Change of variables for functions with invariant integral

[cmichelenstrofer]

Some functions are defined by properties of their integral, and their integral over a differential area must be invariant over a change of variables. That is, a function is defined such that , and the integral over the same area (a-b) needs to be the same constant C even after change of variables. These functions include:

• The elevation variance spectrum, , where is the total variance in the frequency range .
• The energy spectrum, for gravity waves, where the integral gives the total energy in a frequency range.
• The spread function, , where the integral gives the percentage of the total energy in freuquency f that is contained in the range of directions .
• Probability density functions where the integral gives the probability for a range of the random variable (not used in this code but analogous problem and the Wikipedia article has a good explanation.)

If we have a function f such that its integral over a range is a constant C:

and we want to change the variables from x to y, where x and y are related by the invertible function g as

and

The change of variables is

Example: Spread function radians to degrees.
If we have the spread function as a function of angle in radians and want it in terms of angle in degrees the function g is and . This gives

The opposite situation is

Example: Elevation variance spectrum radians to Hz.
By similar arguments the result is

The opposite situation is

These also apply to the energy spectrum, simply carry the factor (for gravity waves).

Example: Elevation variance spectrum frequency to period.
In this case the function g is not a linear function, but applying the same methodology for period T in seconds and frequency f in Hz we get

The opposite situation is

The negative sign is due to the order of the integration bounds being reversed. We could make a definition decision here so that is positive.

TLDR;
For functions whose integral must be invariant over change of variables it is not sufficient to convert the input of the function. For instance when converting a plot of (w, S(w)) to (f, S(f)), it is not only the x-axis that needs to be scaled, the y-axis will have have a scaling too. Similar for changing the spread function between inputs in radians and degrees. The case for converting the spectrum from frequency input to period input is a bit more complicated, see above.

[cmichelenstrofer]

The required scaling is pretty obvious for the cases where it is only a change of units. E.g. for S(f) with f in rad/s to Hz if you consider the units of both the frequency and the spectrum and the fact that the units of frequency times the unit of the spectrum need to give you units of elevation variance (m^2) in the case of the elevation variance spectrum. Similar to the spread function where the product of units needs to be non-dimensional. The conversion between f and T is not as trivial.

[cmichelenstrofer]

change_of_variables.pdf