Civil Engineering Reference
In-Depth Information
3.3.2.2.
Fourier spectra
For seismology, the most commonly used spectral representation resorts to
Fourier transform, an algorithm well-assimilated and quick to use from a numerical
point of view. The Fourier acceleration spectrum is defined by:
f
D
³
³
F (f )
A(f) . exp(iij (f)) =
a(t)exp(
i2ʌft)dt =
a(t) exp(
i2ʌft) dt
a
f
0
where A refers to the modulus, M is the phase and D is the total duration of the
recording.
Similar expressions exist for speed F
v
(f) and displacement F
d
(f) spectra, and the
conventional relations of Fourier transform are quite convenient and useful:
f
f
³
³
N
a(t) =
F (f) . exp(+i2ʌft) df = 2
F (f) . exp (+i2ʌft) df
a
a
f
0
where f
N
is the maximum frequency which can be calculated taking the sampling
time step 't: f
N
= 0.5/'t.
22
F(f)
i. 2ʌf. F (f) =
4 ʌ f
a
v
f
D
³
N
³
2
Fd (f) =
F (f ) df = 1/2
a (t) dt
a
0
0
This last relationship (Parseval's theorem) allows us to link time descriptions
with spectral data: it is particularly used in many “spectral” models that allow the
quantitative parameters of seismic movement to be linked not only to the physical
properties that characterize the seismic source (emitting waves), but also to their
crust (i.e. very deep) propagation and site effects. However, it should be noted that
reconstructing the time signal from its Fourier spectra is only possible if both the
modulus A and the phase M are known. In general terms, only relationships between
the Fourier modulus and various parameters describing both origin and propagation
are well-established.
Therefore, methods and practices for describing the non-stationary time state of a
seismic signal and its consequences for the strength of building works are still
evolving. Some quite interesting tracks have been opened using “group delay time”
or the phase M derivative in relation to frequency f.
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