Intrinsic Random Functions Research Articles

Tangent fields, intrinsic stationarity, and self similarity

This paper studies the local structure of continuous random fields on Rd taking values in a complete separable linear metric space V. Extending seminal work of Falconer, we show that the generalized (1+k)-th order increment tangent fields are self-similar and almost everywhere intrinsically stationary in the sense of Matheron. These results motivate the further study of the structure of V-valued intrinsic random functions of order k (IRFk, k=0,1,…). To this end, we focus on the special case where V is a Hilbert space. Building on the work of Sasvari and Berschneider, we establish the spectral characterization of all second order V-valued IRFk’s, extending the classical Matheron theory. Using these results, we further characterize the class of Gaussian, operator self-similar V-valued IRFk’s, generalizing results of Dobrushin and Didier, Meerschaert and Pipiras, among others. These processes are the Hilbert-space-valued versions of the general k-th order operator fractional Brownian fields and are characterized by their self-similarity operator exponent as well as a finite trace class operator valued spectral measure. We conclude with several examples motivating future applications to probability and statistics.

SecuCode: Intrinsic PUF Entangled Secure Wireless Code Dissemination for Computational RFID Devices

The simplicity of deployment and perpetual operation of energy harvesting devices provides a compelling proposition for a new class of edge devices for the Internet of Things. In particular, Computational Radio Frequency Identification (CRFID) devices are an emerging class of battery-free, computational, sensing enhanced devices that harvest all of their energy for operation. Despite wireless connectivity and powering, secure wireless firmware updates remains an open challenge for CRFID devices due to: intermittent powering, limited computational capabilities, and the absence of a supervisory operating system. We present, for the first time, a secure wireless code dissemination (SecuCode) mechanism for CRFIDs by entangling a device intrinsic hardware security primitive Static Random Access Memory Physical Unclonable Function (SRAM PUF) to a firmware update protocol. The design of SecuCode: i) overcomes the resource-constrained and intermittently powered nature of the CRFID devices; ii) is fully compatible with existing communication protocols employed by CRFID devices in particular, ISO-18000-6C protocol; and ii) is built upon a standard and industry compliant firmware compilation and update method realized by extending a recent framework for firmware updates provided by Texas Instruments. We build an end-to-end SecuCode implementation and conduct extensive experiments to demonstrate standards compliance, evaluate performance and security.

Iterative Atmospheric Phase Screen Compensation for Near-Real-Time Ground-Based InSAR Measurements Over a Mountainous Slope

In this article, an atmospheric phase screen (APS) compensation algorithm for a near real-time ground-based interferometry synthetic aperture radar (GB-InSAR) over a mountainous area is investigated. A novel APS compensation scheme is proposed to compensate the fluctuated APS caused by a spatial 3-D inhomogeneous refractivity index distribution without any a priori knowledge of moving location. The proposed method simultaneously addresses to identify moving pixels by a criterion of absolute velocity estimated by the coherent pixels technique (CPT). The proposed method consists mainly of three steps: 1) the stratified APS compensation; 2) identification of moving pixel candidate; and 3) the residual APS [remained APS after 1)] compensation by Kriging interpolation. The steps mentioned above are iteratively applied in order to increase the accuracy of the whole process. In this framework, we develop the 2-D quadratic polynomial model of the refractivity index with respect to slant range and topographic height for modeling the stratified APS. Furthermore, a prediction of the residual APS is achieved by applying the intrinsic random function of order $k$ (IRF- $k$ ) Kriging interpolation, taking into account the nonstationarity of the residual APS. We evaluate the proposed method using zero-baseline GB-differential InSAR (GB-DInSAR) data over a mountainous area located in Minami-Aso, Kumamoto, Japan, through the near real-time post-landslide measurement campaign.

Modeling 3D soil lithotypes variability through geostatistical data fusion of CPT parameters

The Cone Penetration Test (CPT) measures enable to recognize vertical lithological sequence at each investigated point. From the tip resistance qc and sleeve resistance fs profiles, the Soil Behavior Type index ISBT has been calculated, in order to identify the lithotypes alongside depth. The present study focuses on the combination of different variables to provide a lithological and mechanical subsoil characterization. The main objectives of the paper are: (1) to model the 3D spatial variability structure of the soil lithotypes and mechanical properties using qc and fs profiles; 2) to evaluate the uncertainties of the estimates for designing purposes. 182 CPTs were collected in a 900 km2 area (corresponding to a subsoil volume of about 12 km3) located in the study site in the Bologna province (Italy). The study area is made up of fine-grained soils, silt and clay mixtures that are intercalated at different depths by sandy and gravelly soils. These variations of each soil fraction affect the engineering properties of these alluvial deposits. For 3D modeling, two geostatistical methods, Ordinary Kriging (stationary method), and Intrinsic Random Function theory (non-stationary method) have been used. The results show that the non-stationary method allows to obtain more reliable qc and fs predicted values. The final stochastic mechanical and lithological model enables engineers and geologists to detect the emerging of fan and paleochannels bodies where mean resistance values can abruptly change in terms of bearing capacity, liquefaction potential and static and dynamic settlement occurrence.

Attacking SRAM PUFs using very-low-temperature data remanence

In this work, we extend our previous manuscript regarding a systematic study of data remanence effects on an intrinsic Static Random Access Memory Physical Unclonable Function (SRAM PUF) implemented on a Commercial Off-The-Shelf (COTS) device in the temperature range between −110∘C and −40∘C. As the experimental results of our previous work show, an attack against intrinsic SRAM PUFs, which takes advantage of data remanence effects exhibited due to low temperatures, is possible, resulting in the attacker being able to know the PUF response, with high probability. As demonstrated in our previous work, this attack is highly resistant to memory erasure techniques and can be used to manipulate the cryptographic keys produced by the SRAM PUF. In this work, we examine and discuss potential countermeasures against this attack in more detail, and investigate whether this attack can be performed using an experimental setup that does not guarantee a high degree of thermal isolation. Additionally, we also examine and discuss whether very low temperatures can be used to perform another relevant type of attack against SRAM PUFs, based on whether very low temperature can prevent the SRAM from being overwritten. Finally, we also discuss related works and the generalisation of our results in more detail.

Quantification of methane emissions in a Mediterranean landfill (Southern Spain). A combination of flux chambers and geostatistical methods

In this study, landfill gas emissions from a landfill located in southern Spain were estimated using static surface flux chambers and applying and comparing four geostatistical methods; ordinary kriging, lognormal kriging, intrinsic random functions of order K and indicator kriging. This paper presents the methodology used to calculate methane (and carbon dioxide) emissions from a landfill in southern Spain. Static flux chambers were used to estimate emissions through the sealing layer of a landfill assuming that the geospatial mean best expresses the average value of these emissions. This study considers several geostatistical methods for obtaining the corresponding spatial estimation, using measurements obtained from static flux chambers and finding the best proven results. The most appropriate geostatistical analysis method was found to be indicator kriging and lognormal kriging because of the simplicity of its implementation and the transformation of the flux measurements. Methane surface emissions (100 g·m−2·d−1) and visualization of the hotspots were significant enough to result in the placement of a new cover across the entire landfill. This additional cover had an immediate impact on the effectiveness of the recovery system and increased LFG collection flow rates by 15% with an increase in CH4 concentration in the collected gas from 50% to 60%.

Intrinsic random functions on the sphere

Spatial stochastic processes that are modeled over the entire Earth’s surface require statistical approaches that directly consider the spherical domain. In practice, such processes rarely are second-order stationary — that is, they do not have constant mean values or the covariance function that depends only on their angular distance. Here, in order to model non-stationary processes on the sphere, we extend the notion of intrinsic random functions and show that low-frequency truncation plays an essential role. We show that these developments can be presented through the theory of reproducing kernel Hilbert space. In addition, the link between universal kriging and splines is carefully investigated, whereby we show that thin-plate splines are non-applicable for surface fitting on the sphere.

Intrinsic random functions for mitigation of atmospheric effects in terrestrial radar interferometry

Abstract The benefits of terrestrial radar interferometry (TRI) for deformation monitoring are restricted by the influence of changing meteorological conditions contaminating the potentially highly precise measurements with spurious deformations. This is especially the case when the measurement setup includes long distances between instrument and objects of interest and the topography affecting atmospheric refraction is complex. These situations are typically encountered with geo-monitoring in mountainous regions, e.g. with glaciers, landslides or volcanoes. We propose and explain an approach for the mitigation of atmospheric influences based on the theory of intrinsic random functions of order k (IRF-k) generalizing existing approaches based on ordinary least squares estimation of trend functions. This class of random functions retains convenient computational properties allowing for rigorous statistical inference while still permitting to model stochastic spatial phenomena which are non-stationary in mean and variance. We explore the correspondence between the properties of the IRF-k and the properties of the measurement process. In an exemplary case study, we find that our method reduces the time needed to obtain reliable estimates of glacial movements from 12 h down to 0.5 h compared to simple temporal averaging procedures.

Local intrinsic stationarity and its inference

Dense spatial data are commonplace nowadays, and they provide the impetus for addressing nonstationarity in a general way. This paper extends the notion of intrinsic random function by allowing the stationary component of the covariance to vary with spatial location. A nonparametric estimation procedure based on gridded data is introduced for the case where the covariance function is regularly varying at any location. An asymptotic theory is developed for the procedure on a fixed domain by letting the grid size tend to zero.

A case study of the modeling of a hydrothermal reservoir: Khankala deposit of geothermal waters

In 2013 the Grozny State Oil Technical University together with members of the “Geothermal resources” consortium started a pilot project to build a geothermal plant on the basis of the most promising XIII layer of the Khankala thermal waters deposit. Planned capacity of the projected facility is 22.8GJ/h, which will heat a greenhouse complex. Geostatistical analysis and numerical simulations are required for achieving sustainability in exploitation of the resource and in order to define new drilling sites. In this paper intrinsic random functions of k order kriging (IRF-k) is used for the XIII layer map creation and estimation of the temperature distribution within it. The results obtained by geostatistical approach are used for numerical simulation of the hydro-thermal process relied to reinjection of cold thermal water back into the productive layer. The work carried out allows highlighting the most promising zones for exploitation and shows that the geothermal water temperature in the production well starts to decrease after 6 years.

Intrinsic random functions and universal kriging on the circle

Intrinsic random functions (IRF) provide a versatile approach when the assumption of second-order stationarity is not met. Here, we develop the IRF theory on the circle with its universal kriging application. Unlike IRF in Euclidean spaces, where differential operations are used to achieve stationarity, our result shows that low-frequency truncation of the Fourier series representation of the IRF is required for such processes on the circle. All of these features and developments are presented through the theory of reproducing kernel Hilbert space. In addition, the connection between kriging and splines is also established, demonstrating their equivalence on the circle.

On a class of space–time intrinsic random functions

Power law generalized covariance functions provide a simple model for describing the local behavior of an isotropic random field. This work seeks to extend this class of covariance functions to spatial-temporal processes for which the degree of smoothness in space and in time may differ while maintaining other desirable properties for the covariance functions, including the availability of explicit convergent and asymptotic series expansions.

Spectral representation of intrinsically stationary fields

Transferring the concept of processes with weakly stationary increments to arbitrary locally compact Abelian groups two closely related notions arise: while intrinsically stationary random fields can be seen as a direct analog of intrinsic random functions of order k applied by G. Matheron in geostatistics, stationarizable random fields arise as a natural analog of definitizable functions in harmonic analysis. We concentrate on intrinsically stationary random fields related to finite-dimensional, translation-invariant function spaces, establish an orthogonal decomposition of random fields of this type, and present spectral representations for intrinsically stationary as well as stationarizable random fields using orthogonal vector measures.

Simulation of Gaussian Random Fields With One Derivative

Circulant embedding provides a fast and exact algorithm for simulating stationary Gaussian random fields on grids. However, the method tends to work poorly for differentiable processes with correlation ranges comparable to the size of the simulation domain. This work describes an algorithm for simulating differentiable stationary Gaussian processes, with the focus on processes possessing exactly one derivative. The algorithm works by first simulating a filtered version of the process using circulant embedding and then recovering the original process from the filtered version. This recovery step is the slowest part of the computation, requiring many solutions of large systems of linear equations. Preconditioned conjugate gradient provides a viable method for obtaining essentially exact solutions to these linear equations. The memory requirements are linear in the number of simulation points and if more than one simulation under a model is needed, subsequent simulations are much faster than the first. The algorithm can be modified to simulate certain differentiable nonstationary intrinsic random functions, to which existing approaches to circulant embedding do not apply. A supplemental file contains code for simulating these intrinsic random functions.

A geostatistical and probabilistic spectral image processing methodology for monitoring potential CO 2 leakages on the surface

Remote sensing has demonstrated success in various environmental applications over the past three decades. This is largely attributed to its ability for good areal coverage and continued development in sensor technologies. Carbon dioxide Capture and Storage (CCS) is an emerging climate change mitigation technology where monitoring is vital for its sustainability. This research investigates the use of spectral remote sensing imagery in detecting potential CO 2 occurrences at the surface, should a leakage occur from subsurface reservoirs where CO 2 is stored. Currently, there are no known leakages of CO 2 at industrial storage sites, therefore, this research was carried out at the Latera natural analogue site in Italy, in order to develop the methodology described. This paper describes the use of a popular probabilistic information fusion theory, referred to as the Dempster–Shafer theory of evidence, to analyse outlier pixels (anomalies). Outlier pixels are first determined using a new geostatistical image filtering methodology based on Intrinsic Random Function (IRF), Independent Component Analysis (ICA), and the industry standard parametric Reed–Xiaoli (RX) anomaly detection. Information fusion of detected outlier pixels and indirect surface effects of CO 2 leakage over time, such as stressed vegetation or mineral alterations, assigns a confidence measure per outlier pixel in order to identify potential leakage points. After visual validation using direct field measurements, it was demonstrated that the proposed methodology is able to detect majority of the seepage points at Latera, and holds promise as a new unsupervised CO 2 monitoring methodology.

Interpolating datasets with trends: A modified median polish approach

This article investigates an alternative method to deal with data sets in the presence of trends. Median polish kriging (MPK) was introduced as an alternative solution to universal kriging or intrinsic random functions of order k (IRF-k) for estimation in the presence of trends. The maps obtained using the original MPK algorithm show banding artefacts which do not appear in the reference data set. A modified version of MPK was introduced to attempt to remove the banding artefacts. The results confirm the improvement in quality of estimate using the modified version of MPK (called MPKm), which takes into account the problems of clustered samples and boundary effect associated with the re-addition of the trend along bands. The variation introduced in the median polish algorithm proved to be satisfactory in eliminating the artefacts.

Multivariate Intrinsic Random Functions for Cokriging

In multivariate geostatistics, suppose that we relax the usual second-order-stationarity assumptions and assume that the component processes are intrinsic random functions of general orders. In this article, we introduce a generalized cross-covariance function to describe the spatial cross-dependencies in multivariate intrinsic random functions. A nonparametric method is then proposed for its estimation. Based on this class of generalized cross-covariance functions, we give cokriging equations for multivariate intrinsic random functions in the presence of measurement error. A simulation is presented that demonstrates the accuracy of the proposed nonparametric estimation method. Finally, an application is given to a dataset of plutonium and americium concentrations collected from a region of the Nevada Test Site used for atomic-bomb testing.

Modèles géostatistiques de concentrations ou de débits le long des cours d'eau

Estimating concentrations or flow rates along a stream network requires specific models. Two classes of models, recently proposed in the literature, are generalized, to the intrinsic case in particular. We present a global construction by ‘streams’, i.e. on the whole set of paths between sources and outlet. Combining stationary or intrinsic one-dimensional random functions leads to stationary or intrinsic models on segments, with discontinuities at the forks. A construction from outlet to sources, leads to stationary or intrinsic models on each stream, without any discontinuity at the forks. The linear variogram is found as a particular case. The extension to the linear model of coregionalization is immediate, allowing a multivariate modelling of concentrations. To cite this article: C. de Fouquet, C. Bernard-Michel, C. R. Geoscience 338 (2006).

Study of the spatio-temporal variation of soil moisture under forest using intrinsic random functions of order k

The aim of this study is to describe the spatial pattern and temporal evolution of soil water content observed at the plot scale during the soil drying and wetting cycle. The intrinsic random function of order k (IRF- k) approach was used which decomposes the drift and covariance structure to define models of spatial covariance through increments of a sufficiently high order, so that the drift can be filtered out and stationarity attained. The raw directional variograms of soil water content of the study site in southern Italy showed a clear parabolic increase and zonal anisotropy in the northern and eastern directions. The order of the drift ( k) was 2, whereas the generalized covariance model varied during the soil drying and wetting cycle. The maps of soil water content showed the dynamics of soil water redistribution as a function of the temporal rainfall pattern. The spatial variation of soil water was affected by two main factors: topography and texture. Also the spatial distribution of the kriging standard errors changed over time as a function of the date of soil water content measurement. Finally, a critical analysis on the use of the generalized covariance approach is developed and the results compared with the ones of universal kriging.

Testing the correctness of the sequential algorithm for simulating Gaussian random fields

The sequential algorithm is widely used to simulate Gaussian random fields. However, a rigorous application of this algorithm is impractical and some simplifications are required, in particular a moving neighborhood has to be defined. To examine the effect of such restriction on the quality of the realizations, a reference case is presented and several parameters are reviewed, mainly the histogram, variogram, indicator variograms, as well as the ergodic fluctuations in the first and second-order statistics. The study concludes that, even in a favorable case where the simulated domain is large with respect to the range of the model, the realizations may poorly reproduce the second-order statistics and be inconsistent with the stationarity and ergodicity assumptions. Practical tips such as the ‘multiple-grid strategy’ do not overcome these impediments. Finally, extending the original algorithm by using an ordinary kriging should be avoided, unless an intrinsic random function model is sought after.