundamental, and profound, mistake would be to compute as if the error applied independently to every cell in every DEM. In doing so, one would obtain a ridiculously small estimate of the error of the difference. I suspect that is what might be bothering you. – whuber Dec 18 '18 at 18:55