Space-Wise approach for airborne gravity data modelling

Abstract

Regional gravity field modelling by means of remove-compute-restore procedure is nowadays widely applied in different contexts: it is the most used technique for regional gravimetric geoid determination, and it is also used in exploration geophysics to predict grids of gravity anomalies (Bouguer, free-air, isostatic, etc.), which are useful to understand and map geological structures in a specific region. Considering this last application, due to the required accuracy and resolution, airborne gravity observations are usually adopted. However, due to the relatively high acquisition velocity, presence of atmospheric turbulence, aircraft vibration, instrumental drift, etc., airborne data are usually contaminated by a very high observation error. For this reason, a proper procedure to filter the raw observations in both the low and high frequencies should be applied to recover valuable information. In this work, a software to filter and grid raw airborne observations is presented: the proposed solution consists in a combination of an along-track Wiener filter and a classical Least Squares Collocation technique. Basically, the proposed procedure is an adaptation to airborne gravimetry of the Space-Wise approach, developed by Politecnico di Milano to process data coming from the ESA satellite mission GOCE. Among the main differences with respect to the satellite application of this approach, there is the fact that, while in processing GOCE data the stochastic characteristics of the observation error can be considered a-priori well known, in airborne gravimetry, due to the complex environment in which the observations are acquired, these characteristics are unknown and should be retrieved from the dataset itself. The presented solution is suited for airborne data analysis in order to be able to quickly filter and grid gravity observations in an easy way. Some innovative theoretical aspects focusing in particular on the theoretical covariance modelling are presented too. In the end, the goodness of the procedure is evaluated by means of a test on real data retrieving the gravitational signal with a predicted accuracy of about 0.4 mGal.

Appendix 1

The algorithm implemented to compute the covariance function from a grid of the reduced reference gravitational signal is briefly explained. We will suppose here to have a reduced grid $\delta g\left( \underline{x}\right)$ with zero mean and with homogeneous and isotropic behaviours. Here, $\underline{x}$ represents the two planar coordinates of the grid. The first operation performed consists in computing the bi-dimensional fast Fourier transform $\delta \hat{g}\left( \underline{p}\right)$ of $\delta g$, where $\underline{p}$ represents the frequencies in the directions defined by the $\left( \underline{x}\right)$ axis. We now consider N bins, and for each bin i we compute the following average:

$$\begin{aligned} S_i\left( p\right) =\sum _{\underline{p} \in {\varOmega _p}_i}{\frac{\left| \delta \hat{g}\left( \underline{p}\right) \right| ^2}{n}} \end{aligned}$$

(7)

where $p=\left| \underline{p}\right|$, n is the number of values included within the i-th bin, and $\varOmega _p$ is defined as:

$$\begin{aligned} {\varOmega _p}_i=\bar{p_i}\le p \le \bar{p_i}+\Delta p \end{aligned}$$

(8)

with $\bar{p}_i$ a set of values ranging from 0 to the maximum p with an increment given by $\frac{\hbox {max}\left| \underline{p}\right| }{N-1}$. Note that the final $S_i\left( p\right)$ is a step function that depends only on the radial coordinate p of the plane $\underline{p}$. It should be also observed that if we are interested in a covariance, which is a function on distances between points r only, and this is always the case if the field is considered homogeneous and isotropic, then the covariance $C\left( r\right)$ can be simply inferred from the inverse Fourier transform of $S_i\left( p\right)$. Due to the relation between the bi-dimensional radially symmetric Fourier transform and Hankel transform, we have:

$$\begin{aligned} S\left( p\right) = \int _0^{+\infty } \bar{J}_0\left( p r\right) C \left( r\right) \hbox {d}r \end{aligned}$$

(9)

where the $\bar{J}_0$ function is related to the classical Bessel function of 0 order by the following relation:

$$\begin{aligned} \bar{J}_0\left( pr \right) = 2\pi J_0 \left( 2\pi pr\right) \end{aligned}$$

(10)

Of course $C\left( r\right)$ can be obtained by performing the inverse Henkel transform of eq. 9:

$$\begin{aligned} C \left( r\right) = \int _0^{+\infty } \bar{J}_0\left( p r\right) S\left( p\right) \hbox {d}p. \end{aligned}$$

(11)

We shall also need the following formula (Watson 1995):

$$\begin{aligned} \int _0^{+\infty } J_0 \left( a r \right) \frac{J_1 \left( b r \right) }{r}r\hbox {d}r= \left\{ \begin{matrix} 0 &{}\quad a>b\\ \frac{1}{b} &{}\quad a<b \end{matrix} \right. . \end{aligned}$$

(12)

If we consider $a=2\pi p$, $b=2\pi \bar{p}$, $\bar{J}_0\left( pr \right) =2\pi J_0\left( pr \right)$, and $\bar{J}_1\left( pr \right) =2\pi J_1\left( pr \right)$ we have:

$$\begin{aligned} \frac{\bar{p}}{2\pi }\int _0^{+\infty } \bar{J}_0\left( pr\right) \frac{\bar{J}_1\left( \bar{p}r\right) }{r}r\hbox {d}r=\left\{ \begin{matrix} 0 &{} p>\bar{p}\\ 1 &{} p<\bar{p} \end{matrix} \right. . \end{aligned}$$

(13)

Therefore, combining Eq. 11 and Eq. 13 we have:

$$\begin{aligned}&C\left( r\right) =\sum _{i=0}^n S_i\left[ \frac{\left( i+1 \right) \Delta p}{2\pi }\frac{\bar{J}_1\left( \left( i+1\right) \Delta r\right) }{r}\right. \nonumber \\&\quad \left. - \frac{i\Delta p}{2\pi }\frac{\bar{J}_1\left( i\Delta r\right) }{r}\right] . \end{aligned}$$

(14)

This allows to estimate a covariance function corresponding to a power spectrum described in terms of linear combination of step functions as a linear combination of n Bessel functions of the first order and zero degree.