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Asked by jonathan valle on 30 Nov 2012

by example: NO2=( 1.1 2.4 3.3 4.7 5.9 6.0)' that corresponding to depth: Z=(4.5 6.2 8.4 10.3 12.5 14.8)' I want find d(NO2)/dz and d^2(NO2)/dz^2

Exist some function that calculate this?

Answer by Azzi Abdelmalek on 30 Nov 2012

Edited by Azzi Abdelmalek on 2 Dec 2012

**Edit**

NO2=[1.1 2.4 3.3 4.7 5.9 6.0] Z=[4.5 6.2 8.4 10.3 12.5 14.8] d1=diff(NO2)./diff(Z) d2=diff(NO2,2)./diff(Z(2:end)).^2

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Azzi Abdelmalek on 2 Dec 2012

`diff(y)./diff(t)`is an approximation of the first derivative`g=dy/dt`, In general`diff(t)`is a constant, then`diff(y)./diff(t)=cst*diff(y)`, with`cst=unique(1/diff(t))`- the second derivative
`f=d(dy/dt)/dt`is approximated by`diff(cst*diff(y))./diff(t)=cst*diff(cst*diff(y))=cst^2*diff(diff(y))` - finally
`f=diff(diff(y))./diff(t).^2=diff(y,2)./diff(t).^2`

Jan Simon on 2 Dec 2012

As far as I can see, your approximation is based on the assumption, that Z is equidistant. This is neither the general case, nor does it match the question. Therefore I think, that this approximation in unnecessarily rough, especially if the 2nd derivative is wanted.

Your method, cropped edges:

d2 = [-0.0826, 0.1385, -0.0413, -0.2079]

Suggest 2nd order method, one-sided differences at the edges:

d2 = [-0.0912, -0.0563, 0.0126, -0.0555, -0.1354, -0.1116]

Azzi Abdelmalek on 2 Dec 2012

No, even Z is not equidistant, there is no reason that diff(Z) will change at each point, we are not looking for the variation of Z, it's No2. if the approximation is bad, it's because the distance between Z's value is big. To improve the result, maybe we can interpolate.

Answer by Jan Simon on 1 Dec 2012

Matlab's GRADIENT is accurate in the fist order only for not equidistant input. See FEX: DGradient and FEX: central_difference.

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