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. MODFLOW 3D grid

Conceptual models

Conceptual model A conceptual model is a representation of a system. It consists of concepts used to help people know, understand, or simulate a subject the model represents. In contrast, physical models are physical object such as a toy model that may be asse ...

s 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 A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and express ...

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 using

tributaries 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 drain ...

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

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 physi ...

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 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 In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...

Froude number In continuum mechanics, the Froude number (, after William Froude, ) is a dimensionless number defined as the ratio of the flow inertia to the external field (the latter in many applications simply due to gravity). The Froude number is based on ...

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 between

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 ...

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 t ...

, 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 Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...

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 describe ...

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 solven ...

, 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 measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...

, 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 speci ...

). 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 Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...

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 enc ...

. 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 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, ...

are a type of

mathematical model 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 are commonly used in hydrology to describe data, as well as relationships between data. Using statistical methods, hydrologists develop

empirical relationship In science, an empirical relationship or phenomenological relationship is a relationship or correlation that is supported by experiment and observation but not necessarily supported by theory. Analytical solutions without a theory An empirical rel ...

s 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 '' ar ...

standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...

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 Distribution may refer to: Mathematics * Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a vari ...

, 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 the

Pearson 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 Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...

, 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

stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...

s, 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 Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...

. 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 Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...

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 Conceptual may refer to: Philosophy and Humanities *Concept *Conceptualism *Philosophical analysis (Conceptual analysis) *Theoretical definition (Conceptual definition) * Thinking about Consciousness (Conceptual dualism) *Pragmatism (Conceptual p ...

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 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:

= i(t) - q(t)

which indicates that the change in storage over time is the difference between inflows and outflows. The storage-discharge relationship is:

q(t) = S(t)/K

where K is a constant that indicates how quickly the reservoir drains; a smaller value indicates more rapid outflow. Combining these two equation yields

K  = i - q

and has the solution: NonLin Reserv

q= i(1-e^)

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:

v =  R^ S^

describes steady, one-dimensional groundwater flow using the hydraulic conductivity and the hydraulic gradient:

\vec q = -K \nabla h

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:

T \nabla^2 h = S

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

D  - v  =

Poiseuille's Law describes laminar, steady, one-dimensional fluid flow using the shear stress:

= - \mu

Cauchy's integral is an integral method for solving boundary value problems:

f(a) = \frac \oint_\gamma \frac\, dz

Solution algorithms

Analytic methods

Exact solutions for algebraic, differential, and integral equations can often be found using specified boundary conditions and simplifying assumptions.

Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...

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 In the United States, a SWAT team (special weapons and tactics, originally special weapons assault team) is a police tactical unit that uses specialized or military equipment and tactics. Although they were first created in the 1960s to ...

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 sou ...

, FEFLOW, MIKE SHE, and WEAP.

Model calibration and evaluation

Rainfall_runoff_relation_simulated_by_non-linear_resrvoir_model

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.

References

External links

* http://drought.unl.edu/MonitoringTools/DownloadableSPIProgram.aspx {{Computer modeling Water resources management

Conceptual models

Scale analogs

Process analogs

Statistical models

Moments

Correlation analysis

Conceptual models

Governing equations

Solution algorithms

Analytic methods

Numeric methods

Model calibration and evaluation

See also

References

External links