How can I interpolate different dimension array on a common grid?

Hi everybody,

I find myself with spatialized data coming from different sources that I'd like to compare by interpolating them on a common grid, but I still don't understand how.

I have two sets of data of Longitude, Latitude, Altitude and a scalar value: in the first set all of these variables are vectors (1 row * n columns), while in the second set they are n-dimensional matrixes (for Longitude I have a matrix of m rows * n column * p layers having a longitude value for each [m,n,p] point in this "cube", the same goes for Latitude, ...).

I would like to use the minima and maxima of values of Longitude, Latitude and Altitude to create a common 3D grid on which to interpolate the scalar values.

I can give an example of two different datasets and the steps I did until now along with how I used Matlab's grid interpolation functions (dataset values are purely fictional and way much bigger, it's just to have an example).

Dataset 1:

Lon = [-0.98 -0.97 -0.96] % degrees
Lat = [44.1 44.2 44.3]    % degrees
Alt = [380 400 450]       % meters
Scal = [22.2 35 40]       % concentration

Dataset 2:

 Lon=[-0.982 -0.981;-0.980 -0.975;-0.970 -0.965] % degrees
 Lat= [44.09 44.10;44.11 44.22;44.24 44.23] % degrees
 Alt(:,:,1)= [320.2 330;327 350;322 327] % meters
 ...
 Alt(:,:,4)= [470 447.1;475 480;477 450] % meters

 % Again Scal is a m*n*p matrix where p is equal to 4.

Since Dataset 2 is the one with the most widely spanning data, I've constructed a grid on the basis of those data using ndgrid:

MinLat=min(min(Lat));
MaxLat=max(max(Lat));
StepLat=abs(MaxLat-MinLat)/100;
MinLon=min(min(Lon));
MaxLon=max(max(Lon));
StepLon=abs(MaxLon-MinLon)/100;
MinAlt=min(min(Alt(:,:,1));
MaxAlt=max(max(Alt(:,:,3));
StepAlt=abs(MaxAlt-MinAlt)/100;
LonQueryPoints=MinLon:StepLon:MaxLon;
LatQueryPoints=MinLat:StepLat:MaxLat;
AltQueryPoints=MinAlt:StepAlt:MaxAlt;
Grid=ndgrid(LonQueryPoints,LatQueryPoints,AltQueryPoints);

Grid is now a 100*100*100 matrix made by the query points of Longitude, Latitude and Altitude and it's where I'd like to interpolate my scalar values and I've tried it using the "griddedInterpolant" Matlab function:

% Applied to Dataset 2:
Scalar_Interpolation=griddedInterpolant(Grid,Scal(:,:,1:4))

And I receive the error:

1D Interpolation requires vector input for X and V

Probably I'm not understanding correctly how to employ the griddedInterpolant function, or, maybe, the griddedInterpolant is only able to handle m*n*1 matrixes.

[dpb]

Is this similar to the recent question from Gavin on lat, lon interpolation? It seems awfully close if not.

http://www.mathworks.com/matlabcentral/answers/108164-how-can-i-change-a-lat-long-and-ellipsoid-separation-ascii-file-into-a-matrix

Do these data actually span the ranges or are they, like Gavin's, lacking information on one or more boundaries so that in fact one needs to extrapolate mostly instead of interpolate?

A small sample but representative data set would likely help to envision the situation more precisely.

[dpb]

Indeed griddedinterpolant is for 2D data only; you'd have to do each plane independently. The only builtin for 3D is interp3 which needs plaid input a la ndgrid

You might check on the File Exchange for more flexible routines. In your case since it appears you are in fact interpolating within a grid, it should work. The other posting I referred to wasn't so lucky...

[Federico]

Thanks for the info dpb, I will search on File Exchange for other interpolation routines. I cannot do it on a plane-by-plane basis because my objective is exactly to have it "de-planed" and simply interpolated on a continuous vertical axis (in this case Altitude) so that I can compare it with Dataset 1 data... ...unless I make Dataset 1 more similar to Dataset 2 (i.e.: create altitude planes for that also) and try to compare it plane by plane...

[dpb]

It appeared to me you could use interp3 on the one dataset to obtain points at the locations of the other. It wasn't/isn't totally clear to me what your meaning of compare is here.

[Federico]

The thing is that Dataset 1 is made of field observations of a scalar (made at different Lon/Lat/Alt) and Dataset 2 is made by values of the same scalar but outputted by a model (at different Lon/Lat/Alt): I'd like to be able to interpolate them on a common grid so that, in the end, I'm able to say that the model is correctly evaluating the scalar (or not) and what's the difference, for example, in terms of RMSE between the two.

The problem with interp3 is that

The grid vectors are not strictly monotonic increasing

[dpb]

See the other thread I mentioned wherein the input values were rearranged in ascending order--seems same idea should work here.