### Understanding Mathematical and Statistical Techniques in Hydrology: An Examples-based Approach

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The analogs to fluid flow are the flux of electricity , heat , and solutes , respectively. The analogs to hydraulic conductivity are electrical conductivity , thermal conductivity , and the solute diffusion coefficient.

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An early process analog model was an electrical network model of an aquifer composed of resistors in a grid. Electrical conductivity paper [7] can also be used instead of resistors. Statistical models are a type of mathematical model that are commonly used in hydrology to describe data, as well as relationships between data.

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Statistical moments e. These moments can then be used to determine an appropriate frequency distribution , [12] which can then be used as a probability model. The frequency of extremal events, such as severe droughts and storms, often requires the use of distributions that focus on the tail of the distribution, rather than the data nearest the mean.

These techniques, collectively known as extreme value analysis , provide a methodology for identifying the likelihood and uncertainty of extreme events. The standard method for determining peak discharge uses the log-Pearson Type III log-gamma distribution and observed annual flow peaks.

The degree and nature of correlation may be quantified, by using a method such as the Pearson correlation coefficient , autocorrelation , or the T-test. Regression analysis is used in hydrology to determine whether a relationship may exist between independent and dependent variables. Bivariate diagrams are the most commonly used statistical regression model in the physical sciences, but there are a variety of models available from simplistic to complex. Factor Analysis and Principal Component Analysis are multivariate statistical procedures used to identify relationships between hydrologic variables,.

Convolution is a mathematical operation on two different functions to produce a third function. With respect to hydrologic modeling, convolution can be used to analyze stream discharge's relationship to precipitation.

## Understanding mathematical and statistical techniques in hydrology an…

Convolution is used to predict discharge downstream after a precipitation event. Time-series analysis is used to characterize temporal correlation within a data series as well as between different time series. Many hydrologic phenomena are studied within the context of historical probability. Within a temporal dataset, event frequencies, trends, and comparisons may be made by using the statistical techniques of time series analysis. Markov Chains are a mathematical technique for determine the probability of a state or event based on a previous state or event.

Markov Chains were first used to model rainfall event length in days in , [32] and continues to be used for flood risk assessment and dam management. Conceptual models represent hydrologic systems using physical concepts. The conceptual model is used as the starting point for defining the important model components. The relationships between model components are then specified using algebraic equations , ordinary or partial differential equations , or integral equations. The model is then solved using analytical or numerical procedures. The linear-reservoir model or Nash Model is widely used for rainfall-runoff analysis.

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The model uses a cascade of linear reservoirs along with a constant first-order storage coefficient, K , to predict the outflow from each reservoir which is then used as the input to the next in the series. The model combines continuity and storage-discharge equations, which yields an ordinary differential equation that describes outflow from each reservoir.

The continuity equation for tank models is:. The storage storage-discharge relationship is:. Combining these two equation yields. Instead of using a series of linear reservoirs, also the model of a non-linear reservoir can be used. Governing equations are used to mathematically define the behavior of the system.

Algebraic equations are likely often used for simple systems, while ordinary and partial differential equations are often used for problems that change in space in time. Examples of governing equations include:. Manning's equation is an algebraic equation that predicts stream velocity as a function of channel roughness, the hydraulic radius, and the channel slope:. Darcy's Law describes steady, one-dimensional groundwater flow using the hydraulic conductivity and the hydraulic gradient:. Groundwater flow equation describes time-varying, multidimensional groundwater flow using the aquifer transmissivity and storativity:.

Advection-Dispersion equation describes solute movement in steady, one-dimensional flow using the solute dispersion coefficient and the groundwater velocity:.

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Poiseuille's Law describes laminar, steady, one-dimensional fluid flow using the shear stress:. Cauchy's integral is an integral method for solving boundary value problems:. Exact solutions for algebraic, differential, and integral equations can often be found using specified boundary conditions and simplifying assumptions.

Laplace and Fourier transform methods are widely used to find analytic solutions to differential and integral equations. Many real-world mathematical models are too complex to meet the simplifying assumptions required for an analytic solution. In these cases, the modeler develops a numerical solution that approximates the exact solution. Solution techniques include the finite-difference and finite-element methods, among many others.

Specialized software may also be used to solve sets of equations using a graphical user interface and complex code, such that the solutions are obtained relatively rapidly and the program may be operated by a layperson or an end user without a deep knowledge of the system.

## An Integrated Neuro-Fuzzy-Statistical Approach to Hydrological Modelling

There are model software packages for hundreds of hydrologic purposes, such as surface water flow, nutrient transport and fate, and groundwater flow. Physical models use parameters to characterize the unique aspects of the system being studied. These parameters can be obtained using laboratory and field studies, or estimated by finding the best correspondence between observed and modelled behavior.

Between neighbouring catchments which have physical and hydrological similarities, the model parameters varies smoothly suggesting the spatial transferability of parameters. Model evaluation is used to determine the ability of the calibrated model to meet the needs of the modeler. A commonly used measure of hydrologic model fit is the Nash-Sutcliffe efficiency coefficient. From Wikipedia, the free encyclopedia. Earth Syst. Express deliveries are delivered via Royal Mail 1st Class.

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