Area Volume Under a Surface
Recently there was an email making the rounds at MathWorks about how to calculate the
area volume under a surface. Not surprisingly,
there were several methods chosen, based on each sender's proclivities. Here are some of the ways.
Data Already on a Regular Grid
If you have data already on a regular grid, you can simply call trapz twice, once along the X dimension, and once along Y. Here's an example. Let's first make some random x and y points.
xdata = [0; rand(100,1); 1]; ydata = [0; rand(100,1); 1]; x = sort(xdata); y = sort(ydata);
For now, let's calculate our gridded sampled function. Then use trapz twice.
[X,Y] = meshgrid(x,y); Z = X.^2.*sin(3*(X-Y)); trapz(y,trapz(x,Z,2),1)
ans = 0.13173
Let's compare the result to one using dblquad.
F = @(x,y)(x.^2).*sin(3*(x-y)); dblquad(F,0,1,0,1)
ans = 0.13173
Scattered Data : Finding the Convex Hull
MATLAB has the ability to deal with scattered data in a variety of ways. I've just shown, using meshgrid a technique for integrating a function that you know by sampling, and comparing this result to numerically integrating the same function that generated the samples. I'll take advantage of some of the newer computational geometry functionality in MATLAB in this next attempt.
load seamount minz = min(z); zadj = z-min(z);
Note that I "normalized" the depth data (z), since it is negative, and changed it into the array zadj that is all non-negative. Next I create an interpolant from the scattered data. Then integrate this function using quad2d, but not before trimming the minimum and maximum values for x and y so I won't run into extrapolated values (which are NaN).
F2 = TriScatteredInterp(x,y,zadj); q1 = quad2d(@(x,y) F2(x,y),211.1,211.4,-48.35,-48,'AbsTol',0.01)
q1 = 136.04
Scattered Data : Finding the
If you have access to Curve Fitting Toolbox, you can take advantage of the relatively new capability for fitting surfaces. I'll use the same seamount data as before,
created a fit to the data, and, from the function that comes from the fit, use quad2d again to compute the
f = fit([x, y], zadj, 'linearinterp'); q2 = quad2d(f,211.1,211.4,-48.35,-48,'AbsTol',0.01)
q2 = 136.04
How Do You Work with Scattered Data?
If you have scattered data representing a surface, how do you think about it and work with it - numerically or by fitting a function of some sort? Do you have other techniques? Let me know here.
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