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The inverse problem of determining the flow at the Earth's core-mantle boundary according to an outer core magnetic field and secular variation model has been investigated through a Bayesian formalism. To circumvent the issue arising from the truncated nature of the available fields, we combined two modeling methods. In the first step, we applied a filter on the magnetic field to isolate its large scales by reducing the energy contained in its small scales, we then derived the dynamical equation, referred as filtered frozen flux equation, describing the spatiotemporal evolution of the filtered part of the field. In the second step, we proposed a statistical parametrization of the filtered magnetic field in order to account for both its remaining unresolved scales and its large-scale uncertainties. These two modeling techniques were then included in the Bayesian formulation of the inverse problem. To explore the complex posterior distribution of the velocity field resulting from this development, we numerically implemented an algorithm based on Markov chain Monte Carlo methods. After evaluating our approach on synthetic data and comparing it to previously introduced methods, we applied it to a magnetic field model derived from satellite data for the single epoch 2005.0. We could confirm the existence of specific features already observed in previous studies. In particular, we retrieved the planetary scale eccentric gyre characteristic of flow evaluated under the compressible quasi-geostrophy assumption although this hypothesis was not considered in our study. In addition, through the sampling of the velocity field posterior distribution, we could evaluate the reliability, at any spatial location and at any scale, of the flow we calculated. The flow uncertainties we determined are nevertheless conditioned by the choice of the prior constraints we applied to the velocity field.
We present a new model of the geomagnetic field spanning the last 20 years and called Kalmag. Deriving from the assimilation of CHAMP and Swarm vector field measurements, it separates the different contributions to the observable field through parameterized prior covariance matrices. To make the inverse problem numerically feasible, it has been sequentialized in time through the combination of a Kalman filter and a smoothing algorithm. The model provides reliable estimates of past, present and future mean fields and associated uncertainties. The version presented here is an update of our IGRF candidates; the amount of assimilated data has been doubled and the considered time window has been extended from [2000.5, 2019.74] to [2000.5, 2020.33].
P>We present a statistical analysis of focal mechanism orientations for nine California fault zones with the goal of quantifying variations of fault zone heterogeneity at seismogenic depths. The focal mechanism data are generated from first motion polarities for earthquakes in the time period 1983-2004, magnitude range 0-5, and depth range 0-15 km. Only mechanisms with good quality solutions are used. We define fault zones using 20 km wide rectangles and use summations of normalized potency tensors to describe the distribution of double-couple orientations for each fault zone. Focal mechanism heterogeneity is quantified using two measures computed from the tensors that relate to the scatter in orientations and rotational asymmetry or skewness of the distribution. We illustrate the use of these quantities by showing relative differences in the focal mechanism heterogeneity characteristics for different fault zones. These differences are shown to relate to properties of the fault zone surface traces such that increased scatter correlates with fault trace complexity and rotational asymmetry correlates with the dominant fault trace azimuth. These correlations indicate a link between the long-term evolution of a fault zone over many earthquake cycles and its seismic behaviour over a 20 yr time period. Analysis of the partitioning of San Jacinto fault zone focal mechanisms into different faulting styles further indicates that heterogeneity is dominantly controlled by structural properties of the fault zone, rather than time or magnitude related properties of the seismicity.
The aim of this paper is to estimate the Hurst parameter of Fractional Gaussian Noise (FGN) using Bayesian inference. We propose an estimation technique that takes into account the full correlation structure of this process. Instead of using the integrated time series and then applying an estimator for its Hurst exponent, we propose to use the noise signal directly. As an application we analyze the time series of the Nile River, where we find a posterior distribution which is compatible with previous findings. In addition, our technique provides natural error bars for the Hurst exponent.
During the last 5 Ma the Earth's ocean-atmosphere system passed through several major transitions, many of which are discussed as possible triggers for human evolution. A classic in this context is the possible influence of the closure of the Panama Strait, the intensification of Northern Hemisphere Glaciation, a stepwise increase in aridity in Africa, and the first appearance of the genus Homo about 2.5 - 2.7 Ma ago. Apart from the fact that the correlation between these events does not necessarily imply causality, many attempts to establish a relationship between climate and evolution fail due to the challenge of precisely localizing an a priori unknown number of changes potentially underlying complex climate records. The kernel-based Bayesian inference approach applied here allows inferring the location, generic shape, and temporal scale of multiple transitions in established records of Plio-Pleistocene African climate. By defining a transparent probabilistic analysis strategy, we are able to identify conjoint changes occurring across the investigated terrigenous dust records from Ocean Drilling Programme (ODP) sites in the Atlantic Ocean (ODP 659), Arabian (ODP 721/722) and Mediterranean Sea (ODP 967). The study indicates a two-step transition in the African climate proxy records at (2.35-2.10) Ma and (1.70 - 1.50) Ma, that may be associated with the reorganization of the Hadley-Walker Circulation. .
Bayesian selection of Markov Models for symbol sequences application to microsaccadic eye movements
(2012)
Complex biological dynamics often generate sequences of discrete events which can be described as a Markov process. The order of the underlying Markovian stochastic process is fundamental for characterizing statistical dependencies within sequences. As an example for this class of biological systems, we investigate the Markov order of sequences of microsaccadic eye movements from human observers. We calculate the integrated likelihood of a given sequence for various orders of the Markov process and use this in a Bayesian framework for statistical inference on the Markov order. Our analysis shows that data from most participants are best explained by a first-order Markov process. This is compatible with recent findings of a statistical coupling of subsequent microsaccade orientations. Our method might prove to be useful for a broad class of biological systems.
Context. The theoretically studied impact of rapid rotation on stellar evolution needs to be compared with these results of high-resolution spectroscopy-velocimetry observations. Early-type stars present a perfect laboratory for these studies. The prototype A0 star Vega has been extensively monitored in recent years in spectropolarimetry. A weak surface magnetic field was detected, implying that there might be a (still undetected) structured surface. First indications of the presence of small amplitude stellar radial velocity variations have been reported recently, but the confirmation and in-depth study with the highly stabilized spectrograph SOPHIE/OHP was required.
Aims. The goal of this article is to present a thorough analysis of the line profile variations and associated estimators in the early-type standard star Vega (A0) in order to reveal potential activity tracers, exoplanet companions, and stellar oscillations.
Methods. Vega was monitored in quasi-continuous high-resolution echelle spectroscopy with the highly stabilized velocimeter SOPHIE/OHP. A total of 2588 high signal-to-noise spectra was obtained during 34.7 h on five nights (2 to 6 of August 2012) in high-resolution mode at R = 75 000 and covering the visible domain from 3895 6270 angstrom. For each reduced spectrum, least square deconvolved equivalent photospheric profiles were calculated with a T-eff = 9500 and log g = 4.0 spectral line mask. Several methods were applied to study the dynamic behaviour of the profile variations (evolution of radial velocity, bisectors, vspan, 2D profiles, amongst others).
Results. We present the discovery of a spotted stellar surface on an A-type standard star (Vega) with very faint spot amplitudes Delta F/Fc similar to 5 x 10(-4). A rotational modulation of spectral lines with a period of rotation P = 0.68 d has clearly been exhibited, unambiguously confirming the results of previous spectropolarimetric studies. Most of these brightness inhomogeneities seem to be located in lower equatorial latitudes. Either a very thin convective layer can be responsible for magnetic field generation at small amplitudes, or a new mechanism has to be invoked to explain the existence of activity tracing starspots. At this stage it is difficult to disentangle a rotational from a stellar pulsational origin for the existing higher frequency periodic variations.
Conclusions. This first strong evidence that standard A-type stars can show surface structures opens a new field of research and ask about a potential link with the recently discovered weak magnetic field discoveries in this category of stars.
constraints
(2016)
Prior information in ill-posed inverse problem is of critical importance because it is conditioning the posterior solution and its associated variability. The problem of determining the flow evolving at the Earth's core-mantle boundary through magnetic field models derived from satellite or observatory data is no exception to the rule. This study aims to estimate what information can be extracted on the velocity field at the core-mantle boundary, when the frozen flux equation is inverted under very weakly informative, but realistic, prior constraints. Instead of imposing a converging spectrum to the flow, we simply assume that its poloidal and toroidal energy spectra are characterized by power laws. The parameters of the spectra, namely, their magnitudes, and slopes are unknown. The connection between the velocity field, its spectra parameters, and the magnetic field model is established through the Bayesian formulation of the problem. Working in two steps, we determined the time-averaged spectra of the flow within the 2001–2009.5 period, as well as the flow itself and its associated uncertainties in 2005.0. According to the spectra we obtained, we can conclude that the large-scale approximation of the velocity field is not an appropriate assumption within the time window we considered. For the flow itself, we show that although it is dominated by its equatorial symmetric component, it is very unlikely to be perfectly symmetric. We also demonstrate that its geostrophic state is questioned in different locations of the outer core.
We propose a reduced dynamical system describing the coupled evolution of fluid flow and magnetic field at the top of the Earth's core between the years 1900 and 2014. The flow evolution is modeled with a first-order autoregressive process, while the magnetic field obeys the classical frozen flux equation. An ensemble Kalman filter algorithm serves to constrain the dynamics with the geomagnetic field and its secular variation given by the COV-OBS.x1 model. Using a large ensemble with 40,000 members provides meaningful statistics including reliable error estimates. The model highlights two distinct flow scales. Slowly varying large-scale elements include the already documented eccentric gyre. Localized short-lived structures include distinctly ageostophic features like the high-latitude polar jet on the Northern Hemisphere. Comparisons with independent observations of the length-of-day variations not only validate the flow estimates but also suggest an acceleration of the geostrophic flows over the last century. Hindcasting tests show that our model outperforms simpler predictions bases (linear extrapolation and stationary flow). The predictability limit, of about 2,000 years for the magnetic dipole component, is mostly determined by the random fast varying dynamics of the flow and much less by the geomagnetic data quality or lack of small-scale information.
Potential fields are classically represented on the sphere using spherical harmonics. However, this decomposition leads to numerical difficulties when data to be modelled are irregularly distributed or cover a regional zone. To overcome this drawback, we develop a new representation of the magnetic and the gravity fields based on wavelet frames. In this paper, we first describe how to build wavelet frames on the sphere. The chosen frames are based on the Poisson multipole wavelets, which are of special interest for geophysical modelling, since their scaling parameter is linked to the multipole depth (Holschneider et al.). The implementation of wavelet frames results from a discretization of the continuous wavelet transform in space and scale. We also build different frames using two kinds of spherical meshes and various scale sequences. We then validate the mathematical method through simple fits of scalar functions on the sphere, named 'scalar models'. Moreover, we propose magnetic and gravity models, referred to as 'vectorial models', taking into account geophysical constraints. We then discuss the representation of the Earth's magnetic and gravity fields from data regularly or irregularly distributed. Comparisons of the obtained wavelet models with the initial spherical harmonic models point out the advantages of wavelet modelling when the used magnetic or gravity data are sparsely distributed or cover just a very local zone
In this paper we present a Bayesian framework for interpolating data in a reproducing kernel Hilbert space associated with a random subdivision scheme, where not only approximations of the values of a function at some missing points can be obtained, but also uncertainty estimates for such predicted values. This random scheme generalizes the usual subdivision by taking into account, at each level, some uncertainty given in terms of suitably scaled noise sequences of i.i.d. Gaussian random variables with zero mean and given variance, and generating, in the limit, a Gaussian process whose correlation structure is characterized and used for computing realizations of the conditional posterior distribution. The hierarchical nature of the procedure may be exploited to reduce the computational cost compared to standard techniques in the case where many prediction points need to be considered.
In this paper we propose a procedure which allows the construction of a large family of FIR d x d matrix wavelet filters by exploiting the one-to-one correspondence between QMF systems and orthogonal operators which commute with the shifts by two. A characterization of the class of filters of full rank type that can be obtained with such procedure is given. In particular, we restrict our attention to a special construction based on the representation of SO(2d) in terms of the elements of its Lie algebra. Explicit expressions for the filters in the case d = 2 are given, as a result of a local analysis of the parameterization obtained from perturbing the Haar system.
We introduce a method for computing instantaneous-polarization attributes from multicomponent signals. This is an improvement on the standard covariance method (SCM) because it does not depend on the window size used to compute the standard covariance matrix. We overcome the window-size problem by deriving an approximate analytical formula for the cross-energy matrix in which we automatically and adaptively determine the time window. The proposed method uses polarization analysis as applied to multicomponent seismic by waveform separation and filtering.
We introduce a method of wavefield separation from multicomponent data sets based on the use of the continuous wavelet transform. Our method is a further generalization of the approach proposed by Morozov and Smithson, in that by using the continuous wavelet transform, we can achieve a better separation of wave types by designing the filter in the time-frequency domain. Furthermore, using the instantaneous polarization attributes defined in the wavelet domain, we show how to construct filters tailored to separate different wave types (elliptically or linearly polarized), followed by an inverse wavelet transform to obtain the desired wave type in the time domain. Using synthetic and experimental data, we show how the present method can be used for wavefield separation
Characterization of polarization attributes of seismic waves using continuous wavelet transforms
(2006)
Complex-trace analysis is the method of choice for analyzing polarized data. Because particle motion can be represented by instantaneous attributes that show distinct features for waves of different polarization characteristics, it can be used to separate and characterize these waves. Traditional methods of complex-trace analysis only give the instantaneous attributes as a function of time or frequency. However. for transient wave types or seismic events that overlap in time, an estimate of the polarization parameters requires analysis of the time-frequency dependence of these attributes. We propose a method to map instantaneous polarization attributes of seismic signals in the wavelet domain and explicitly relate these attributes with the wavelet-transform coefficients of the analyzed signal. We compare our method with traditional complex-trace analysis using numerical examples. An advantage of our method is its possibility of performing the complete wave-mode separation/ filtering process in the wavelet domain and its ability to provide the frequency dependence of ellipticity, which contains important information on the subsurface structure. Furthermore, using 2-C synthetic and real seismic shot gathers, we show how to use the method to separate different wave types and identify zones of interfering wave modes
The Gutenberg-Richter relation for earthquake magnitudes is the most famous empirical law in seismology. It states that the frequency of earthquake magnitudes follows an exponential distribution; this has been found to be a robust feature of seismicity above the completeness magnitude, and it is independent of whether global, regional, or local seismicity is analyzed. However, the exponent b of the distribution varies significantly in space and time, which is important for process understanding and seismic hazard assessment; this is particularly true because of the fact that the Gutenberg-Richter b-value acts as a proxy for the stress state and quantifies the ratio of large-to-small earthquakes. In our work, we focus on the automatic detection of statistically significant temporal changes of the b-value in seismicity data. In our approach, we use Bayes factors for model selection and estimate multiple change-points of the frequency-magnitude distribution in time. The method is first applied to synthetic data, showing its capability to detect change-points as function of the size of the sample and the b-value contrast. Finally, we apply this approach to examples of observational data sets for which b-value changes have previously been stated. Our analysis of foreshock and after-shock sequences related to mainshocks, as well as earthquake swarms, shows that only a portion of the b-value changes is statistically significant.
Earthquake rates are driven by tectonic stress buildup, earthquake-induced stress changes, and transient aseismic processes. Although the origin of the first two sources is known, transient aseismic processes are more difficult to detect. However, the knowledge of the associated changes of the earthquake activity is of great interest, because it might help identify natural aseismic deformation patterns such as slow-slip events, as well as the occurrence of induced seismicity related to human activities. For this goal, we develop a Bayesian approach to identify change-points in seismicity data automatically. Using the Bayes factor, we select a suitable model, estimate possible change-points, and we additionally use a likelihood ratio test to calculate the significance of the change of the intensity. The approach is extended to spatiotemporal data to detect the area in which the changes occur. The method is first applied to synthetic data showing its capability to detect real change-points. Finally, we apply this approach to observational data from Oklahoma and observe statistical significant changes of seismicity in space and time.
Borehole logs provide in situ information about the fluctuations of petrophysical properties with depth and thus allow the characterization of the crustal heterogeneities. A detailed investigation of these measurements may lead to extract features of the geological media. In this study, we suggest a regularity analysis based on the continuous wavelet transform to examine sonic logs data. The description of the local behavior of the logs at each depth is carried out using the local Hurst exponent estimated by two (02) approaches: the local wavelet approach and the average-local wavelet approach. Firstly, a synthetic log, generated using the random midpoints displacement algorithm, is processed by the regularity analysis. The obtained Hurst curves allowed the discernment of the different layers composing the simulated geological model. Next, this analysis is extended to real sonic logs data recorded at the Kontinentales Tiefbohrprogramm (KTB) pilot borehole (Continental Deep Drilling Program, Germany). The results show a significant correlation between the estimated Hurst exponents and the lithological discontinuities crossed by the well. Hence, the Hurst exponent can be used as a tool to characterize underground heterogeneities.
Borehole logs provide geological information about the rocks crossed by the wells. Several properties of rocks can be interpreted in terms of lithology, type and quantity of the fluid filling the pores and fractures. Here, the logs are assumed to be nonhomogeneous Brownian motions (nhBms) which are generalized fractional Brownian motions (fBms) indexed by depth-dependent Hurst parameters H(z). Three techniques, the local wavelet approach (LWA), the average-local wavelet approach (ALWA), and Peltier Algorithm (PA), are suggested to estimate the Hurst functions (or the regularity profiles) from the logs. First, two synthetic sonic logs with different parameters, shaped by the successive random additions (SRA) algorithm, are used to demonstrate the potential of the proposed methods. The obtained Hurst functions are close to the theoretical Hurst functions. Besides, the transitions between the modeled layers are marked by Hurst values discontinuities. It is also shown that PA leads to the best Hurst value estimations. Second, we investigate the multifractional property of sonic logs data recorded at two scientific deep boreholes: the pilot hole VB and the ultra deep main hole HB, drilled for the German Continental Deep Drilling Program (KTB). All the regularity profiles independently obtained for the logs provide a clear correlation with lithology, and from each regularity profile, we derive a similar segmentation in terms of lithological units. The lithological discontinuities (strata' bounds and faults contacts) are located at the local extrema of the Hurst functions. Moreover, the regularity profiles are compared with the KTB estimated porosity logs, showing a significant relation between the local extrema of the Hurst functions and the fluid-filled fractures. The Hurst function may then constitute a tool to characterize underground heterogeneities.
We construct a family of admissible analysis reconstruction pairs of wavelet families on the sphere. The construction is an extension of the isotropic Poisson wavelets. Similar to those, the directional wavelets allow a finite expansion in terms of off-center multipoles. Unlike the isotropic case, the directional wavelets are not a tight frame. However, at small scales, they almost behave like a tight frame. We give an explicit formula for the pseudodifferential operator given by the combination analysis-synthesis with respect to these wavelets. The Euclidean limit is shown to exist and an explicit formula is given. This allows us to quantify the asymptotic angular resolution of the wavelets.