second_derivative shows different results from that of applying first_derivative twice

[664787022]

What went wrong?

I want to calculate second derivative of a variable, but I find second_derivative shows different results from that of applying first_derivative twice.
There is a big difference between the two results. Which one is correct to abtain second derivative? Thank you in advance！

import xarray as xr
import matplotlib.pyplot as plt
from metpy.calc import first_derivative, second_derivative

z = xr.DataArray([5,3,4,8,9,6,4,3,1,5,6,4,5,2,4,1,3,4,1,2,5,3,5,1,1], coords={'x':np.arange(25)}) 

dzdx = second_derivative(z, axis='x').values 
dzdx2 = first_derivative(first_derivative(z, axis='x'), axis='x').values 

plt.plot(dzdx, label='second_derivative')
plt.plot(dzdx2, label='twice first_derivative')
plt.legend()

Operating System

Windows

Version

1.3.1

Python Version

3.9.13

Code to Reproduce

import xarray as xr
import matplotlib.pyplot as plt
from metpy.calc import first_derivative, second_derivative

z = xr.DataArray([5,3,4,8,9,6,4,3,1,5,6,4,5,2,4,1,3,4,1,2,5,3,5,1,1], coords={'x':np.arange(25)}) 

dzdx = second_derivative(z, axis='x').values 
dzdx2 = first_derivative(first_derivative(z, axis='x'), axis='x').values 

plt.plot(dzdx, label='second_derivative')
plt.plot(dzdx2, label='twice first_derivative')
plt.legend()

Errors, Traceback, and Logs

No response

[kgoebber]

I'll refer you to the discussion that happened in closed issue #1389 that contains information about how the two calculations are fundamentally different. Essentially when applying the first derivative twice, you get a smoothed first pass. Additionally, the discussion of issue #893 might be relevant to the conversation around some of the mathematics at play here. Hope this helps distinguish the differences that occur when employing these particular MetPy calculations.

[dopplershift]

To add to what @kgoebber posted above, if you neglect the formulation around non-uniform spacing that MetPy uses, the centered, "3-point" approximation in first_derivative is:
$ f'(x) \approx (f_{i+1} - f_{i - 1}) / 2 \Delta x$

If you apply this twice, you end up with:
$ f''(x) \approx (f_{i+2} - 2f_i + f_{i-2}) / 4 \Delta x^2$

This is the same form as the centered 3-point approximation used in second_derivative:
$f''(x) \approx (f_{i+1} - 2f_i + f_{i-1}) / \Delta x^2$

except the repeated application ends up using further neighbors and a $2 \Delta x$ spacing. Thus the smoothing effect you can see in the graph as you're using a less precise estimate (twice the point spacing).

TLDR; Use second_derivative directly. While calculus indicates that the second derivative is just the first derivative applied twice, numerical approximations do not necessarily stack in the same way.