Deriving the frequency spectrum graph from the graph of a spectrogram and vice versa? I have a signal's spectrogram and frequency spectrum (which, if i'm not mistaken, is its the signal's fourier transform) as given below:

I understand that the spectrogram plots the frequency (y-axis) vs time (x-axis) and that the colors represent the amplitude.  I also understand the frequency spectrum as the plot of amplitude vs. frequency.  But I'm not quite sure how to produce one graph from the other.   Since there is no time axis on the frequency spectrum, surmise that it is impossible to create the spectrogram from a frequency spectrum.  It seems plausible to derive the frequency spectrum from the spectrogram, but I'm not quite sure how to remove the time parameter.
Any help with this would be greatly appreciated.
The data that when colored produces one vertical slice of your spectrogram image is a spectrum.  If you have the color mapping function, and it is invertible, you could take the array of colors from one vertical column in a spectrogram and attempt to produce the original spectrum for that point in time.  But the inversion process would likely be lossy, since there are usually less quantized colors than spectrum amplitudes.

Since a spectrum is only one vertical slice of a spectrogram, a spectrogram produced from just one spectrum would only be something like 1 pixel wide, thus not very useful.  But a bunch of spectrums at evenly spaced time increments could be used to produce a wider spectrogram.

Also note that due to time-frequency-resolution trade-offs, the width of time data used for each FFT/DFT spectrum array can be different between spectrum and spectrogram analysis presentations.