Ute Mueller

The U-WEDGE Transformation Method for Multivariate Geostatistical Simulation

To speed up multivariate geostatistical simulation it is common to transform the set of attributes into spatially uncorrelated factors that can be simulated independently. Spatial decorrelation methods are usually based on the diagonalisation of the variance/covariance and semivariogram matrices of the set of attributes for a chosen family of lag spacings. These matrices are symmetric and there are several efficient methods for the approximate joint diagonalisation of a family of symmetric matrices. One of these is the uniformly weighted exhaustive diagonalisation with Gauss iterations (U-WEDGE) method. In contrast to the method of minimum/maximum autocorrelation factors (MAF), where a two structure linear model of coregionalisation is assumed, U-WEDGE can be applied directly to the set of experimental semivariogram matrices without having to place restrictions on the number of structures in the linear model of coregionalisation, thus removing one of the restrictions placed on the subsequent modelling of the spatial structure of the factors. We use an iron-ore data set to illustrate the method and present a comparison between the simulated attributes obtained from U-WEDGE and MAF with the full co-simulation of the attributes.

Direct minimum/maximum autocorrelation factors within the framework of a two structure linear model of coregionalisation

In this paper, we present an approach to the method of minimum/maximum autocorrelation factors (MAF) that involves the derivation of the factors in the space of the sample data. The usual approach to MAF begins with an a priori normal score transformation of each attribute. However, as the MAF method is based on principal component analysis (PCA) this initial transformation is unnecessary. Since our method derives the MAF directly in the space of the sample data, we refer to it as direct minimum/maximum autocorrelation factors (DMAF). We present a theoretical derivation of DMAF that simplifies the multi-Gaussian approach. The DMAF method is particularly advantageous when the factors are simulated using a direct simulation algorithm, as no further transformation is required. We demonstrate the DMAF method by means of the simulation of attributes from a multivariate soil data set and show that this method successfully transforms the sample attributes into uncorrelated factors for all lag spacings and is useful for multivariate simulation.

The impact of the graphics calculator on the assessment of calculus and modelling

In this paper we consider some changes that the use of graphics calculators imposes on the assessment of calculus and mathematical modelling at an undergraduate level. For a student who is proficient in the use of a graphics calculator many standard questions involving differentiation, numerical integration or the graphing of functions do not necessarily test the mathematical skills traditionally regarded as fundamental. Furthermore, the use of a graphics calculator requires new skills to be acquired and employed effectively. Any assessment will therefore need to take into account such items as transcription skills, the limitations of the calculator and the students ability to frame problems. We indicate some of the ways in which the assessment of mathematical tasks can be modified as the mechanics of calculation become routine and questions of analysis and interpretation assume greater importance. Particular references will be made throughout to the HP 38G and the HP 48G graphics calculators.

The dynamics of phase farming in dryland salinity abatement

In Australia, soil salinisation has become a major concern. One way to deal with the problem is for farmers to plant trees or regenerate native bush. However, doing so raises several questions which involve optimal switching times, when switching involves a cost in the form of up-front investments. Optimality conditions are derived for the three-stage problem, and applied to dryland salinity control in Western Australia. Optimal management practices are found to be very sensitive to farmers’ discount rates and to the speed at which the watertable rises or falls.

A marine heatwave drives massive losses from the world’s largest seagrass carbon stocks

Seagrass ecosystems contain globally significant organic carbon (C) stocks. However, climate change and increasing frequency of extreme events threaten their preservation. Shark Bay, Western Australia, has the largest C stock reported for a seagrass ecosystem, containing up to 1.3% of the total C stored within the top metre of seagrass sediments worldwide. On the basis of field studies and satellite imagery, we estimate that 36% of Shark Bay’s seagrass meadows were damaged following a marine heatwave in 2010/2011. Assuming that 10 to 50% of the seagrass sediment C stock was exposed to oxic conditions after disturbance, between 2 and 9 Tg CO2 could have been released to the atmosphere during the following three years, increasing emissions from land-use change in Australia by 4–21% per annum. With heatwaves predicted to increase with further climate warming, conservation of seagrass ecosystems is essential to avoid adverse feedbacks on the climate system.Marine ecosystems and their stored carbon are threatened by warming and marine heatwaves. During a 2010–2011 heatwave, around a third of a Western Australian seagrass ecosystem suffered damage, potentially releasing 2–9 Tg CO2 in the following years.

An Affine Equivariant Multivariate Normal Score Transform for Compositional Data

The geostatistical treatment of continuous variables often includes a transformation to normal scores. In the case of analysing a composition, it has been suggested that standard methods can be applied to (isometric) logratio transformed compositions. Several logratio transformations are available and invariance of the final results under the choice of logratio transform is desirable. However, a geostatistical procedure which includes marginal normal scores transformations of the individual logratio scores via quantile matching will not have this invariance property, nor will the resulting vectors of scores show a joint multivariate normal distribution. In this paper an affine-equivariant normal score transform is proposed. The method is based on a continuous deformation of the underlying logratio space to a Gaussian space. The properties and performance of this method are illustrated and compared with existing alternatives using a simulated setting and a case study from a banded iron formation ore mining operation from Western Australia. The proposed method is also suitable for the study of other multivariate non-compositional cases.

Health at the Sub-catchment Scale: Typhoid and Its Environmental Determinants in Central Division, Fiji.

The impact of environmental change on transmission patterns of waterborne enteric diseases is a major public health concern. This study concerns the burden and spatial nature of enteric fever, attributable to Salmonella Typhi infection in the Central Division, Republic of Fiji at a sub-catchment scale over 30-months (2013–2015). Quantitative spatial analysis suggested relationships between environmental conditions of sub-catchments and incidence and recurrence of typhoid fever. Average incidence per inhabited sub-catchment for the Central Division was high at 205.9/100,000, with cases recurring in each calendar year in 26% of sub-catchments. Although the numbers of cases were highest within dense, urban coastal sub-catchments, the incidence was highest in low-density mountainous rural areas. Significant environmental determinants at this scale suggest increased risk of exposure where sediment yields increase following runoff. The study suggests that populations living on large systems that broaden into meandering mid-reaches and floodplains with alluvial deposition are at a greater risk compared to small populations living near small, erosional, high-energy headwaters and small streams unconnected to large hydrological networks. This study suggests that anthropogenic alteration of land cover and hydrology (particularly via fragmentation of riparian forest and connectivity between road and river networks) facilitates increased transmission of typhoid fever and that environmental transmission of typhoid fever is important in Fiji.

Variability in the spatial and temporal distribution of the saucer scallop, Amusium balloti, in Shark Bay – management implications

The present paper is the first description using geostatistical modelling of recruitment (0+) and residual (1+) scallop variability from an annual survey in a semi-tropical embayment. Geostatistical modelling provides a useful tool to explore and interpret distribution patterns and can provide information to determine potential behaviour of fishers. It may also aid in determining the time it will take from the beginning of the season to reach a catch-rate threshold, which is the management strategy implemented in the Shark Bay scallop fishery since 2004. High variability in recruit abundance and spatial distribution was observed among years, whereas patterns of residual abundance and distribution were less variable because of the fishing patterns of both the scallop and prawn fleets. Comparisons of commercial catch patterns indicated that high survey-abundance areas correlate with higher catches, validating that survey results are a good tool for fishers to utilise to target their fishing practices to optimise and maximise catch efficiencies. The study highlighted the inherent annual variability of scallop recruitment abundance and distribution that are primarily considered to be environmentally driven. However, both recruits and residual scallops contribute to the whole catch, so retaining residual scallops from year to year is important.

Multivariate Estimation Using Log Ratios: A Worked Alternative

Common implementations of geostatistical methods, kriging and simulation, ignore the fact that geochemical data are usually reported in weight percent and are thus compositional data. Compositional geostatistics is an approach developed to ensure that the constant sum constraint is respected in estimation and simulation. The compositional geostatistical framework was implemented to test its applicability to an iron ore mine in Western Australia. Cross-validation was used to compare the results from ordinary cokriging of the additive log ratio variables with those from conventional ordinary cokriging. Two methods were used to back-transform the additive logratio estimates, the additive generalized logistic back transformation and Gauss-Hermite Quadrature approximation. Both the Aitchison distance and the Euclidean distance were used to quantify the error between the estimates and original sample values. The results follow the required constraints and produce better estimates when considering the Aitchison distance. When the Euclidean errors are considered the conventional ordinary cokriging estimates are less biased but the distribution of errors for the additive logratio estimates appear to be superior to the conventional ordinary cokriging estimates.

Trend analysis and short-term forecast of incident HIV infection in Ghana.

The study uses time-series modelling to determine and predict trends in incident HIV infection in Ghana among specific age groups. The HIV data for Ghana were grouped according to northern and southern spatial sectors as they exhibited slightly different data collection formats. The trend of the epidemic is modelled using moving-average smoothing techniques, and the Box-Jenkins ARIMA model is used to forecast cases of newly acquired (incident) HIV infection. Trend analysis of past growth patterns reveals an increase in new cases of HIV infection in the northern sector, with the greatest increase occurring among persons aged 30 years and over. The epidemic in the southern sector appears to have levelled off. However, incident HIV infection in the 20–39-year-old age group of females in the sector is estimated to increase in the next three years. Moreover, the estimates suggest a higher increase in incident cases than that predicted by the National AIDS Control Programme. Nevertheless, incident HIV infection among persons aged 19 and below is found to be relatively stable. Thus, if efforts are made to reduce or prevent an increase in the number of new infections in the northern sector, and for the 20–39 years age group in the southern sector, Ghana will have a brighter future with regard to its response to the HIV epidemic. These findings can assist with developing strategic-intervention policy planning for Ghana and other countries in sub-Saharan Africa.