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A hydrologic model is a simplification of a real-world system (e.g., surface water, soil water, wetland, groundwater, estuary) that aids in understanding, predicting, and managing water resources. Both the flow and quality of water are commonly studied using hydrologic models.
Conceptual models
Conceptual models are commonly used to represent the important components (e.g., features, events, and processes) that relate hydrologic inputs to outputs. These components describe the important functions of the system of interest, and are often constructed using entities (stores of water) and relationships between these entitites (flows or fluxes between stores). The conceptual model is coupled with scenarios to describe specific events (either input or outcome scenarios). For example, a watershed model could be represented usingtributaries
A tributary, or affluent, is a stream or river that flows into a larger stream or main stem (or parent) river or a lake. A tributary does not flow directly into a sea or ocean. Tributaries and the main stem river drain the surrounding drainag ...
as boxes with arrows pointing toward a box that represents the main river. The conceptual model would then specify the important watershed features (e.g., land use, land cover, soils, subsoils, geology, wetlands, lakes), atmospheric exchanges (e.g., precipitation, evapotranspiration), human uses (e.g., agricultural, municipal, industrial, navigation, thermo- and hydro-electric power generation), flow processes (e.g., overland, interflow, baseflow, channel flow), transport processes (e.g.,
sediments, nutrients, pathogens), and events (e.g., low-, flood-, and mean-flow conditions).
Model scope and complexity is dependent on modeling objectives, with greater detail required if human or environmental systems are subject to greater risk. Systems modeling
Systems modeling or system modeling is the interdisciplinary study of the use of models to conceptualize and construct systems in business and IT development.analog models to simulate flow and transport systems. Unlike
Scale models offer a useful approximation of physical or chemical processes at a size that allows for greater ease of visualization. The model may be created in one (core, column), two (plan, profile), or three dimensions, and can be designed to represent a variety of specific initial and boundary conditions as needed to answer a question.
Scale models commonly use physical properties that are similar to their natural counterparts (e.g., gravity, temperature). Yet, maintaining some properties at their natural values can lead to erroneous predictions. Properties such as viscosity, friction, and surface area must be adjusted to maintain appropriate flow and transport behavior. This usually involves matching dimensionless ratios (e.g., Reynolds number, Froude number).
Groundwater flow can be visualized using a scale model built of acrylic and filled with sand, silt, and clay. Water and tracer dye may be pumped through this system to represent the flow of the simulated groundwater. Some physical aquifer models are between two and three dimensions, with simplified boundary conditions simulated using pumps and barriers.
Example 1
The linear-reservoir model (or Nash Model) is widely used for rainfall-runoff analysis. 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:
which indicates that the change in storage over time is the difference between inflows and outflows. The storage-discharge relationship is:
where K is a constant that indicates how quickly the reservoir drains; a smaller value indicates more rapid outflow. Combining these two equation yields
and has the solution:
Example 2
Instead of using a series of linear reservoirs, also the model of a non-linear reservoir can be used.
In such a model the constant K in the above equation, that may also be called ''reaction factor'', needs to be replaced by another symbol, say α (Alpha), to indicate the dependence of this factor on storage (S) and discharge (q).
In the left figure the relation is quadratic:
α = 0.0123 q2 + 0.138 q - 0.112
Physical models use
mathematical models
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
that use equations to describe, predict, and manage hydrologic systems, analog models use non-mathematical approaches to simulate hydrology.
Two general categories of analog models are common; scale analogs that use miniaturized versions of the physical system and process analogs
A process is a series or set of activities that interact to produce a result; it may occur once-only or be recurrent or periodic.
Things called a process include:
Business and management
*Business process, activities that produce a specific se ...
that use comparable physics (e.g., electricity, heat, diffusion) to mimic the system of interest.
Scale analogs
Scale models offer a useful approximation of physical or chemical processes at a size that allows for greater ease of visualization. The model may be created in one (core, column), two (plan, profile), or three dimensions, and can be designed to represent a variety of specific initial and boundary conditions as needed to answer a question.
Scale models commonly use physical properties that are similar to their natural counterparts (e.g., gravity, temperature). Yet, maintaining some properties at their natural values can lead to erroneous predictions. Properties such as viscosity, friction, and surface area must be adjusted to maintain appropriate flow and transport behavior. This usually involves matching dimensionless ratios (e.g., Reynolds number, Froude number).
Groundwater flow can be visualized using a scale model built of acrylic and filled with sand, silt, and clay. Water and tracer dye may be pumped through this system to represent the flow of the simulated groundwater. Some physical aquifer models are between two and three dimensions, with simplified boundary conditions simulated using pumps and barriers.
Process analogs
Process analogs are used in hydrology to represent fluid flow using the similarity betweenDarcy's Law
Darcy's law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of ...
, Ohms Law
Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equa ...
, Fourier's Law
Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted .
Heat spontaneously flows along a te ...
, and Fick's Law
Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion equ ...
. The analogs to fluid flow are the flux of electricity
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as describ ...
, heat
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
, and solutes
In chemistry, a solution is a special type of homogeneous mixture composed of two or more substances. In such a mixture, a solute is a substance dissolved in another substance, known as a solvent. If the attractive forces between the solvent ...
, respectively. The corresponding analogs to fluid potential are voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
, temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer.
Thermometers are calibrated in various Conversion of units of temperature, temp ...
, and solute concentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', ''molar concentration'', '' number concentration'', ...
(or chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
). The analogs to hydraulic conductivity
Hydraulic conductivity, symbolically represented as (unit: m/s), is a property of porous materials, soils and rocks, that describes the ease with which a fluid (usually water) can move through the pore space, or fractures network. It depends on ...
are electrical conductivity, thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
, and the solute diffusion coefficient
Diffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is enco ...
.
An early process analog model was an electrical network model of an aquifer composed of resistors in a grid. Voltages were assigned along the outer boundary, and then measured within the domain. Electrical conductivity paper can also be used instead of resistors.
Statistical models
Statistical models are a type of mathematical model that are commonly used in hydrology to describe data, as well as relationships between data. Using statistical methods, hydrologists develop empirical relationships between observed variables, find trends in historical data, or forecast probable storm or drought events.Moments
Statistical moments (e.g.,mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
, standard deviation, skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimodal ...
, kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurt ...
) are used to describe the information content of data. These moments can then be used to determine an appropriate frequency distribution, which can then be used as a probability model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, ...
. Two common techniques include L-moment ratios and Moment-Ratio Diagrams.
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. Examples of extreme value distributions include the Gumbel, Pearson Pearson may refer to:
Organizations Education
*Lester B. Pearson College, Victoria, British Columbia, Canada
*Pearson College (UK), London, owned by Pearson PLC
*Lester B. Pearson High School (disambiguation)
Companies
*Pearson PLC, a UK-based int ...
, and Generalized Extreme Value. The standard method for determining peak discharge uses the log-Pearson Type III (log-gamma) distribution and observed annual flow peaks.
Correlation analysis
The degree and nature of correlation may be quantified, by using a method such as thePearson correlation coefficient
In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ...
, autocorrelation, or the T-test
A ''t''-test is any statistical hypothesis test in which the test statistic follows a Student's ''t''-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of ...
. The degree of randomness or uncertainty in the model may also be estimated using stochastics, or residual analysis. These techniques may be used in the identification of flood dynamics, storm characterization, and groundwater flow in karst systems.
Regression analysis
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
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. In a bivariate diagram, a linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
or higher-order model may be fitted to the data.
Factor Analysis
Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. For example, it is possible that variations in six observed ...
and Principal Component Analysis are multivariate
Multivariate may refer to:
In mathematics
* Multivariable calculus
* Multivariate function
* Multivariate polynomial
In computing
* Multivariate cryptography
* Multivariate division algorithm
* Multivariate interpolation
* Multivariate optical c ...
statistical procedures used to identify relationships between hydrologic variables,.
Convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''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. Convolution is used to predict discharge downstream after a precipitation event. This type of model would be considered a “lag convolution”, because of the predicting of the “lag time” as water moves through the watershed using this method of modeling.
Time-series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Ex ...
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. The questions that are answered through these techniques are often important for municipal planning, civil engineering, and risk assessments.
Markov Chains
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happen ...
are a mathematical technique for determine the probability of a state or event based on a previous state or event. The event must be dependent, such as rainy weather. Markov Chains were first used to model rainfall event length in days in 1976, and continues to be used for flood risk assessment and dam management.
Conceptual models
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 usingalgebraic equations
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
, ordinary or partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
, or integral equations
In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ...
. The model is then solved using analytical or numerical procedures.
Example 1
The linear-reservoir model (or Nash Model) is widely used for rainfall-runoff analysis. 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:
which indicates that the change in storage over time is the difference between inflows and outflows. The storage-discharge relationship is:
where K is a constant that indicates how quickly the reservoir drains; a smaller value indicates more rapid outflow. Combining these two equation yields
and has the solution:
Example 2
Instead of using a series of linear reservoirs, also the model of a non-linear reservoir can be used.
In such a model the constant K in the above equation, that may also be called ''reaction factor'', needs to be replaced by another symbol, say α (Alpha), to indicate the dependence of this factor on storage (S) and discharge (q).
In the left figure the relation is quadratic:
α = 0.0123 q2 + 0.138 q - 0.112
Governing equations
Governing equation The governing equations of a mathematical model describe how the values of the unknown variables (i.e. the dependent variables) change when one or more of the known (i.e. independent) variables change.
Mass balance
A mass balance, also called ...
s 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
Darcy's law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of ...
describes steady, one-dimensional groundwater flow using the hydraulic conductivity and the hydraulic gradient:
Groundwater flow equation Used in hydrogeology, the groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through an aquifer. The transient flow of groundwater is described by a form of the diffusion equation, similar ...
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:
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:
Solution algorithms
Analytic methods
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.Numeric methods
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. There are model software packages for hundreds of hydrologic purposes, such as surface water flow, nutrient transport and fate, and groundwater flow. Commonly used numerical models include SWAT,MODFLOW
MODFLOW is the U.S. Geological Survey modular finite-difference flow model, which is a computer code that solves the groundwater flow equation. The program is used by hydrogeologists to simulate the flow of groundwater through aquifers. The so ...
, FEFLOW
FEFLOW (Finite Element subsurface FLOW system) is a computer program for simulating groundwater flow, mass transfer and heat transfer in porous media and fractured media. The program uses finite element analysis to solve the groundwater flow eq ...
, MIKE SHE, and WEAP.
Model calibration and evaluation
Physical models use parameters
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
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
Evaluation is a
systematic determination and assessment of a subject's merit, worth and significance, using criteria governed by a set of standards. It can assist an organization, program, design, project or any other intervention or initiative to ...
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.
See also
*Hydrological optimization Hydrological optimization applies mathematical optimization techniques (such as dynamic programming, linear programming, integer programming, or quadratic programming) to water-related problems. These problems may be for surface water, groundwate ...
*Scientific modelling
Scientific modelling is a scientific activity, the aim of which is to make a particular part or feature of the world easier to understand, define, quantify, visualize, or simulate by referencing it to existing and usually commonly accepted ...
* Soil and Water Assessment Tool
References
External links
* http://drought.unl.edu/MonitoringTools/DownloadableSPIProgram.aspx {{Computer modeling Water resources management