The basic low-cut filter

Many geophysical measurements contain very low-frequency noise called ``drift.'' For example, it might take some months to survey the depth of a lake. Meanwhile, rainfall or evaporation could change the lake level so that new survey lines become inconsistent with old ones. Likewise, gravimeters are sensitive to atmospheric pressure, which changes with the weather. A magnetic survey of an archeological site would need to contend with the fact that the earth's main magnetic field is changing randomly through time while the survey is being done. Such noises are sometimes called ``secular noise.''

The simplest way to eliminate low frequency noise is to take a time derivative. A disadvantage is that the derivative changes the waveform from a pulse to a doublet (finite difference). Here we examine the most basic low-cut filter. It preserves the waveform at high frequencies; it has an adjustable parameter for choosing the bandwidth of the low cut; and it is causal (uses the past but not the future).

We make our causal lowcut filter (highpass filter) by two stages which can be done in either order.

1.

Apply a time derivative, actually a finite difference, convolving the data with (1,-1).

2.

Integrate, actually to do a leaky integration, to deconvolve with $(1,-\rho)$ where numerically, $\rho$ is slightly less than unity.

The convolution ensures that the zero frequency is removed. The leaky integration almost undoes the differentiation (but does not restore the zero frequency). Adjusting the numerical value of $\rho$ adjusts the cutoff frequency of the filter. To learn the impulse response of the combined processes, we need to convolve the finite difference (1,-1) with the leaky integration $(1, \rho, \rho^2, \rho^3, \rho^4, \cdots)$.The result is $(1, \rho, \rho^2, \rho^3, \rho^4, \cdots)$ minus $(0, 1, \rho, \rho^2, \rho^3, \cdots)$.We can think of this as $(1, 0, 0, 0, 0, \cdots)$ minus $(1-\rho) (1, \rho, \rho^2, \rho^3, \cdots)$.In other words the impulse response is an impulse followed by the negative of a weak $(1-\rho)$ decaying exponential $\rho^t$.Roughly speaking, the cutoff frequency of the filter corresponds to matching one wavelength to the exponential decay time.

Some exercise with Fourier transforms or Z-transforms, shows the Fourier transform of this highpass filter filter to be  

$$H(Z) \eq {1-Z \over 1-\rho Z} \eq 1 -(1-\rho) [ Z^1 +\rho Z^2 +\rho^2 Z^3 +\rho^3 Z^4 \cdots ]$$ (25)

where the unit-delay operator is $Z=e^{i\omega\Delta t}$and where $\omega$ is the frequency. A symmetical (noncausal) lowcut filter would filter once forward with H(Z) and once backwards (adjoint) with H(1/Z). This is not the place for a detailed Fourier analysis of this filter but it is the place to mention that a cutoff filter is typically specified by its cutoff frequency, a frequency that separates the pass and reject region. For this filter, the cutoff frequency $\omega_0$would correspond to matching a quarter wavelength of a sinusoid to the exponential decay length of $\rho^k$, namely, say the value of k for which $\rho^k \approx 1/2$

Seismological data is more complex. A single ``measurement'' consists of an explosion and echo signals recorded at many locations. As before, a complete survey is a track (or tracks) of explosion locations. Thus, in seismology, data space is higher dimensional. Its most troublesome noise is not simply low frequency; it is low velocity. We will do more with multidimensional data in later chapters.

 

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Figure 6 The depth of the Sea of Galilee after roughening.