212. 
ways. In one method, called PMA1, he computes the slopes by the moving surface 
method, using here for each local polynomial the 16 nodes surrounding the grid 
element. In the second method, called PMA2, he computes each slope at a node from 
the heights at the node itself and at the two nearest ones in the required 
direction. Presumably, therefore, in the second method the slopes are simply the 
slopes of the chords connecting the adjacent nodes. In Leberl's experiments, the 
first method gives more accurate results at the centres of the grid elements. 
However, only the second method makes the total surface smooth at the nodes and it 
should give better results in the neighborhood of those points. 
Alternatively, but less desirably, each local polynomial surface can be 
made to have the specified heights at the 12 nearest nodes surrounding its grid 
element. In this case, the total surface will still be continuous but it will 
suffer a partial loss of the smoothness at the nodes. 
If also the terms with a3; and a:3 were omitted, the polynomial would 
reduce to an ordinary cubic polynomial. In a DTM called TS1, developed by 
Nakamura [62] and described also by Linkwitz [3] and by Benner and Schult [19], 
this is done. The 10 parameters of this polynomial are not sufficient to ensure 
identical profiles at common boundaries of the local polynomial surfaces and 
therefore a continuous total surface is not possible. 
6.3 An _8-term biquadratic polynomial 
A still simpler interpolation is obtained by making use of a biquadratic 
polynomial. Its equation follows from eq. (6) by omitting the terms with x3 and 
those with y? and, therefore, the fourth row and the fourth column of the matrix 
A. Usually, the term with x?y? is omitted also and the remaining eight parameters 
are computed by making the local surface fit the heights at the corners and at the 
middle of the sides of the grid element. The equation becomes here: 
ago Zo1 202||l 
h = [1 xx°]|ja10 ax1 ally 
a20 431 O JU 
Because profiles along the boundaries of the elements are quadratic 
polynomials, each uniquely determined by three points, the total interpolation 
surface is continuous. Smoothness is not enforced anymore anywhere along the 
boundaries. 
It is interesting to note that, here also, it is convenient to use the 
matrix formalism to write the parameters as linear functions of the heights. As 
before, 
A= XH HT (11b) 
Here, all matrices are of order three, the matrix H contains the heights of the 
eight points and the unknown height of the centre point, and the matrices X and Y 
contain the coordinates of these points and their squares, all in the proper 
arrangement. By specifying a2270, the last height can be written as a function of 
the other eight and be eliminated. This computation can be very easily performed 
by hand. The eight parameters turn out to be extremely simple functions of the 
eight heights. 
At the IAGB, University of Stuttgart, this method is used next to the 
one by Nakamura, see [19]. The heights in the middle of the sides are each 
computed from the heights of the four nearest points on the grid line on which 
the point is located. With sequential numbering of these heights, one finds, both