Spatial Interpolation
1. Definition
        a procedure of estimating the values of properties at un-sampled
        sites

        the property must be interval/ratio values

        the rational behind is that points close together in space
        are more likely to have similar values than points far apart

2. Terminology
        point/line/areal interpolation
        point - point,  point - line, point - areal

        global/local interpolation
        global - apply a single function across the entire region
        local - apply an algorithm to a small portion at a time

        exact/approximate interpolation
        exact - honor the original points
        approximate - when uncertainty is involved in the data

        gradual/abrupt
 
3. Interpolation methods
        Proximal - by Thiesson polygon approach
                local, exact, abrupt
                perpendicular bisector of a line connecting two points
                best for nominal data

        B-splines - piecewise polynomial approach
                local, exact, gradual
                pieces a series of smooth patches into a smooth surface
                that has continuous first and second derivatives
                best for very smooth surfaces e.g. French curves

        Trend surface - polynomial approach
                global, approximate, gradual
                linear (1st order): z = a + bx + cy
                quadratic (2nd order): z = a +bx + cy + dx² + exy + fy²
                cubic etc.

        Moving average/distance weighted average
                local, approximate, gradual
                       S  W_iZ_i              1
                Z = --------,   W = -----  or  W = e ^-kd  etc.
                       S   W_i               d^k

       Fourier series - sine and cosine approach
                global, approximate, gradual
                overlay of a series of sine and cosine curves
                best for data showing periodicity

        Kriging - semivariogram approach, D.G. Krige
                local, exact, gradual

        regionalized variable theory, by Georges Matheron
                a situation between truly random and deterministic
                spatial dependence
                stationary vs non-stationary

        semivariogram
                                1        N
                 g(h) = ------      S     (Z_i - Z_i+h)²
                              2N      i=1

        sill, range, nugget
 
        isotropy vs. anisotropy
 
 
 

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