extrapolate to min point between grid points using slope trends

I have a lat/lon grid with height values at each evenly spaced grid point. My grid has a minimum height value at its center grid point, but I want to extrapolate to the minimum height (and associated lat/lon) that lies somewhere between grid points based on the slope trends about the minimum central point.

My first thought was to use surf(lon,lat,height) to fit a surface and then find where the second derivative is >0 for the local min, though as I understand the surf function would use the minimum height of my grid as the minimum height of the surface and not extrapolate to a lower value than what appears on the grid.

Is there a matlab function or recommended process to extrapolate to a minimum point on a grid based on the surrounding slope trends? Thank you.

Answer 1

[Jamie Rodgers]

Do You have the curve fitting toolbox?

If so:

Possible Strategy: use 'fit' to create a 'fit object' using a suitable input array and a selected fit type, then apply this to an output array ofthe required precision in the Region of interest. (You could arrive at this by a coarse uninterpolated min) Then simply take the 'min' and index

e.g. here is a surface with a minimum - at lowish resolution

[lat,lon]=meshgrid([-10:1:10],[-5:1:5]);
height=((lat-3.65).^2+(lon+2.22).^2+lat.*2)+0.251;
surf(lat,lon,height);
hold on

Now create a surface fit object to the surface

myfit = fit( [lat(:), lon(:)], height(:), 'poly23', 'Robust', 'LAR' );

I have selected a poly23 interpolation

There are other options. Use 'doc fit' to explore

now apply 'myfit' in the region you want to interpolate at high resolution: in this case between where lat =1 and 3 and lon = -3 and -1

[Newlat,Newlon]=meshgrid(1:1e-3:3,-3:1e-3:-1);
newheight=myfit([Newlat],[Newlon]);

Take the minimim from this interpolation

minheight=min(min(newheight));

index to find lat and lon that correspond,

[r,c]=find(newheight==min(minheight));
interpolated_min=[Newlat(r,c),Newlon(r,c),minheight];

 bullseye!  

scatter3(interpolated_min(1),interpolated_min(2),interpolated_min(3),'ro','filled')

[Walter Roberson]

Hi, Jamie,

The places that are code should be proceeded with two spaces, and the places that are human-readable text should not be proceeded with spaces. You had used the other way around. I have adjusted your Answer.

[Jamie Rodgers]

Thanks... new and not used to this formatting

[Walter Roberson]

somehow it got flipped back again. I fixed it again.

[Nina]

Thanks, Jamie. Will give it a go. Just to verify, this approach should allow me to obtain an extrapolated of the surface minimim that could be lower than my lowest gridpoint value?

Answer 2

[Nina]

Here is how I actually ended up solving this problem and interpolating to values lower than the range of my gridded data. Used function griddata and a cubic interpolation. As I am new to this forum, I do not know how the formatting works, apologies! Just wanted to share this solution for anyone else who may need it!

figure

[loni,lati]=meshgrid(lonmin:0.25:lonmax,latmin:0.25:latmax);  %where lonmin, latmin are the min,max values of original gridded data and 0.25 is the step between them used to create the finer scale grid [loni,lati]

heighti=griddata(testplot(:,1),testplot(:,2),testplot(:,3), loni,lati,'cubic'); %heighti=interpolated height values; testplot(:, 1, 2, and 3) are lon, lat, height values

surf(loni,lati,heighti); %creates surface from interpolated values

xlabel('Longitude'),ylabel('Latitude'),zlabel('Height (gpm)');

minz=min(testplot(:,3)); %min value of gridded height field=5370 gpm in this example

minzi=min(min(heighti)); % min value of interp height field=5336 gpm

[a,b]=find(heighti==minzi); %find indicies of (lon,lat) for min height on interpolated  grid

Answer 3

[Jamie Rodgers]

Nina,

Yes it most certainly can - BUT this will depend upon the type of surface fit you employ: Also you need to be aware that if you do higher polynomial fits based on inputs that are too far away from the ROI, strange results may be seen. Choose a fit that gives good agreement (see documentation) and be sensible in creating the fit from the data near the ROI... Not much point in trying to extrapolate the height of Everest from a contour map of Scotland!