Model Covariance Functions Research Articles

Free energy in multi-species mixed p-spin spherical models

We prove a Parisi formula for the limiting free energy of multi-species spherical spin glasses with mixed $p$-spin interactions. The upper bound involves a Guerra-style interpolation and requires a convexity assumption on the model's covariance function. Meanwhile, the lower bound adapts the cavity method of Chen so that it can be combined with the synchronization technique of Panchenko; this part requires no convexity assumption. In order to guarantee that the resulting Parisi formula has a minimizer, we formalize the pairing of synchronization maps with overlap measures so that the constraint set is a compact metric space. This space is not related to the model's spherical structure and can be carried over to other multi-species settings.

Probabilistic Linear Inversion of Satellite Gravity Gradient Data Applied to the Northeast Atlantic

AbstractWe explore the mantle density structure of the northeast Atlantic region using constrained linear inversion of the satellite gravity gradient data based on statistical prior information and assuming a Gaussian model. The uncertainty of the residual gravity gradient signal is characterized by a covariance matrix obtained using geostatistical analysis of controlled‐source seismic data. The forward modeling of the gravity gradients in the 3D reference crustal model is performed using a global spherical harmonics analysis. We estimate the model covariance function in the radial and angular directions using a variogram method. We compute volumetric gravity gradient kernels for a spherical shell covering the northeast Atlantic region down to the mantle transition zone (410 km depth). The solution of the linear inverse problem in the form of the mean density model and the posterior covariance matrix follows a least squares approach. The results indicate that on average the seismic velocity variation is proportional to the density variation in the northeast Atlantic region. However, a noticeable mismatch or anti‐correlation exists in some areas, such as the Greenland‐Iceland‐Faroe ridge and southwestern Norway. The predicted low‐density anomalies at depths of 100–150 km underneath the northeast Atlantic Ocean are correlated with the distribution of Cenozoic submarine volcanoes and seamount‐like features of the seafloor.

Surface Wave Tomography with Spatially Varying Smoothing Based on Continuous Model Regionalization

Surface wave tomography based on continuous regionalization of model parameters is widely used to invert for 2-D phase or group velocity maps. An inevitable problem is that the distribution of ray paths is far from homogeneous due to the spatially uneven distribution of stations and seismic events, which often affects the spatial resolution of the tomographic model. We present an improved tomographic method with a spatially varying smoothing scheme that is based on the continuous regionalization approach. The smoothness of the inverted model is constrained by the Gaussian a priori model covariance function with spatially varying correlation lengths based on ray path density. In addition, a two-step inversion procedure is used to suppress the effects of data outliers on tomographic models. Both synthetic and real data are used to evaluate this newly developed tomographic algorithm. In the synthetic tests, when the contrived model has different scales of anomalies but with uneven ray path distribution, we compare the performance of our spatially varying smoothing method with the traditional inversion method, and show that the new method is capable of improving the recovery in regions of dense ray sampling. For real data applications, the resulting phase velocity maps of Rayleigh waves in SE Tibet produced using the spatially varying smoothing method show similar features to the results with the traditional method. However, the new results contain more detailed structures and appears to better resolve the amplitude of anomalies. From both synthetic and real data tests we demonstrate that our new approach is useful to achieve spatially varying resolution when used in regions with heterogeneous ray path distribution.

Assessing random field models in finite element analysis: a case study

The buckling load of thin-walled structures is sensitive to the presence of imperfections. Random field models constitute an established approach to accounting for these imperfections in numerical computations. Current evaluations of finite element models including random fields are usually done by comparison with the experimental buckling loads. This paper proposes an alternative approach to account for discrepancies inherent to numerical modelling. We use this to judge the shape of fitted correlation functions and the sensitivity of the computed buckling load with respect to the number of eigenvectors included in the expansion of the random field. Further, the applicability of fitted model covariance functions is contrasted with a principal component analysis approach as well as with artificially created local random imperfections. The results show that random field models have to be used with greatest care in order to avoid erroneous predictions of the buckling load.

Breeding value accuracy estimates for growth traits using random regression and multi-trait models in Nelore cattle

We quantified the potential increase in accuracy of expected breeding value for weights of Nelore cattle, from birth to mature age, using multi-trait and random regression models on Legendre polynomials and B-spline functions. A total of 87,712 weight records from 8144 females were used, recorded every three months from birth to mature age from the Nelore Brazil Program. For random regression analyses, all female weight records from birth to eight years of age (data set I) were considered. From this general data set, a subset was created (data set II), which included only nine weight records: at birth, weaning, 365 and 550 days of age, and 2, 3, 4, 5, and 6 years of age. Data set II was analyzed using random regression and multi-trait models. The model of analysis included the contemporary group as fixed effects and age of dam as a linear and quadratic covariable. In the random regression analyses, average growth trends were modeled using a cubic regression on orthogonal polynomials of age. Residual variances were modeled by a step function with five classes. Legendre polynomials of fourth and sixth order were utilized to model the direct genetic and animal permanent environmental effects, respectively, while third-order Legendre polynomials were considered for maternal genetic and maternal permanent environmental effects. Quadratic polynomials were applied to model all random effects in random regression models on B-spline functions. Direct genetic and animal permanent environmental effects were modeled using three segments or five coefficients, and genetic maternal and maternal permanent environmental effects were modeled with one segment or three coefficients in the random regression models on B-spline functions. For both data sets (I and II), animals ranked differently according to expected breeding value obtained by random regression or multi-trait models. With random regression models, the highest gains in accuracy were obtained at ages with a low number of weight records. The results indicate that random regression models provide more accurate expected breeding values than the traditionally finite multi-trait models. Thus, higher genetic responses are expected for beef cattle growth traits by replacing a multi-trait model with random regression models for genetic evaluation. B-spline functions could be applied as an alternative to Legendre polynomials to model covariance functions for weights from birth to mature age.

Gravity field recovery from airborne gradiometer data using collaction and taking into account correlated errors

Least squares collocation is a very comprehensive method for gravity field modelling, since it may use known noise characteristics of the data. In many earlier applications the errors affecting the data were considered uncorrelated, mainly due to the difficulty in estimating the systematic character of such kind of errors. In this study, error covariance functions of airborne gravity gradiometer data are estimated by comparing model covariance functions with empirical covariance functions of the gravity gradiometer data. The model covariance functions were estimated from accurate surface gravity data and continuated upward to the height of the airborne measurements using the covariance propagation law. The estimated error covariance functions were modeled as finite ones and used as an additional information for the prediction of gravity anomalies from gravity gradiometer data. The assessment of the prediction results was made by comparing the gravity values predicted from the airborne gradient data and showed up to 25% improvement compared to not using correlated errors.

Stochastic regularization: Smoothness or similarity?

Inversions of geophysical data often involve solving large‐scale underdetermined systems of equations that require regularization, preferably through incorporation of a priori information. Since many natural phenomena exhibit complex random behavior, statistical properties offer important a priori constraints. Inversion constrained by model covariance functions, a form of stochastic regularization, is formally equivalent to imposing simultaneously the auxiliary constraints of (i) model correlation (smoothness) and (ii) similarity with a preferred model (damping). We show that a priori stochastic information defines uniquely the relative contributions of smoothing and damping, such that the higher the fractal dimension the greater the damping contribution. However, if the model discretization interval exceeds the characteristic scale length of the parameters to be resolved, stochastic regularization artificially reduces to only damping constraints.

Gravity Field Continuation of Irregularly Spaced Data Using Least Squares Collocation

SUMMARY The calculation of Bouguer anomalies in high mountains requires a gravity field continuation from irregularly spaced gravity data on the topography to an equipotential surface. Least squares collocation (LSC) is successfully used for geodetic purposes and offers an elegant solution to this problem. A great advantage of this method is the possibility to give estimates of the accuracy of the approximated gravity field. Of central importance in LSC is the covariance function. The influence on the result of the model covariance function (MCF), which replaces in the calculation the empirical covariance function (ECF) of the data, is studied. The synthetic model used for these studies is as realistic as possible, including topography, data gaps and noise. The quality of the predicted error seems to be much more sensitive to the choice of the MCF than the quality of the approximation. The improvement of the field continuation to an equipotential surface compared to simple interpolation is remarkable. An example of such a field continuation is given. The Bouguer anomaly in the Andes of northern Chile is continued to the sea-level by the combined application of collocation and simple models, describing regional parts of the gravity field.

Gravity anomalies and sea surface heights derived from a combined GEOS 3/Seasat altimeter data set

Gravity anomalies and sea surface heights have been computed on a 0.125° grid in the ocean areas from a combined GEOS 3/Seasat altimeter data set. The basic estimation procedure used least squares collocation where model covariance functions are tailored to individual areas through altimeter residual variance scaling. Preliminary tests led to production prediction procedures using a reference model defined by a set of potential coefficients complete to degree 180. Comparisons of the predicted anomalies with ship‐derived values showed agreement varying from ±9 to ±30 mGal. No correction to the altimeter‐implied sea surface heights was made for sea surface topography effects. The maximum anomaly predicted is 396 mGal near Hawaii and the most negative anomaly is −361 mGal over the Puerto Rican Trench. The actual resolution of the predicted quantities was estimated to be about 0.19° based on a power spectrum analysis. The resolution is limited by our data selection process which uses 300 data points from a thinned altimeter data set for the prediction of the 0.125° in a 3° × 3° block with one data selection and matrix inversion. The predicted data are used to compute 104 potential coefficient spectrums using flat earth approximations developed by Forsberg. The spectra were classified in terms of smooth, mild, or rough areas. The great majority (78) of the spectra were in the smooth classification. Good agreement was found from spectra computed from topographically reduced land data and the mean ocean data, up to degree 800. We found that the spectra decayed as l−3.6 as compared to the decay of l−3 implied by the Kaula model. The gridded data was used to compute 1° × 1° and 0.5° × 0.5° mean values. The 1° mean anomalies were compared to terrestrial data where an rms difference of ±7 mGal was found in comparing 10,139 values. These new values have allowed us to identify 1° mean anomalies (based on terrestrial estimates) that are in substantial error.

A Stochastic-Dynamic Model for the Spatial Structure of Forecast Error Statistics

A simple model that yields the spatial correlation structure of global atmospheric mass-field forecast errors is derived. The model states that the relative potential vorticity of the forecast error is forced by spatially multi-dimensional white noise. The forecast error equation contains a nondimensional parameter c0, which depends on the Rossby radius of deformation. From this stochastic-dynamic equation, a deterministic equation for the spatial covariance function of the 500 mb geopotential error field is obtained. Three methods of solution are examined: 1) an analytic method based on spherical harmonics, 2) a numerical method based on stratified sampling of Monte-Carlo realizations of the stochastic-dynamic equation, and 3) a combined analytic-numerical method based on two successive applications of a fast Poisson solver to the deterministic covariance equation. The three methods are compared for accuracy and efficiency, and the third (combined) method is found to be clearly superior. The model's covariance function is compared with global correlation data of forecast-minus-observed geopoteniial fields for the DST-6 period February–March 1976. The data are based on the GLAS forecast-assimilation system in use at that lime (Ghil et al., 1979). The model correlations agree well with the latitude dependence of the data correlations. The fit between model and data confirms that the forecast error between 24 and 36 h is largely random, rather than systematic; the value of the parameter c0 which gives the best fit suggests that much of this error can be attributed to baroclinic, rather than barotropic effects. Deterministic influences not included in the model appear at 12 and 48 h. They suggest possibilities of improving the forecast system by a better objective analysis and initialization procedure, and a better treatment of planetary-wave propagation, respectively. An analytic formula is obtained which locally approximates well the model's global correlations. This formula is convenient to use in the calculation of weighting coefficients for analysis and assimilation schemes. It shows that Gaussian functions are a poor approximation for the forecast error correlations of the mass field, and their derivatives an even poorer approximation to wind field correlations.

Underwater Acoustic Scattering from Lake Surfaces. I. Theory, Experiment, and Validation of the Data

A statistical analysis, based directly on ensemble averages, is made of reverberation data obtained from the wind-driven surface of a fresh water lake. To produce the reverberation, two types of signals consisting of gated (linear “chirp”) FM or CW pulses, are transmitted with a carrier frequency of 110 kHz. In the CW cases the pulse durations are 0.1 and 1.0 msec. The bandwidth of the FM signals (pulse durations 1.25 msec) is 9.6 kHz, and operation in all instances is monostatic (R@T). In the CW experiment the grazing angle is 22°, and in the FM experiment the grazing angle is 0°. Particular care is taken to insure that a valid sample ensemble is obtained by testing each sample ensemble for independence (from a one-sample runs test) and homogeneity (by a suitable Kolmogorov-Smirnov test). A recently developed point scattering model is introduced and adapted to the present experimental geometry to provide a theoretical description of the reverberation and a basis for later comparisons with the data. From this model covariance functions and intensities of the scattered signal are determined and quantitative insight obtained regarding the nonstationarity of the reverberation process.