KEY CONCEPTS & TECHNIQUES IN GIS Part 7 potx

66 KEY CONCEPTS AND TECHNIQUES IN GIS

the calculation is repeated for every cell for which we don’t have a measurement.

The implementation of IDW differs among software packages, but most of them

allow specification of the number and or distance of known values to be included,

and in order to function properly they must allow for the user to specify the rate at

which a location’s weight decreases over distance. The differences lie in how sophis￾ticated that distance–decay function can be. Because IDW calculates new values

only for points for which no measurements exist, it does not touch the values of

known locations and hence is an exact interpolator.

10.1.2 Global and local polynomials

Most readers will remember polynomials from their high school geometry classes.

These are equations that we use to fit a line or curve through a number of known

points. We encountered them in their simplest form in the calculation of slope, usu￾ally described in the form y = a + bx. Here we fit a straight line between two points,

which works perfectly well in a raster GIS, where the distance from one elevation

value to the next is minimal.

If the distance between the measured point locations is large, however, then a straight

line is unlikely to adequately represent the surface; it would also be highly unusual for

all the measured points to line up along a straight line (see Figure 53). Polynomials of

second or higher degree (the number of plus or minus signs in the equation determines

the degree of a polynomial) represent the actual surface much better.

Increasingly higher degrees have two disadvantages. First, the math to solve higher

degree polynomials is quite complicated (remember your geometry class?). Second,

even more importantly, a very sophisticated equation is likely to be an overfit. An over￾fit occurs when the equation is made to fit one particular set of input points but gets

thrown off when that set changes or even when just one other point is added. In prac￾tice, polynomials of second or third degree have proven to strike the best balance.

We distinguish between so-called local and global polynomials, depending on

whether we attempt to derive a surface for all our data or for only parts of it. By their

very nature, local polynomials are more accurate within their local realm. It depends

on our knowledge of what the data is supposed to represent, whether a single global

P = 0

P = 1

P = 2

2015105

Relative Weight

Distance

0

1.0

0.8

0.6

0.4

0.2

0.0

Figure 52 Inverse distance weighting

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