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A hydrologic model is a simplification of a real world system that aids in understanding, testing, and prediction. In hydrology and hydrogeology, models are used as aids in studying, predicting, and managing surface flow, groundwater flow, and the transport of particles and solutes.

Conceptual Models
Conceptual models are representations of other systems using concepts in the system. These models can span a range of scales and can also be physical models, like toy models. These models can also be described as a generalization process and convey the basic functionality of the system it is representing. Conceptual models are used in a variety of other subjects, not just hydrology.

Bucket Model
The bucket model is a simple approach to the soil moisture and water budget model. Used by Manabe (1969), this conceptual model made the leap into the atmospheric world through becoming a simplified description of the hydrologic cycle and is thought to be the “first generation” of Land Surface Models. The Bucket model calculates potential and actual water losses throughout the system, by separating different areas of the watershed into buckets. Each bucket has a certain “depth” before it over flows and goes into the following bucket representing runoff. For more information and to experiment with the parameters for the Bucket model visit this page ( http://biocycle.atmos.colostate.edu/shiny/bucket/ )

Black Box Model
A black box model is used most commonly in engineering. The basic understanding of a black box model is that there is an input and an output. The input undergoes some kind of transformation in the “black box” to become the output. The reason that it is called the “black box” is because there is no full understanding of the inner workings of the box to create the outputs; its mechanisms are opaque.

Feature, Event, Process Models
Feature, Event, Process Models (FEPs) combine a few different conceptual models together. The features aspect of the model, can be described as a “box and arrows” model. If looking at a watershed, the tributaries can be labeled as boxes with arrows pointing into a different box that would be the main river. Other features to put in this section would be runoff, precipitation, and sediment loading. In the event section of the model, there would be different scenarios listed that would affect different features. These scenarios would include droughts and floods. The processes aspect of the model would include a list of different processes acting on the system that can be physical, chemical, or biological. An example of a process would be lake turnover.

Graphical Models
Graphical models are used in hydrology to visualize systems and relationships between variables. Some graphical models are used to obtain an initial view of a solution that can be verified analytically. Where precise detail is not necessary, the graphical approach may offer insight without intensive time or  computational resources.

See also:

Flownet

Graph Theory

Physical Models
Physical modeling uses a simplified representation of a real world phenomena to understand the behaviour of a complex system. Physical modeling can be implemented in all aspects of hydrology, from surface to groundwater flow. One such example of physical modeling done is by John D. Hewlett, who built a series of sluices at the Cowetta Experimental Forest in Asheville, North Carolina. These concrete sluices allowed Hewlett to model the quantity and speed of model flowing through groundwater down a slope. These concrete sluices are still at Cowetta today and still being used for hillslope hydrology and biogeochemistry experiments today. Physical modeling can be linked to groundwater equations that can tell us more information about how water is moving. Some models may tell us how the groundwater is moving particulates such as pollutants or sediment, or otherwise interacting with topography in the watershed.

There are two types of physical models in hydrology, Scale Models and Analog models.

Scale Models
Scale models may be constructed which offer a useful approximation of physical or chemical hydrologic processes at a size which allows for greater ease of visualization. The model may be set at two or three dimensions, and can be designed to represent a variety of specific initial and boundary conditions as needed to answer a question. The model will have a similar physical properties to the natural system, such as gravity or temperature. However, several forces do not transfer well to varying scales. Friction, viscosity, and surface area may be difficult to scale perfectly, requiring use of alternative materials to represent the natural system. For instance, water in a basin model may be too viscous when flowing in very small channels to represent the flow in a river. The dimensionless number values within these systems, such as Reynolds number (Re) or Froud’s number (Fr), should be taken into consideration.

2-D Scale Models
Scale models may be built to represent systems in two dimensions and are commonly used to visualize groundwater flow paths. These models are often in the x-z plane, representing a vertical slice of the subsurface.

Physical Aquifer Model


Groundwater flow can be modeled 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 visualize 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.

Freshwater Lens Model
Groundwater flow-meteoric water interactions show complex variable-density dynamics. In islands, saline groundwater can create a base, above which will sit freshwater from infiltration of rainwater, creating a local supply or lens of freshwater resources. The geometry of these lenses have been studied using physical models similar to the general Physical Aquifer Model, in acrylic tanks filled with sand and induced saltwater and freshwater flow.

3-D Scale Models
Physically modeling hydrologic processes in three dimensions can be useful for surface water visualization. Examples of large scale 3-D scale models include the Mississippi River Basin Model, the San Francisco Bay Model, and the Chesapeake Bay Model.

Process Models
Process Analog Models are used in hydrology to visualize groundwater flow, using the principle behind Field Theory: Darcy and Ohm’s Law s both describe the same empirical relationship, whether electrical, solute, thermal, or groundwater flow.

Statistical Models
Hydrologists use statistical modeling when it is beneficial to describe or understand whole populations of data, rather than individual or specific data. Using statistical methods, hydrologists may investigate relationships between parameters, find trends in historical data , or forecast probable storm or drought events.

Moments
In statistical modeling, mathematical moments are used to describe distribution shape of data. The first and second moments, mean and variance respectively, are commonly used in hydrologic statistic modeling, along with higher-order moments such as in moment ratios, for determining the most appropriate distribution probability model to use. The two common techniques include L-moment ratios and Moment-Ratio Diagrams.

Regression Analysis
Regression analysis is used in hydrology to determine whether a relationship may exist between independant and dependant 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 or higher-order model may be fitted to the data.

Correlation Analysis
The degree and nature of correlation may be quantified, by using a method such as the Pearson Correlation Coefficient, autocorrelation, or the T-test. 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.

Factor Analysis
Complex hydrologic systems may require multivariate statistical analysis. Relationships may be analyzed for relative relationships by means of factor analysis.

Principal Component Analysis
Many hydrologic systems show relationships between a set of variables, rather than a single pair, necessitating a multivariate approach.

Convolution
In terms of mathematics, 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.

Lag calculation
Convolution modeling, is used to predict the change in discharge downstream in watershed typically 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.

See also:
Correlation and dependence

Correlation matrix

Time Series Analysis
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.

Standard Precipitation Index
The Standard Precipitation Index is a time series obtained from monthly historical precipitation data and used for the local characterization and identification of drought conditions. The SPI may be run at a variety of temporal scales, each of which is beneficial for differing analyses. The precipitation data are fitted to a gamma distribution, after which distribution parameters are estimated and the cumulative probabilities for each event is calculated. The result is a value from 3 to -3 for each historical time step, with values above 2 being unusually wet, and -2 being unusually dry for that location.

Palmer Drought Severity Index (Palmer Drought Index)
The Palmer Drought Severity Index is a statistical assessment of drought conditions for a given month, used for national and state monitoring by NOAA and the USGS.

Markov Chains
Markov Chains 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.

Extremal Analysis
The hydrology of statistically unusual events, such as severe droughts and storms, often requires the use of distributions which focus on the tail of the distribution, rather than the data nearest the mean. These techniques, collectively known as extremal analysis, allow for event identification and uncertainty analysis. Examples of extreme value distribution types include Gumbel, Pearson, and Generalized Extreme Value.

See Also:

 * Transformation
 * ANOVA
 * Kruskal-Wallis
 * Aikake Information Criteria

Mathematical Models
Mathematical model is a representation of a system or process using mathematical equations. It can be represented as a function relationship as shown in the figure. We can classify mathematical models based on their defining equation and solution algorithm they use. The defining equations may be algebraic, differential, or integral in nature, while the solution algorithms may solve for or approximate the solution using analytical, numeric, or computational methods.

Hydrologists use mathematical modelling in a variety of fields and applications, such as nutrient fate and transport, or rainfull/runoff relationships. In the application to nutrient transport, non-point source s of pollution may be analyzed to address agricultural and urban runoff and or land use change and their detrimental effect on local water quality. Running mathematical models on these systems using an appropriate defining equation and solution algorithm may provide insight on how best to mitigate these inputs.

We can classify mathematical models based on their defining equation and solution algorithm they use. It can be represented as a function relationship, with a dependent and an independent variable.

Algebraic
Algebraic equations may be used when a system can be represented by simple linear mathematical equations with reasonable accuracy. Because of its algebraic form, the equation usually produces an exact solution.

A good example in hydrology is the equation for Darcy’s Law, which defines the ability of a fluid to flow through a porous media.

Darcy’s Law:

$$q = K{\Delta h \over L}$$

Differential
In a real world not all systems can be represented using simple algebraic equation and most physical phenomena are best characterized by their rate of change, that is differential equations.

Differential equations are composed of an unknown function and its derivative.

Ordinary Differential Equation:
ODEs are used when a system is represented by a differential function, that involves one independent variable.

Examples: Tank Model

The Tank Model is a very simple model composed of a series of tanks laid vertically one below the other. Precipitation is added into the top tank and evaporation is subtracted from each tank. It incorporates an outlet in each level which simulates the runoff. It has the ability to model a hydrologic response of a wide range of watersheds.

Nash Model

The Nash Model is also called linear cascade is a series of linear reservoirs with the same storage coefficient K, such that the output of the first one is used as an input for the next one. Linear Reservoir is the most widely used for rainfall-runoff analysis. It is based on a combined continuity and storage-discharge equations, which resulted in linear differential equation representing the linear reservoir.

Continuity equation: $${dS(t) \over dt} = i(t) - q(t) Or \Delta S = I - O$$

Storage-discharge equation: $$ q(t) = S(t)/K $$ ; K is time constant for the model.

Combining the two equation we get: $$ K {dq \over dt} = i - q $$    which has a solution,  $$ q= i(1-e^{-t/k}) $$

Poiseuille’s Law

Also known as Hagen–Poiseuille equation, Poiseuille's Law describes pressure change in laminar fluid flow.

Partial Differential Equation:
PDEs are used when two or more independent variables are present in the function which represents the system.

Examples: $$q_x = -K {\delta h \over\delta x},     q_y = -K {\delta h \over\delta y},    q_z = -K {\delta h \over\delta z} $$ $$D {\partial^2 C \over\partial x^2} - \nu {\partial C \over\partial x} = {\partial C \over\partial t}$$
 * Darcy
 * Groundwater Flow
 * Advection-Dispersion
 * Manning’s Equation

Integral
Integral models allow the user to bring a large area into a small, more manageable area that can be observed. Integrated models are capable of looking at two separate parameters at the same time, in hydrology this could be combining hydrologic and groundwater/surface water data.

Analytic Models:
Uses Analytical or exact solution for the equation. It can be used when it is possible to get the solution which satisfies the equation.

Numerical Models:
In many mathematical models equations cannot be solved exactly, necessitating the  development of a numerical solution which approximates the exact solution. An example of this is can be found in advection-dispersion, which must be modeled using numerical solutions such as finite elements https://en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation]. See also: [10.1029/2012WR012134] 10.1002/0470044209.ch7

Computational Models:
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.

Examples:
SCS TR-20

SWAT

MODFLOW

FEFLOW

Quotes

“All models are wrong, but some are useful.” -George Box

“It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.” - Albert Einstein

“Your model is incorrect.”-Jim Freer, Bristol University