We study certain aspects of the sample path behaviour of χ
2 processes; in particular, problems related to the behaviour of these processes at their local extrema. Emphasis is placed on behaviour that is qualitatively different to that observed for Gaussian processes, rather than on phenomena common to both classes of processes, such as previously studied (global) extremal type results.
An understanding of the mechanisms controlling the P concentrations in the Upper Jordan River is necessary to develop strategies to avoid eutrophication of Lake Kinneret. This study examines the solubility of orthophosphate and CaCO3 in the calcareous Upper Jordan Basin via a thermodynamic model. The variation in alkalinity, pH, Ca, and phosphate concentrations in the surface waters of the Jordan River are consistent with an equilibrium between the solution and a metastable Ca phosphocarbonate phase with an apparent composition of Ca2(HCO3)2HPO4. This phase, CBP2, has not been observed in sediments or suspended particles. However, it rapidly forms on calcite surfaces in laboratory experiments. The CBP2 is replaced after a few days by a more stable compound, CBP3, with an apparent composition of Ca3(HCO3)3PO4. The apparent solubility product of CBP2 is 10−19.96±0.87. The consistency of the observed water chemistry collected from a variety of stations in the Upper Jordan Watershed over a period of 20 yr with this simple metastable “equilibrium” model suggests that CBP2 is rapidly formed in the Jordan River waters and its tributaries and needs to be considered in models of fluvial P transport.
We study the sample path properties of χ
2 random surfaces, in particular in the neighbourhood of their extrema. We show that, as is the case for their Gaussian counterparts, χ
2 surfaces at high levels follow the form of certain deterministic paraboloids, but that, unlike their Gaussian counterparts, at low levels their form is much more random. This has a number of interesting implications in the modelling of rough surfaces and the study of the ‘robustness' of Gaussian field models. The general approach of the paper is the study of extrema via the ‘Slepian model process', which, for χ
2 fields, is tractable only at asymptotically high or low levels.
We study certain aspects of the sample path behaviour of χ2 processes; in particular, problems related to the behaviour of these processes at their local extrema. Emphasis is placed on behaviour that is qualitatively different to that observed for Gaussian processes, rather than on phenomena common to both classes of processes, such as previously studied (global) extremal type results.
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