GRAVIMETRIC DATA FILTERING

From [Panteleev (1983)], when just the vertical component of $g$ has to be determined by a gravimeter, and the Eötvös and cross coupling effects may be neglected, the following method to estract the gravity signal from vertical accelerations can be used. Let the gravimeter be described by a linear system of the first-order and the input signal being the sum of gravity $g(t)$ and of platform vertical accelerations $u(t)$:

$$\tau \dot{x}(t) + x(t) = g(t) + u(t)$$ (1)

It is therefore possible to detect field gravity variations from the background system noise due to difference in their frequency spectra. We want to design a filter with impulse response $w_0$ and output

$$\tilde{g}(t) = \int_{-\infty}^{+\infty} w_0 (\xi)\bigg[ g(t+\xi)+w(t+\xi) \bigg]d\xi$$ (2)

and where the filter minimizes the error $e^2(t)=[g(t)-\tilde{g}(t)]^2$.

Taking into account that $w_0(\xi)$ tends to zero as $\xi$ increases it is possible to transfer a signal from the output of a dynamic system to its input using 1 and 2:

$$\tilde{g}(t) = \int_{-\infty}^{+\infty} w_0 (\xi) \big[ x(t... ...\infty} \big[ w_0(\xi) -\tau \dot{w}_0(\xi)\big] x(t+\xi)d\xi$$ (3)

The impulse response of the filter is defined now by transformation

$$w(t) = w_0 (t) - \tau \dot{w}_0 (t)$$ (4)

using the Panteleev's normalized weight function

$$w_0 (t) = \frac{\omega_0}{8} exp \bigg( -\omega_0 \frac{\ve... ...{2}}\bigg) \sin \frac{\omega_0 \vert t\vert}{\sqrt{2}} \bigg]$$ (5)

with frequency characteristic response

$$A( \omega ) = \frac{ \omega_0^8}{( \omega^4 + \omega_0^4)^2}$$ (6)

where $\omega$ is the angular frequency and

$$\omega_0 = 0.0045 \frac{1}{sec}$$ (7)

hence the filter becomes

$$w(t) = \frac{\omega_0}{8}exp \bigg( \frac{-\omega_0 \vert t... ...+ \omega_0 (t -2\tau) \sin \frac{\omega_0 t}{\sqrt{2}} \bigg]$$ (8)

This method was used to restore the gravity signal from the observed data.