The Trend Surface operation is a point interpolation which requires a point map as input and returns a raster map as output. One polynomial surface is calculated by a global least squares fit approaching all point values in the map. The calculated surface values are assigned to the output pixels.
Below the functions and surface types are listed, as well as the absolute minimum number of points that are mathematically required to fit such a surface. You will always need more points than this absolute mathematical minimum to obtain good results.
In general, the use of simple surfaces is preferred, as these will produce the least artificial extreme values.
Plane: |
the surface is a plane; formula: z = a + bx + cy Minimum number of points required: 3 |
2nd degree Linear: |
the surface is planar but tilted, i.e. first order plane; formula: z = a + bx + cy + dxy Minimum number of points required: 4 |
2nd degree Parabolic: |
the surface is a second order polynomial surface; formula: z = a + bx + cy + ex2 + fy2 Minimum number of points required: 5 |
2nd degree: |
the surface is a full second order polynomial surface; formula: z = a + bx + cy + dxy + ex2 + fy2 Minimum number of points required: 6 |
3rd degree: |
the surface is a third order polynomial surface; formula: z = a + ... + gx3 + hx2y + ixy2 + jy3 Minimum number of points required: 10 |
4th degree: |
the surface is a fourth order polynomial surface; formula: z = a + ... + kx4 + lx3y + mx2y2 + nxy3 + oy4 Minimum number of points required: 15 |
5th degree: |
the surface is a fifth order polynomial surface; formula: z = a + ... +px5 + qx4y +rx3y2 + ... + uy5 Minimum number of points required: 21 |
6th degree: |
the surface is a sixth order polynomial surface z = a + ... + vx6+... Minimum number of points required: 28 |
See also: