Nicolas Florsch

Bayesian estimation of the free core nutation parameters from the analysis of precise tidal gravity data

Abstract This paper is devoted to a new approach to the problem of estimating the free core nutation (FCN) parameters from tidal gravity data. We first review different models of the FCN resonance parameters (period, damping, strength), as well as different inversion techniques used in gravimetry (analytical, linearised least squares, non-linear generalised inverse, stochastic). We propose then a new Bayesian inversion method since such a probabilistic view allows the most complete and reliable information on the FCN resonance to be obtained. We show the consequences of introducing the positivity of the quality factor Q and, even more important, we show that the null information criterion on this factor allows us to better understand why Q is often underestimated in gravity studies (or even found to be negative) with standard least squares techniques. We apply the Bayesian estimation method to a set of gravity data originating from a 3000-day record (1988–1996) of the superconducting gravimeter (SG T005) near Strasbourg (France), after correction for pressure effects and ocean loading with the help of a recent model derived from satellite altimetry. The marginal distribution found for the eigenperiod is nearly Gaussian and leads to a most probable value of 428 days, in agreement with previous gravity studies. The Q distribution is found to be highly asymmetrical with a flat maximum probability that Q exceeds 10 5 , in agreement with the high values derived from very large baseline interferometry (VLBI) studies. The joint probability laws of the parameters show that strong correlations exist between some couples of parameters. Finally, the important impact of inaccurate ocean loading corrections on the determination of the damping of the eigenmode is pointed out; it appears that wrong ocean corrections may lead to underestimation of Q or, even worse, to generation of negative Q when using standard least squares retrieval techniques applied to gravity data.

Compaction of quartz sands by pressure solution using a Cole‐Cole distribution of relaxation times

Stressed water-infiltrated silica rocks may deform by pervasive pressure solution transfer (PPST), which involves dissolution of the grain-to-grain contacts, transport by diffusion of the solute, and precipitation on the free surfaces of the grains. A fundamental question regarding this process is how to model rheological behavior at stresses and temperatures typical of the crust of the Earth. A Voigt-type poroviscoplastic model is modified by using a Cole-Cole distribution of relaxation times rather than a Dirac distribution used previously. The motivation of this choice is to account for the distribution of the grain size in the compaction of the porous aggregate assuming that this distribution obeys approximately a log normal distribution. This grain size distribution depends upon the initial grain size distribution and cataclasis in the early stage of compaction. We compared this modified viscoplastic model with the full set of experimental data obtained in various conditions of mean grain size, effective stress, and temperature by Niemeijer et al. (2002). These data provide tests of all aspects of the model, which can be considered to have no free parameters. We show the experiments of Niemeijer et al. (2002) on PPST are primarily diffusion-limited. The grain size distributions observed for three samples imply that the distribution of the relaxation time covers 5 orders of magnitude in grain size.

Extracting coherent regional information from local measurements with Karhunen‐Loève transform: Case study of an alluvial aquifer (Rhine valley, France and Germany)

We investigate the ability of combining the Karhunen-Loeve transform (KLT) with the kriging method to extract regional information from a set of point measurements. This method was applied to a set of 195 piezometric head time series over a period of 17 years from observation wells distributed within the French and German area of the Rhine valley alluvial groundwater body. Piezometric head time series are analyzed with KLT in order to highlight characteristic temporal signals, classified from the most energetic (global) to the least energetic (local) signals. The first five signals amount to 80% of the global variance of the system and are inferred to represent different hydrological contributions (exchanges with rivers and rainfall), but they also represent a significant anthropogenic component. Kriging is then used to regionalize the signals and to build a reconstruction model of the behavior of the whole aquifer containing only filtered information coming from identified source signals.

Crustal thickness in Antarctica from CHAMP gravimetry

Abstract CHAMP, flying at an altitude of about 400 km, is the first of a new generation of satellites dedicated to Earth gravity field observation. The high-quality data have generated new gravity field models: EIGEN-1S in 2001, and EIGEN-2S more recently. The gravitational potential is decomposed into spherical harmonic coefficients and in this study we use the free air gravity anomalies reconstituted up to degree 60, at zero altitude. The anomalies for the Antarctic continent range from −57 to 65 mGal. We have modeled the gravity effect from the ice, the ocean and the bedrock, using a 666 km cut-off filter to simulate the resolution obtained by CHAMP. Computing the differences between this terrain effect and the CHAMP map provides a map of the Bouguer anomalies. Because of the dominant influence of the crust, we first used a crustal thickness model from seismology. This gives a map of the mantle Bouguer anomalies, the range of which is still large (between −255 and 216 mGal) indicating imperfections in the crust model. By appealing to isostasy we then imposed the condition that this mantle Bouguer anomaly should vanish and therefore solve for a new resulting crustal thickness. This gravity-based crust model gives thicknesses from 8.5 to 42.6 km in the zone of interest. There is a good general agreement with seismological models, but our models shows more detail, particularly in the western part of the continent. These details are in agreement with geological studies.

A physical model of the low‐frequency electrical polarization of clay rocks

Low-frequency (0.18 Hz to 1.5 kHz) effective dielectric spectra have been measured on a set of near-saturated samples of argillite. The measured spectra of the real part of the complex apparent permittivity did not show significant correlation with cation exchange capacity (CEC) per unit mass of rock values. They satisfied a power law relationship with the frequency, at least for samples with CEC values lower than 10 cmol/kg. The Maxwell-Wagner-Hanai-Bruggeman formulation used for a two-phase mixture has been modified to account for mutual polarization between the pockets of water located in the micropores and those located in the macropores. The results of the modeling calculations illustrate (1) the ability of this new formulation to reproduce the power law relationships of the measured spectra of the real and imaginary components of the complex permittivity and (2) the strong impact of the pore electrical conductivity.

The search for weak harmonic signals in a spectrum with application to gravity data

Abstract When considering the search for discovery or amplitude estimation of a spectral line with a probabilistic approach, great attention must be paid to the meaning of each step. We give the probability law for the amplitude of a spectral peak in the presence of random noise appearing in a periodogram and discuss the effective probability of the existence of the corresponding wave. We find that the estimated amplitude of a spectral peak is biased and should be corrected when the signal-to-noise ratio is small. As a first application to gravity data, it results in a re-estimation of the gravimetric amplitude factors (delta factors) provided by least-squares tidal analysis. We also estimate the probability of observing a spectral line above a given level in the spectrum of a purely random noise. This allows us to compute for given spectrum the number of peaks expected to overcross the classical levels used in statistical analysis (like nσ, where σ is the standard deviation of the temporal noise distribution and n is an integer with typical values equal to 2 or 3). A specific application to real data is investigating the gravity spectrum derived from a 5 year record of the French superconducting gravimeter and we show that the predicted statistics are indeed in agreement with the observations. We also show the statistical consequence of using longer observing periods to obtain the spectral estimations. The problem of detecting translational motion of the Earths solid inner core (Slichter modes) in a gravity spectrum is analyzed and the probabilities of having a triplet of random peaks thresholding specific levels in a given frequency window are computed. We show that, in the case of a typical gravity spectrum (1 year of hourly data and a frequency window of 0.03 cycle h−1), the probability of having a random set of three peaks exceeding a level of 3 σ, is very high. This emphasizes the need for a very careful analysis of spectral lines before inferring the existence of a true physical signal.

Preliminary spectral analysis of the residual signal of a superconducting gravimeter for periods shorter than one day

We show for the first time an analysis of a 1.5 year data set from the superconducting gravimeter in Strasbourg. For periods between 7 and 24 h, a residual gravity signal is first generated by removal of the main tidal components by a standard least squares fit; this signal is then treated to suppress any remaining transient signals (de-spiking). At shorter periods, between 7 h and 10 min, a simple high-pass filter is applied before de-spiking. In both cases, the noise level is estimated in different frequency bands; spectra of the gravity residuals are shown. We also analysed by two different methods the local barometric pressure, which contains harmonic components of the S1 wave related to air pressure changes — driven by diurnal solar heating — superimposed on the frequency-dependent meteorological noise. After removal of the daily harmonics which are also present in the gravity residual spectrum, a few remaining spectral lines of possible geophysical interest are discussed.

Observation gravimétrique des surcharges océaniques : premières expériences en Bretagne

Abstract Ocean loading involves both the dynamic and geometric effects that result from the action of ocean water masses on the crust. Gravity variations are generated by the direct attraction of the ocean masses, by the deformation of the crust and by the redistribution potential of the masses. Such effects are still perceptible far inland. A Greens function formalism, using loading Love numbers, allows to predict the loading effects. We present here the first French experiment in coastal areas devoted to the study of this phenomenon. The campaign took place in Brest in March 1998. The absolute gravimeter of the French community recorded during four days the gravity changes. Once the standard contributions (body tides, atmospheric pressure effect, polar motion) are removed, the residual variations are mainly due to the ocean loading and they can be confronted to the models. The observed gravity variations exceed by 16 % the theoretical predictions, and we impute this fact to the raw spatial resolution of the global ocean tide models, with a strong consequence near the coasts. Improvements are needed in all the geodetic features of coastal type (tide gauge links, vertical displacement displacement, motion of reference geodetic stations), and also in the validation of the hydrodynamic models by using gravity as an integrating quantity.

Complex conductivity of soils

The complex conductivity of soils remains poorly known despite the growing importance of this method in hydrogeophysics. In order to fill this gap of knowledge, we investigate the complex conductivity of 71 soils samples (including four peat samples) and one clean sand in the frequency range 0.1 Hz to 45 kHz. The soil samples are saturated with six different NaCl brines with conductivities (0.031, 0.53, 1.15, 5.7, 14.7, and 22 S m21, NaCl, 258C) in order to determine their intrinsic formation factor and surface conductivity. This data set is used to test the predictions of the dynamic Stern polarization model of porous media in terms of relationship between the quadrature conductivity and the surface conductivity. We also investigate the relationship between the normalized chargeability (the difference of in-phase conductivity between two frequencies) and the quadrature conductivity at the geometric mean frequency. This data set confirms the relationships between the surface conductivity, the quadrature conductivity, and the normalized chargeability. The normalized chargeability depends linearly on the cation exchange capacity and specific surface area while the chargeability shows no dependence on these parameters. These new data and the dynamic Stern layer polarization model are observed to be mutually consistent. Traditionally, in hydrogeophysics, surface conductivity is neglected in the analysis of resistivity data. The relationships we have developed can be used in field conditions to avoid neglecting surface conductivity in the interpretation of DC resistivity tomograms. We also investigate the effects of temperature and saturation and, here again, the dynamic Stern layer predictions and the experimental observations are mutually consistent.

A rapid electrical sounding method: The «three-point» method: a Bayesian approach

Only three independent measurements of apparent resistivity are theoretically sufficient to retrieve the parameters of a geoelectrical model with two layers, i.e. the resistivities of the two layers and the thickness of the first one. The Three-Point Method (TPM) presented here is a rapid electrical sounding field procedure based on that principle. It consists of measuring the apparent resistivity using only three interelectrode spacings instead of some tens as for a usual vertical electrical sounding. To invert these data and check out the limits of this simplified field procedure, we use a Bayesian probabilistic formalism. Such an approach appears more effective than other traditional inversion methods because it allows to calculate exhaustively a probabilistic description of the equivalent solutions. Both synthetic and field data validate TPM when a model with two layers correctly describes the geoelectrical structure of the ground. Particularly, TPM combined with the Bayesian inversion method is a tool well-designed to rapidly characterise the overburden thickness in sedimentary contexts, or the water table depth in the case of sufficient resistivity contrast between saturated and nonsaturated materials.