A History of Regression and Related Model-Fitting in the Earth Sciences (1636?-2000)

The (statistical) modeling of the behavior of a dependent variate as a function of one or more predictors provides examples of model-fitting which span the development of the earth sciences from the 17th Century to the present. The historical development of these methods and their subsequent application is reviewed. Bond's predictions (c. 1636 and 1668) of change in the magnetic declination at London may be the earliest attempt to fit such models to geophysical data. Following publication of Newton's theory of gravitation in 1726, analysis of data on the length of a 1° meridian arc, and the length of a pendulum beating seconds, as a function of sin²(latitude), was used to determine the ellipticity of the oblate spheroid defining the Figure of the Earth. The pioneering computational methods of Mayer in 1750, Boscovich in 1755, and Lambert in 1765, and the subsequent independent discoveries of the principle of least squares by Gauss in 1799, Legendre in 1805, and Adrain in 1808, and its later substantiation on the basis of probability theory by Gauss in 1809 were all applied to the analysis of such geodetic and geophysical data. Notable later applications include: the geomagnetic survey of Ireland by Lloyd, Sabine, and Ross in 1836, Gauss's model of the terrestrial magnetic field in 1838, and Airy's 1845 analysis of the residuals from a fit to pendulum lengths, from which he recognized the anomalous character of measurements of gravitational force which had been made on islands. In the early 20th Century applications to geological topics proliferated, but the computational burden effectively held back applications of multivariate analysis. Following World War II, the arrival of digital computers in universities in the 1950s facilitated computation, and fitting linear or polynomial models as a function of geographic coordinates, trend surface analysis, became popular during the 1950–60s. The inception of geostatistics in France at this time by Matheron had its roots in meeting the evident need for improved estimators in spatial interpolation. Technical advances in regression analysis during the 1970s embraced the development of regression diagnostics and consequent attention to outliers; the recognition of problems caused by correlated predictors, and the subsequent introduction of ridge regression to overcome them; and techniques for fitting errors-in-variables and mixture models. Improvements in computational power have enabled ever more computer-intensive methods to be applied. These include algorithms which are robust in the presence of outliers, for example Rousseeuw's 1984 Least Median Squares; nonparametric smoothing methods, such as kernel-functions, splines and Cleveland's 1979 LOcally WEighted Scatterplot Smoother (LOWESS); and the Classification and Regression Tree (CART) technique of Breiman and others in 1984. Despite a continuing improvement in the rate of technology-transfer from the statistical to the earth-science community, despite an abrupt drop to a time-lag of about 10 years following the introduction of digital computers, these more recent developments are only just beginning to penetrate beyond the research community of earth scientists. Examples of applications to problem-solving in the earth sciences are given.     

Disjunctive kriging revisited: Part I

Difficulties in applying disjunctive kriging (D.K.) with an anamorphosis to a normal distribution have led to an interest in D.K. based on other distributions. After reviewing Gaussian D.K., this paper reviews other types of D.K. based on other infinitely divisible distributions (gamma, Poisson, and negative binomial).     

Misspecifications in interaction model distance decay relations: A spatial structure effect

 An exclusively statistical approach is proposed to address the spatial structure effects of general interaction models. It is shown that the spatial heterogeneity in the estimated region-specific distance decay parameters may in part be due to the combination of two factors: (a) a functional mis-specification of the global distance decay relationship; and (b) the heterogeneity in the region-specific conditional distance distributions. A properly specified global distance decay function allows controlling for these spatially induced biases in the local distance decay parameters. However, inherent multicollinearities between the set of region specific distance decay parameters and other estimated model parameters prevent an unambiguous interpretation. A key conclusion is that a proper model specification, in particular, the specification of the global distance decay relationship, is of paramount importance in interaction modeling and for accessibility studies. Received: September 2002 / Accepted: January 2003     

Uppermost Tortonian to Quaternary depocentre migration related with segmentation of the strike-slip Palomares Fault Zone,Vera Basin (SE Spain)

The Palomares Fault Zone (PFZ) is one of the main strike-slip brittle shear zones found in the Betics. It is segmented in several faults that have been active between the Upper Tortonian and present day. Data from drill cores in the Palomares area have permitted us to define the geometry and location of sedimentary depocentres related with the PFZ. These data show an eastward displacement between the Upper Tortonian to Messinian and the Pliocene–Quaternary sedimentary depocentres, towards the presently active Arteal fault, which bounds the western mountain front of Sierra Almagrera, showing that deformation along this fault zone has migrated towards the east, from the Palomares segment, with its main activity during the Upper Tortonian and Messinian, towards the Arteal fault, active during the Pliocene and Quaternary. To cite this article: G. Booth-Rea et al., C. R. Geoscience 335 (2003).     

Improved stress release model: Application to the study of earthquake prediction in Taiwan area

Introduction Stress release model (SRM) was proposed by Vere-Jones (1978) for statistical study of seismicity. Physically it is a stochastic version of the elastic rebound theory of earthquake genesis. The classical elastic rebound model suggests that the stress has been slowly accumulating until the burst of an earthquake occurrence for stress release. This can be simulated by the jump Markov process in stochastic field, and SRM was developed on the basis of Knopoff (s Markov model (Knop…     

Foundation settlement statistics via finite element analysis

The dispersion observed in soil data comes both from the spatial variability which greatly influences the behavior of large structures and from errors in testing. Thus, the geotechnical engineering deals with uncertainties for which deterministic approaches are not suitable. The resort to probabilistic techniques, enables modeling uncertainties by analyzing their dispersion effect on the global behavior of the structure. The scope of this paper is analyzing settlement and differential settlement variability of a pair of foundations on random heterogeneous medium. The random soil properties of interest are the elastic modulus, and the Poisson ratio. The elastic modulus is modeled as a spatially random field by adopting the lognormal distribution, which enables analyzing its large variability. Because soil Poisson ratio is bounded in practice between two extreme values, its random field is obtained by using the Beta distribution. In this study, one proposes for the Beta field determination, a mapping technique on the probability distribution function diagram, by solving a non-linear equation. However, the mean and variance are unchanged through the mapping operation. Because the soil Poisson ratio is a positive parameter, one prefers to perform the mapping operation with the probability function of the lognormal distribution. Also, the proposed technique can be used for other bounded soil properties such as the porosity. In this paper, settlement and differential settlement statistics prediction are carried out using Monte Carlo simulations combined with deterministic finite element method (DFEM). A performed parametric study shows the following: (i) as the variability of the elastic modulus increases as settlement and differential settlement statistics are important, also, settlement statistics decreases as the Poisson ratio variability increases, and differential settlement statistics do not seem be affected by its variability. (ii) settlement and differential settlement statistics are important for positive inter-property correlation. (iii) a great influence of the correlation lengths on settlement and differential settlement statistics.     

Transpressional deformation in northwestern Sardinia (Italy): insights on the tectonic evolution of the Variscan BeltDéformation transpressive dans le Nord-Ouest de la Sardaigne (Italie) : considérations sur l'évolution tectonique de la chaı̂ne Varisque

Structural investigations in northwestern Sardinia highlight the occurrence of a regional D2 transpressional deformation related both to NNE–SSW direction of compression and to a NW–SE shear displacement. The deformation is continuous and heterogeneous, showing a northward strain increase, indicated by progressively tighter folds and occurrence of F2 sheath folds. D2 transpression is characterised by the presence of a crustal-scale shear deformation overprinting previous D1 structures, related to nappe stacking and top-to-the-south and southwest thrusting. The L2 prominent stretching lineation points to an orogen-parallel extension and to a change in the tectonic transport from D1 to D2. It is attributed to the position of Sardinia close to the northeastern part of the Cantabrian indenter during the progressive evolution of the Ibero-Armorican Arc. D1 phase developed during initial frontal collision, whereas D2 deformation characterised the progressive effect of horizontal displacement during the increasing curvature of the arc. To cite this article: R. Carosi, G. Oggiano, C. R. Geoscience 334 (2002) 287–294.     

Les sillons plissés du Burundien supérieur dans la chaı̂ne Kibarienne d'Afrique centrale

The Kibaran belt that extends all over the central Africa, from the Katanga to the southern Uganda, straddles the African great lakes area. The Kibaran stratigraphy is one of the most debated questions. Some geologists favoured two different systems: the Ruzizian system (1800–2000 Ma) and the Burundian or Kibaran system (1600–960 Ma). Some others favoured a single Burundian system with large areas granitised and metamorphosed during several Burundian tectonic events and with a lot of sedimentary fold belts of the Burundian age. Recent geological data and new remote sensing interpretations allow us to favour the second hypothesis but with a major disconformity within the Burundian system which is separating the Upper and the Lower Burundian. To cite this article: M. Villeneuve, J. Chorowicz, C. R. Geoscience 336 (2004).     

两种新参数在银川—河套和祁连山—六盘山地震带地震预报中的应用研究

采用极值理论求得岗贝尔分布b值,可以克服目前广泛使用的求b值的方法对资料要求高的弱点。根据中强地震发生前小震出现偏离泊松分布的现象,构造泊松分布符合度Yn值。将这两种参数具体应用于银川-河套地震带和祁连山-六盘山地震带进行预测检验。结果表明在两个区域岗贝尔分布b值和泊松分布符合度Yn值对中强地震映震效果均较好。