How to integrate a modified cfit funciton?

Please find the input variables c (sound speed m/s) and z (depth m) at end. Where sound speed is a function of depth c=c(z) I can get the fitted function f by

   figure(1)
   f=fit(z,c,'smoothingspline');
   plot(f,z,c)
   xlabel('Depth z ')
   ylabel('Spund Speed C(z)')

I can do the integration from 30m to 50m for instance, which is

 integrate(f,30,50);

But how can I modify the fitted function f then integrate like

 integrate(sqrt(1./f^2-1/1800^2),30,50);

Is there any means to achieve this? Thank you!!!!!

c= [1532.35812332183;1532.09931142412;1531.55663714730;1531.13879204333;1527.17235792085;1524.06757234096;1520.77750440136;1500.98252643213;1494.68127723941;1490.39364113755;1487.89565341648;1487.62521870344;1495.01065240456;1503.30879683038;1503.53436491052];
 z= [2.47551587249922;4.48813040298639;6.50074493347357;7.50705219871715;11.0039567548543;13.3491225832363;15.1491225832363;18.8991225832363;22.6491225832363;26.3991225832363;33.8991225832363;41.3991225832363;56.3991225832363;71.3991225832363;78.5991225832363];

[Matt J]

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Answer 1

[David Goodmanson]

Hi Tsuwei,

Here are two related methods. The first way just does a smoothing spline on the new function that you have defined:

c0 = 1800;
g = fit(z,(1./c.^2 - 1/c0^2),'smoothingspline');
I1 = integrate(g,50,30)

Because of the wonky syntax for the integrate function, the limits 30 and 50 have been reversed so that you get a positive result.

Since sound speed is the primary data, you may feel that it's better to fit sound speed with a smoothing spline rather than some derived quantity. In that case

I2 = integral(@(x) (1./f(x).^2 - 1/c0^2)', 30, 50)

works (note that a transpose sign ' is necessary). Since the original function c is so good, the two methods agree to within a couple of parts per million.

[Tsuwei Tan]

Thank you David! I used your second method and writes a function as output then I can integrate. I also use Simpson's method to do the integral with a very accurate result. Thank you!