2D Interpolation irregular grid fortran

Question

How can I implement 2-dimensional interpolation in FORTRAN where the data looks like as shown below. x and y are two co ordinates and z is a value dependent on them x is spaced evenly but y is not uniformly spaced and the maximum value of y corresponding to uniform values of x keep on increasing. Without losing much of accuracy-

• What is the simplest method to obtain a value of z based on a given x and y?
• What is the fastest method to obtain a value of z based on a given x and y?

Thanks SM

x    y    z
-----------
0   0     -
0   0.014 -
0   0.02  -

.....
....

0.1 0     -
0.1 0.02  -
0.1 0.03  - 

.......
.....

1.0  0     -
1.0  0.05  -
1.0  0.08  -

.......
.......

Answer 1

I am going to assume you have already read your data into an array N x 3, following the format you gave. I am assuming you don't know ahead of time what the X spacing is - you definitely don't know the Y spacing as it varies. Thus I would recommend the following strategy:

• Figure out the X spacing: start at the first row, and go through the X elements until you see a change in value. You now know XSTART and XSTEP - you will need those later.
• Do a binary search in your array for value X, until you find a value XFOUND such that XFOUND < X < XFOUND + XSTART
• Assuming you are pointing "somewhere in the list", you find the corresponding Y value - depending on whether it's bigger or smaller than the value you need, you step up or down the array until you find the first entry < Y. The corresponding values are X11, Y11, Z11. The next line in the array has X12 Y12 and Z12.
• You need two more points before you can do the interpolation - repeat this process, looking for the "next larger value of X". This will give you XYZ21 and XYZ22
• Now you can think about calculating the interpolated Z value. In general there are different techniques, with different accuracies:
• "Nearest neighbor": find the closest point, and use its value of Z (simplest, least accurate)
• "linear interpolation": find the three closest points, and do a linear interpolation of values based on the relative distances
• "higher order estimates": to do this, you typically need to create a full connectivity mapping of the grid points, so you can do spline interpolation and get a smooth interpolation that will typically be more accurate at points between the grid points (assuming that the function being described by the samples is in fact a smooth function!)

My FORTRAN is a bit rusty - hope this is some help.

PS - potentially a simpler approach is to use the fact that the X values are already evenly spaced. This allows you to make a better interpolation. See this picture: