Additivity of envelope entropy to spectral entropy

In some of my efforts to fill in missing data with entropy criteria, I have often based the entropy on the spectrum and then found that the envelope would misbehave. I have come to believe that the definition of entropy should involve both the spectrum and the envelope. To get started, let us assume that the power of a seismic signal is the product of an envelope function times a spectral function, say  

$$u(\omega, t) \eq p(\omega) e(t)$$ (20)

Notice that this separability assumption resembles the stationarity concept. I am not defending the assumption (20), only suggesting that it is an improvement over each term separately. Let us examine some of the algebraic consequences. First evaluate the intensive entropy:

$$\begin{eqnarray} S'_{\rm intensive}\ &=&\ {\sum_t \sum_\omega u \ln u \over \s... ...\ -\ \ln {1\over N} \sum e \right) \ &=&\ S(p)\ +\ S(e) \geq\ 0\end{eqnarray}$$ (21) (22) (23) (24) (25)

It is remarkable that all the cross terms have disappeared and that the resulting entropy is the sum of the two parts.

Now we will tackle the same calculation with the geometric inequality:

$$\begin{eqnarray} G &=&\ \ln {1\over N^2}\sum\sum u\ -\ {1\over N^2}\sum\sum \... ...omega \ln p\ -\ {1\over N} \sum_t \ln e \ &=&\ G(t)\ +\ G(\omega)\end{eqnarray}$$ (26) (27) (28) (29) (30)

Again all the cross terms disappear, and the resulting entropy is the sum of the two parts. I wonder if this result applies for the other Jensen inequalities.

In conclusion, although this book is dominated by model building using the method of least squares, Jensen inequalities suggest many interesting alternatives.