Parallel Monte Carlo sampling scheme for sphere and hemisphere

The sampling of certain solid angle is a fundamental operation in realistic image synthesis, where the rendering equation describing the light propagation in closed domains is solved. Monte Carlo methods for solving the rendering equation use sampling of the solid angle subtended by unit hemisphere or unit sphere in order to perform the numerical integration of the rendering equation. In this work we consider the problem for generation of uniformly distributed random samples over hemisphere and sphere. Our aim is to construct and study the parallel sampling scheme for hemisphere and sphere. First we apply the symmetry property for partitioning of hemisphere and sphere. The domain of solid angle subtended by a hemisphere is divided into a number of equal sub-domains. Each sub-domain represents solid angle subtended by orthogonal spherical triangle with fixed vertices and computable parameters. Then we introduce two new algorithms for sampling of orthogonal spherical triangles. Both algorithms are based on a transformation of the unit square. Similarly to the Arvo's algorithm for sampling of arbitrary spherical triangle the suggested algorithms accommodate the stratified sampling. We derive the necessary transformations for the algorithms. The first sampling algorithm generates a sample by mapping of the unit square onto orthogonal spherical triangle. The second algorithm directly compute the unit radius vector of a sampling point inside to the orthogonal spherical triangle. The sampling of total hemisphere and sphere is performed in parallel for all sub-domains simultaneously by using the symmetry property of partitioning. The applicability of the corresponding parallel sampling scheme for Monte Carlo and Quasi-D/lonte Carlo solving of rendering equation is discussed.

Random orthogonal matrix simulation

This paper introduces a method for simulating multivariate samples that have exact means, covariances, skewness and kurtosis. We introduce a new class of rectangular orthogonal matrix which is fundamental to the methodology and we call these matrices L matrices. They may be deterministic, parametric or data specific in nature. The target moments determine the L matrix then infinitely many random samples with the same exact moments may be generated by multiplying the L matrix by arbitrary random orthogonal matrices. This methodology is thus termed “ROM simulation”. Considering certain elementary types of random orthogonal matrices we demonstrate that they generate samples with different characteristics. ROM simulation has applications to many problems that are resolved using standard Monte Carlo methods. But no parametric assumptions are required (unless parametric L matrices are used) so there is no sampling error caused by the discrete approximation of a continuous distribution, which is a major source of error in standard Monte Carlo simulations. For illustration, we apply ROM simulation to determine the value-at-risk of a stock portfolio.

PDFOS: PDF estimation based over-sampling for imbalanced two-class problems

This contribution proposes a novel probability density function (PDF) estimation based over-sampling (PDFOS) approach for two-class imbalanced classification problems. The classical Parzen-window kernel function is adopted to estimate the PDF of the positive class. Then according to the estimated PDF, synthetic instances are generated as the additional training data. The essential concept is to re-balance the class distribution of the original imbalanced data set under the principle that synthetic data sample follows the same statistical properties. Based on the over-sampled training data, the radial basis function (RBF) classifier is constructed by applying the orthogonal forward selection procedure, in which the classifier’s structure and the parameters of RBF kernels are determined using a particle swarm optimisation algorithm based on the criterion of minimising the leave-one-out misclassification rate. The effectiveness of the proposed PDFOS approach is demonstrated by the empirical study on several imbalanced data sets.

A three-dimensional magnetometer for motion sensing of a balloon-carried atmospheric measurement package

An instrument is described which carries three orthogonal geomagnetic field sensors on a standard meteorological balloon package, to sense rapid motion and position changes during ascent through the atmosphere. Because of the finite data bandwidth available over the UHF radio link, a burst sampling strategy is adopted. Bursts of 9s of measurements at 3.6Hz are interleaved with periods of slow data telemetry lasting 25s. Calculation of the variability in each channel is used to determine position changes, a method robust to periods of poor radio signals. During three balloon ascents, variability was found repeatedly at similar altitudes, simultaneously in each of three orthogonal sensors carried. This variability is attributed to atmospheric motions. It is found that the vertical sensor is least prone to stray motions, and that the use of two horizontal sensors provides no additional information over a single horizontal sensor

Variability of the Atlantic thermohaline circulation described by three-dimensional empirical orthogonal functions

We describe the use of bivariate 3d empirical orthogonal functions (EOFs) in characterising low frequency variability of the Atlantic thermohaline circulation (THC) in the Hadley Centre global climate model, HadCM3. We find that the leading two modes are well correlated with an index of the meridional overturning circulation (MOC) on decadal timescales, with the leading mode alone accounting for 54% of the decadal variance. Episodes of coherent oscillations in the sub-space of the leading EOFs are identified; these episodes are of great interest for the predictability of the THC, and could indicate the existence of different regimes of natural variability. The mechanism identified for the multi-decadal variability is an internal ocean mode, dominated by changes in convection in the Nordic Seas, which lead the changes in the MOC by a few years. Variations in salinity transports from the Arctic and from the North Atlantic are the main feedbacks which control the oscillation. This mode has a weak feedback onto the atmosphere and hence a surface climatic influence. Interestingly, some of these climate impacts lead the changes in the overturning. There are also similarities to observed multi-decadal climate variability.

Sampling uncertainties in surface radiation budget calculations in RADAGAST

In the Radiative Atmospheric Divergence Using ARM Mobile Facility GERB and AMMA Stations (RADAGAST) project we calculate the divergence of radiative flux across the atmosphere by comparing fluxes measured at each end of an atmospheric column above Niamey, in the African Sahel region. The combination of broadband flux measurements from geostationary orbit and the deployment for over 12 months of a comprehensive suite of active and passive instrumentation at the surface eliminates a number of sampling issues that could otherwise affect divergence calculations of this sort. However, one sampling issue that challenges the project is the fact that the surface flux data are essentially measurements made at a point, while the top-of-atmosphere values are taken over a solid angle that corresponds to an area at the surface of some 2500 km2. Variability of cloud cover and aerosol loading in the atmosphere mean that the downwelling fluxes, even when averaged over a day, will not be an exact match to the area-averaged value over that larger area, although we might expect that it is an unbiased estimate thereof. The heterogeneity of the surface, for example, fixed variations in albedo, further means that there is a likely systematic difference in the corresponding upwelling fluxes. In this paper we characterize and quantify this spatial sampling problem. We bound the root-mean-square error in the downwelling fluxes by exploiting a second set of surface flux measurements from a site that was run in parallel with the main deployment. The differences in the two sets of fluxes lead us to an upper bound to the sampling uncertainty, and their correlation leads to another which is probably optimistic as it requires certain other conditions to be met. For the upwelling fluxes we use data products from a number of satellite instruments to characterize the relevant heterogeneities and so estimate the systematic effects that arise from the flux measurements having to be taken at a single point. The sampling uncertainties vary with the season, being higher during the monsoon period. We find that the sampling errors for the daily average flux are small for the shortwave irradiance, generally less than 5 W m−2, under relatively clear skies, but these increase to about 10 W m−2 during the monsoon. For the upwelling fluxes, again taking daily averages, systematic errors are of order 10 W m−2 as a result of albedo variability. The uncertainty on the longwave component of the surface radiation budget is smaller than that on the shortwave component, in all conditions, but a bias of 4 W m−2 is calculated to exist in the surface leaving longwave flux.

A new set of orthogonal patterns in weather and climate: Optimally interpolated patterns.

A new spectral-based approach is presented to find orthogonal patterns from gridded weather/climate data. The method is based on optimizing the interpolation error variance. The optimally interpolated patterns (OIP) are then given by the eigenvectors of the interpolation error covariance matrix, obtained using the cross-spectral matrix. The formulation of the approach is presented, and the application to low-dimension stochastic toy models and to various reanalyses datasets is performed. In particular, it is found that the lowest-frequency patterns correspond to largest eigenvalues, that is, variances, of the interpolation error matrix. The approach has been applied to the Northern Hemispheric (NH) and tropical sea level pressure (SLP) and to the Indian Ocean sea surface temperature (SST). Two main OIP patterns are found for the NH SLP representing respectively the North Atlantic Oscillation and the North Pacific pattern. The leading tropical SLP OIP represents the Southern Oscillation. For the Indian Ocean SST, the leading OIP pattern shows a tripole-like structure having one sign over the eastern and north- and southwestern parts and an opposite sign in the remaining parts of the basin. The pattern is also found to have a high lagged correlation with the Niño-3 index with 6-months lag.

Estimating the spatial scales of regionalized variables by nested sampling, hierarchical analysis of variance and residual maximum likelihood

The variogram is essential for local estimation and mapping of any variable by kriging. The variogram itself must usually be estimated from sample data. The sampling density is a compromise between precision and cost, but it must be sufficiently dense to encompass the principal spatial sources of variance. A nested, multi-stage, sampling with separating distances increasing in geometric progression from stage to stage will do that. The data may then be analyzed by a hierarchical analysis of variance to estimate the components of variance for every stage, and hence lag. By accumulating the components starting from the shortest lag one obtains a rough variogram for modest effort. For balanced designs the analysis of variance is optimal; for unbalanced ones, however, these estimators are not necessarily the best, and the analysis by residual maximum likelihood (REML) will usually be preferable. The paper summarizes the underlying theory and illustrates its application with data from three surveys, one in which the design had four stages and was balanced and two implemented with unbalanced designs to economize when there were more stages. A Fortran program is available for the analysis of variance, and code for the REML analysis is listed in the paper. (c) 2005 Elsevier Ltd. All rights reserved.

The Representative Soil Sampling Scheme of England and Wales: the spatial variation of topsoil nutrient status and pH between 1971 and 2001

The Representative Soil Sampling Scheme (RSSS) has monitored the soil of agricultural land in England and Wales since 1969. Here we describe the first spatial analysis of the data from these surveys using geostatistics. Four years of data (1971, 1981, 1991 and 2001) were chosen to examine the nutrient (available K, Mg and P) and pH status of the soil. At each farm, four fields were sampled; however, for the earlier years, coordinates were available for the farm only and not for each field. The averaged data for each farm were used for spatial analysis and the variograms showed spatial structure even with the smaller sample size. These variograms provide a reasonable summary of the larger scale of variation identified from the data of the more intensively sampled National Soil Inventory. Maps of kriged predictions of K generally show larger values in the central and southeastern areas (above 200 mg L-1) and an increase in values in the west over time, whereas Mg is fairly stable over time. The kriged predictions of P show a decline over time, particularly in the east, and those of pH show an increase in the east over time. Disjunctive kriging was used to examine temporal changes in available P using probabilities less than given thresholds of this element. The RSSS was not designed for spatial analysis, but the results show that the data from these surveys are suitable for this purpose. The results of the spatial analysis, together with those of the statistical analyses, provide a comprehensive view of the RSSS database as a basis for monitoring the soil. These data should be taken into account when future national soil monitoring schemes are designed.

Uncertainties in annual riverine phosphorus load estimation: Impact of load estimation methodology, sampling frequency, baseflow index and catchment population density

Models developed to identify the rates and origins of nutrient export from land to stream require an accurate assessment of the nutrient load present in the water body in order to calibrate model parameters and structure. These data are rarely available at a representative scale and in an appropriate chemical form except in research catchments. Observational errors associated with nutrient load estimates based on these data lead to a high degree of uncertainty in modelling and nutrient budgeting studies. Here, daily paired instantaneous P and flow data for 17 UK research catchments covering a total of 39 water years (WY) have been used to explore the nature and extent of the observational error associated with nutrient flux estimates based on partial fractions and infrequent sampling. The daily records were artificially decimated to create 7 stratified sampling records, 7 weekly records, and 30 monthly records from each WY and catchment. These were used to evaluate the impact of sampling frequency on load estimate uncertainty. The analysis underlines the high uncertainty of load estimates based on monthly data and individual P fractions rather than total P. Catchments with a high baseflow index and/or low population density were found to return a lower RMSE on load estimates when sampled infrequently than those with a tow baseflow index and high population density. Catchment size was not shown to be important, though a limitation of this study is that daily records may fail to capture the full range of P export behaviour in smaller catchments with flashy hydrographs, leading to an underestimate of uncertainty in Load estimates for such catchments. Further analysis of sub-daily records is needed to investigate this fully. Here, recommendations are given on load estimation methodologies for different catchment types sampled at different frequencies, and the ways in which this analysis can be used to identify observational error and uncertainty for model calibration and nutrient budgeting studies. (c) 2006 Elsevier B.V. All rights reserved.

Variograms of ancillary data to aid sampling for soil surveys

To provide reliable estimates for mapping soil properties for precision agriculture requires intensive sampling and costly laboratory analyses. If the spatial structure of ancillary data, such as yield, digital information from aerial photographs, and soil electrical conductivity (EC) measurements, relates to that of soil properties they could be used to guide the sampling intensity for soil surveys. Variograins of permanent soil properties at two study sites on different parent materials were compared with each other and with those for ancillary data. The ranges of spatial dependence identified by the variograms of both sets of properties are of similar orders of magnitude for each study site, Maps of the ancillary data appear to show similar patterns of variation and these seem to relate to those of the permanent properties of the soil. Correlation analysis has confirmed these relations. Maps of kriged estimates from sub-sampled data and the original variograrns showed that the main patterns of variation were preserved when a sampling interval of less than half the average variogram range of ancillary data was used. Digital data from aerial photographs for different years and EC appear to show a more consistent relation with the soil properties than does yield. Aerial photographs, in particular those of bare soil, seem to be the most useful ancillary data and they are often cheaper to obtain than yield and EC data.

Comparing sampling needs for variograms of soil properties computed by the method of moments and residual maximum likelihood

It has been generally accepted that the method of moments (MoM) variogram, which has been widely applied in soil science, requires about 100 sites at an appropriate interval apart to describe the variation adequately. This sample size is often larger than can be afforded for soil surveys of agricultural fields or contaminated sites. Furthermore, it might be a much larger sample size than is needed where the scale of variation is large. A possible alternative in such situations is the residual maximum likelihood (REML) variogram because fewer data appear to be required. The REML method is parametric and is considered reliable where there is trend in the data because it is based on generalized increments that filter trend out and only the covariance parameters are estimated. Previous research has suggested that fewer data are needed to compute a reliable variogram using a maximum likelihood approach such as REML, however, the results can vary according to the nature of the spatial variation. There remain issues to examine: how many fewer data can be used, how should the sampling sites be distributed over the site of interest, and how do different degrees of spatial variation affect the data requirements? The soil of four field sites of different size, physiography, parent material and soil type was sampled intensively, and MoM and REML variograms were calculated for clay content. The data were then sub-sampled to give different sample sizes and distributions of sites and the variograms were computed again. The model parameters for the sets of variograms for each site were used for cross-validation. Predictions based on REML variograms were generally more accurate than those from MoM variograms with fewer than 100 sampling sites. A sample size of around 50 sites at an appropriate distance apart, possibly determined from variograms of ancillary data, appears adequate to compute REML variograms for kriging soil properties for precision agriculture and contaminated sites. (C) 2007 Elsevier B.V. All rights reserved.