Trend Surface Basics

 Hanging Rock DEM, and 1st to 8th order trend surfaces.  These use every point in the initial DEM.

The program uses a least squares fit after Davis (1973). It will fit up to a eighth order surface. The terms in the fit include:

• z = C1 + C2*x + C3*y
• + C4*x^2 + C5*x*y + C6*y^2
• + C7*x^3 + C8*x^2*y + C9*x*y^2 + C10*y^3
• + C11*x^4 + C12*x^3*y + C13*x^2*y^2 + C14*x*y^3 + C15*y^4
• ........ similar progression up to eighth order

The first order surface is a plane and includes terms C1 to C3 (only terms in x and y); the second order surface adds terms C4 to C6 (terms in x² and xy and y²); the third order surface adds terms C7 to C10 (terms in x³ and x²y and xy² and y ³); and the fourth order surface adds terms C11 to C15. After the first order surface, each additional order adds another inflection point to the surface, and more terms to the equation.

The higher the order of the trend surface, the "better" the fit. With a high enough order, and number of terms in the equation (one term per point in the DEM) you could fit the earth's surface exactly.  That is not generally the purpose of the trend surface, which seeks a simple surface in the absence of noise.

Last revision 10/23/2012