if the corresponding waveforms are single-mode plane waves (then A(ω,r)
will vary only slowly with r).
If they are not, the outcome of phase-difference techniques is
The phase φ(ω,r)=p(ω)ωr
of a single plane mode is a linear function of the offset.
The derivation of its absolute value from the Fourier coefficients is
ambiguous by an additive constant of an integer multiple of 2*π*.
This is due to the non-uniqueness of the involved arctan- or
However, for a wide frequency-range the phase increment from r,,l,, to
r,,l+1,,'' is certainly less than 2''π.
For this reason we use the advantage of a dense geophone spread and
derive the phase traveltime
is minimized (linear regression).
Some offsets at both ends of the spread may be discarded if this improves
The gradient p(ω) is the sought phase-slowness at angular frequency
If the wave does not consist of a single mode, T(ω,r_l) as a
function of r will not appear like a straight line. The outcome of the
analysis then is undefined.
If receiver distances r_l-r_(l-1) are too large, signals are
aliased. The actual phase difference of the Fourier signals between
r_l and r_(l-1) then is larger than 2π. The algorithm
will then not be able to obtain the correct phase slowness. Presenting
phase slowness p(ω) as a function of ω it will have