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I have a flat, rectangular surface with an evenly-spaced grid. Let's say it is $100 \times 100$ units (in the $X$ and $Y$ directions, defining the coordinates of each grid cell), and $Z = 0$ everywhere. I have placed $10$ randomly-located normal vectors on this surface in the positive $Z$ direction (orthogonal to the surface, and none at the boundaries, and I know there locations with respect to $X$ and $Y$).

Then, the surface experiences slight deformations such that $X$ and $Y$ coordinates do not change, but $Z$ changes slightly (imagine the surface being stretched rubber, and I've placed some random objects - say, balls and cylinders - on the surface to stretch it). The surface remains smooth (has a real derivative in $X$ and $Y$ directions everywhere). I don't know the locations of the random objects. However, the $10$ surface vectors remain orthogonal to the new surface and I can measure their new direction (and change from the original direction is small, lets say all less than $1$ degree).

Is there a process to determine the new elevations $Z$ for each cell in the grid based on the known locations and change in direction of the normal vectors?

Eric Krantz
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    This is underdetermined, of course: you are trying to infer $10^4$ values from $10$ or so measurements. Presumably you are looking for the "smoothest" surface in some sense that has these normals, is that right? –  Nov 05 '19 at 04:52
  • @Rahul Yes, that is correct. I realize that there is no single unique solution. A smooth surface is welcome. Perhaps some sort of spline solution using inverse distance weighting or nearest neighbors? – Eric Krantz Nov 05 '19 at 20:58
  • (EDIT - Deleted previous similar comment due to confusion) The real-world application: I have 10 biaxial tiltmeters installed at the surface over an area that experiences subsidence. (At each tiltmeter, I can measure tilt direction in 360 degrees and magnitude). When the instruments were installed, they were all set at zero. As the land subsides in an unknown pattern (perhaps a few bowls), the tiltmeters will measure tilt while remaining normal to the "new" surface. – Eric Krantz Nov 05 '19 at 21:06
  • If you had measured the heights at the 10 points rather than the normals, the smoothest solution would be given by the thin plate spline. Maybe you can adapt the derivation given here to impose constraints on the gradient instead of the value at the known points, but I haven't thought through it carefully. –  Nov 07 '19 at 07:13
  • Yes, heights are the goal. The issue with that is to get the elevations (heights) we would need to send a surveyor out quarterly and would not have real-time data (which we have with the tiltmeters - we are collecting a data point every 10 minutes to also look for quantum movement). We could install GPS equipment at each of the 10 sites, but then we lose accuracy and the cost goes way up. Thanks for the suggestion! – Eric Krantz Nov 07 '19 at 15:43

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