Martin M. Fogel

Elevation effects on rainfall: A stochastic model

Abstract The variation in point precipitation with elevation is investigated using an event-based stochastic model of thunderstorm rainfall and empirical data. Parameters of the model correspond to the number of events per unit of time and the depth of rainfall per event. An increase in precipitation with elevation may be due to an increase in the number of events, in the amount of rainfall per event or to some combination of both possibilities. The distribution of the number of events per season is assumed to be a Poisson variate while the distribution of point rainfall depths may be taken as geometric. The summation of a random number of random variables is used to represent seasonal point precipitation. Assuming that the two parameters of the model increase linearly with elevation, then total seasonal rainfall increases as a quadratic polynomial with elevation. The use of the model allows one to obtain the return period of storm rainfall of a given magnitude despite a short historical record. An independent set of data was used to verify the procedure.

An event-based stochastic model of phosphorus loading into a lake

Abstract In most lakes, phosphorus (P) is the nutrient controlling the trophic state. Thus, for effective control of eutrophication, the uncertainty in P-loading should be encoded as a probability density function (pdf). Specifically, the pdf of P-loading Y from non-point agricultural sources is sought by means of an event-based stochastic model. P-loading events are triggered by precipitation events ( X 1 , X 2 , T ), in which X 1 is the rainfall amount, X 2 the duration, and T the interarrival time between events. ( X 1 , X 2 ) are dependent random variables, while T is assumed to be exponentially distributed. The precipitation event causes runoff, which carries dissolved P into the lake with a concentration C 1 and sediment yield, Z , which carries fixed or sorbed P into the lake in a fraction C 2 of Z . Seasonal loading of P is calculated by adding random numbers of random variables. The model accounts separately for dissolved P and sorbed P. Explicit expressions are given for the mean and variance of each type of P-loadings. The case study of a sub-watershed of Lake Balaton, Hungary, is used to illustrate the methodology. Precipitation data, empirical rainfall-runoff-sediment yield relationships and a small number of observations of events are used to calibrate the model and estimate the means and variances of loading per event and per season. Then a simulation method is used to estimate complete pdf of these random variables. Use of the model for alternative methods of controlling P-loading is briefly discussed, as well as the economics of control.

Lake ecosystems: A polyhedral dynamics representation

Abstract For water quality problems involving global changes in the structure of the aquatic ecosystem, differential equation representations are often inadequate. A technique for modeling the global structure and dynamic behavior of such systems, called polyhedral dynamics, is presented by means of a simple yet realistic example. The lake ecosystem, composed of organic and inorganic nutrients, light, phytoplankton, zooplankton and fish, is represented as a multidimensional “graph” using a technique labeled “Q-analysis”, then a flow pattern is superimposed upon the “Q-analysis” to describe the dynamic behavior of the system. The importance of each element may be measured by its connectivity level with the other elements of the system. In this way, a strong correlation is found between the results obtained through Q-analysis and traditional ecological thinking. In other words, a Q-analysis can be used to gain insight into a complex process such as lake eutrophication. Also, consideration of the dynamic behavior gives a quantitative basis to the qualitative changes which an ecosystem undergoes as a result of increased nutrient loading. The approach may thus be used to assess the effectiveness of alternative eutrophication control methods.

UNCERTAINTY IN THE RETURN PERIOD OF MAXIMUM HYDROLOGIC EVENTS: A BAYESIAN APPROACH

The effect of uncertainty in parameters of rainfall inputs on the return period of maximum rainfall amounts and runoff volumes in a thunderstorm season is ascertained. First, the return period T sub R of the maximum seasonal rainfall amount is considered using an event-based rainfall model which possesses the following features: (1) the distribution of the number of events per season N is Poisson with mean m; (2) the distribution of point rainfall amount R per event is exponential with mean 1/u; and (3) N and R are independent. Then, the case of uncertainty in the return period T sub Q of the maximum seasonal runoff volume Q for an ungaged streamsite is studied. T sub Q is calcualted on the basis of the above event-based rainfall model coupled with a linear rainfall--runoff relationship. A correct distribution function for the return period T sub R and T sub Q is obtained by integrating out the uncertainty in m and u, which is encoded in prior conjugate distribution. It is found that the expected values of the return periods possess a certain bias, and that the variances are substantial even for a record length of twenty years. This demonstrates the advantage of following this approach for a decision- theoretic analysis of a water-resource design problem. The approach enables us furthermore to design structrures, relying only on rainfall data on watersheds with ungaged streams, by taking into account uncertainty of design site parameters. Also, the design can be tailored to a specific problem rather than rely on the strict use of a prespecified T-year design flood. /Auther/

RESERVOIR SEDIMENTATION UNDER UNCERTAINTY: ANALYTIC APPROACH VERSUS SIMULATION / Sédimentation des réservoirs en cas de l'incertitude: méthode analytique contre la méthode par simulation

Abstract Two methods are presented of estimating accumulated sediment yield stemming from erosion in a semiarid climate during a given time span, and the methods are compared from the viewpoint of the economic consequences evaluated within a Bayesian framework. The design of reservoirs requires the estimation of the random sediment volume Z accumulated over the lifetime of the project. An analytic and a simulation method are used to estimate the density function of Z and to calculate an economically optimal sediment storage space D*. A case study in Arizona, USA, is used to illustrate the methodology for the typical situation when only rainfall records are available. Both methods prove to be superior to the traditional procedure leaning on mean values. For the example considered, the two methods yield commensurate results. It appears that, in a semiarid basin, the effect of parameter uncertainty on both Z and D* may be considerably higher than that of natural uncertainty.