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| Subject: | Re: GISList: Contour Interval Question |
| Date: |
07/31/2001 09:43:43 AM |
| From: |
Quantitative Decisions |
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At 11:40 AM 7/31/01 +0930, David.Irving@santos.com.au wrote:
>If your accuracy is +/- 5ft, you can't justify a finer interval than 10ft.
in response to Christopher Weaver, who asked:
>If I have a series of points, covering many miles, showing elevation spread
>out at a distance of 25 feet from each other on the horizontal -- and
>they have +/- 5 foot vertical accuracy
>
>what is the recommended contour interval to use? .... and if I choose to
>use a "finer" contour interval what is the interpolated accuracy curve
>involved? ( i.e. how fine can i get away with and not have it be total BS
>:-) )
>
>... or is it possible that because a contour is generally interpreted by
>the
>algorithm, that there is only a small amount of difference in accuracy that
>a 2 foot contour is only slightly worse than say a 10ft ... because error
>is
>inherent regardless.
In some cases you can justify extremely fine intervals. It all depends on
how the data were generated, what their statistical characteristics are,
and what use you will make of them.
Evidently a complete answer would take a lot of space--a textbook
perhaps--so I will just sketch some possibilities.
First, Weaver has not indicated whether the accuracy is absolute or
relative. If it is absolute, then the relative accuracy is probably much,
much better than five feet. Thus extremely fine contour gradations are
warranted, understanding that the resulting map may very accurately reflect
fine undulations in the surface, but will be subject to an overall
"secular" inaccuracy of about five feet.
For a good example of this situation, look at USGS topographic
data. Frequently these have much larger absolute vertical (in)accuracies
than the contour intervals shown on them.
Let's now consider the case Irving must have been thinking of, where the
accuracy is relative. What that conventionally means is each point
elevation is subject to a random error and that 95% of a sample of points
that were verified were found to be within five feet of the correct value.
In the simplest case, these errors will be Normally distributed (with a
standard deviation of about 5/2 feet) and independent of each other.
Many contouring methods begin by interpolating the data onto a fine regular
grid of points. Usually these methods estimate the value at a grid point
by using a weighted average of values at nearby data points. Letting w be
a weight and z be an elevation, this is a simple formula of the form
z(grid point) = Sum over data points of z(data point) * w(data point)
If the weights have been appropriately chosen (this is where things get
complex--read a good geostatistics book for one approach to choosing the
w), then the variance of z(grid point) is equal to
Var(z) = Sum over data points of (5/2)^2 * w^2
This typically is small compared to the variances of the original
data. Suppose for the sake of illustration that 10 neighboring points are
used and a straight average (all w's are 1/10) is the best estimate of
z(grid point). Then
Var(z) = Sum over the 10 data points of (5/2)^2 * (1/10)^2
which, as you can easily work out, is (5/2)^2 / 10. In short, the variance
of the elevation at the grid point is just one-tenth the variance of the
data points. By taking square roots and doubling we can conclude that the
inaccuracy of the estimated elevation is just 1.6 feet, not 5 feet.
This is just a way of saying that combining independent observations in an
average can create a more precise result. This conclusion was hotly
debated by physical scientists until the late 18th century, but has been
accepted and widely used since then.
Thus, the interpolated estimates used for contouring could be quite a bit
more accurate than the individual raw data.
This example is an idealization: no real elevation data set will conform to
all the simplifying assumptions I have made. But it does illustrate what
can go on and, I hope, demonstrates two things:
(1) You need to know exactly how "accuracy" is defined and applied to
your data set before you can correctly answer questions about accuracy of
derived data, such as contour lines.
(2) The accuracy of derived data can actually be better than the
accuracy of the original data. Geostatistical methods are useful for
estimating the accuracy of the derived data.
Finally, I have said nothing yet about how these results will be used. If
the contour lines will be used to show the outlines of steep mountains,
little will be accomplished by using two-foot intervals instead of ten-foot
intervals. If the contour lines will be used to evaluate drainage slopes
and directions at individual locations, then possibly using ten-foo
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