From the book Basic theory of seismic trace (Costain and Coruh) we find something about the time-shifting theorem:

Short abstract:

I a source wavelet were to propagate in the earth without distortion, then its Fourier spectrum would have to change only by addition of the linear term – wτ to its phase spectrum. The amplitude spectrum would remain unchanged. This is not possible for propagating seismic wavelets except in rocks with extremely high value of Q (small attenuation from absorption). Intrinsic damping and the absorption of higher frequencies changes the shape of the wavelet.

The time-shifting theorem is important:

If the phase angle spectrum does not change in a linear fashion as a real seismic wavelet propagates in time, then an inverse Fourier transform at a point in time (or at a point in space) will reveal that the wavelet  shape is different for different times.

Amplitude is:

An = (an2 + bn2) ½

As the wavelet is shifted in time, the time shifting theorem tells us that an and bn both change and so does:

tan -1(bn/an)

that is the phase angle. If something is shifted in time and the time-shifting theorem is not obeyed, then some other physical process must be involved, such as intrinsic damping in the case of a real seismic wavelet.

Figure 1.a gives an unshifted function f(t). Figure 1.b gives an shifted function f(t).

 

 

As a wavelet propagates toward increasing time t its phase spectrum is continuously changing by virtue of the term exp(-iwτ) that multiplies the wavelet Fourier transform and therefore adds –wt to the wavelet’s phase spectrum. The quantity – wt is a phase lag because multiplication of the wavelet spectrum by exp(-iwt) delays the time-domain pulse by an amount t.

Full Article

Time-shifting theorem

This thoughts are taken from the book:  Basic theory of  exploration seismology. By Costain and Coruh

A function f(t) shifted toward positive time by an amount τ is denoted by f(t- τ).  The Fourier transform of this time-shifted function is, by definition:

G(w) =

And this can be written:

 

 

And this expression can be written:

 

 

Where F(w) is the Fourier transform of the unshifted function.

 

This can also be written:

 

G(w) = 

 

Where F(w) is the modulus sqrt( a(w) 2 + b(w)2 )

 

A(w) is the real part of F and b is the imaginary part. We have therefore:

 

 

That is, the Fourier transform of a time-shifted function always has the same modulus F(w). But the phase spectrum is continually changing as the amount -wτ = - 2πwτ is added to its phase spectrum after a traveltime of τ time units. In practice, we will see in other papers about Q that intrinsic damping and the attenuation of higher frequencies by internal friction not only results in a continuous change in wavelet shape, but also introduces body wave dispersion, where different frequencies travel at slightly different body wave velocities.

The effect on the Fourier spectrum of multiplication by e-iwτ is shown in Figure 1. The Fourier trans­form of a propagating seismic pulse that has traveled for a time τ has the same modulus but a different phase spectrum. Said another way, if the shape of the propagating seismic wavelet is time-invariant, then its Fourier transform must be described by Equation (3.50).

Implications

This conclusion states that if a source wavelet were to propagate in the Earth without distortion, then its Fourier spectrum (the word "spectrum" by itself refers to both amplitude and phase) would have to change only by addition of the linear term -wτ to its phase spectrum. The amplitude spectrum would remain unchanged. In reality, this is not possible for propagating seismic wavelets except in rocks that have an extremely high value of Q with constant velocity. Intrinsic damping and the absorption of higher frequencies changes the shape of the wavelet.

The time-shifting theorem is important. If the phase angle spectrum does not change in a linear fashion as a real seismic wavelet propagates in time, then an inverse Fourier transform at a point in time (or at a point in space) will reveal that the wavelet shape is different for different times.

The amplitude spectrum of the seismic wavelet is the square root of the sum of the squares of the real and imaginary parts of the Fourier transform:

sqrt( a(w) 2 + b(w)2 )

As the wavelet is shifted in time, the time-shifting theorem tells us that an and bn both change, and so does

i.e., the phase angle also changes, but An does not. If something is shifted in time (or space), and the time-shifting theorem is not obeyed, then some other physical process must be involved, such as intrinsic damping in the case of a real seismic wavelet.

Phase spectrum versus phase lag spectrum

As a wavelet propagates toward increasing time t its phase spectrum  is continuously changing by virtue of the term e-iwτ that multiplies the wavelet Fourier transform and therefore adds -wτ to the wavelet's phase spectrum, whatever that might be. The quantity -wτ  is a phase lag because multipli­cation of the wavelet spectrum by e-iwτ delays the time-domain pulse by an amount τ. That is, for the spectrum of an impulse (unity).

Δ(t-τ) =

and the Dirac delta function is seen to be delayed by an amount τ. Delay of a signal in the time domain is equivalent to a phase lag in the frequency domain. If the phase spectrum is wτ then - wτ is the phase lag spectrum. So in general given a phase spectrum θ(w) then - θ(w) is the phase lag spectrum.