Model Classification

Models take a variety of forms. Physical models are reduced-dimension representations of real world systems. A physical model of a watershed, such as the model constructed in the lab at Colorado State University, is a large surface with overhead sprinkling devices that simulate the precipitation input. The surface can be altered to simulate various land uses, soil types, surface slopes, and so on; and the rainfall rate can be controlled. The runoff can be measured, as the system is closed. A more common application of a physical model is simulation of open channel flow. The Corps of Engineers San Francisco District maintains and operates the Bay-Delta Model to provide information for answering questions about complex hydraulic flow in the San Francisco Bay and upstream watershed.

Table 2.What is a mathematical model?

Researchers have also developed analog models that represent the flow of water with the flow of electricity in a circuit. With those models, the input is controlled by adjusting the amperage, and the output is measured with a voltmeter. Historically, analog models have been used to calculate subsurface flow.

The HEC-HMS program includes models in a third category—mathematical models. In this manual, that term defines an equation or a set of equations that represent the response of a hydrologic system component to a change in hydrometeorological conditions. Table 2 shows some other definitions of mathematical models; each of these applies to the models included in the program.
Mathematical models, including those that are included in the program, can be classified using a number of different criteria. These focus on the mechanics of the model: how it deals with time, how it addresses randomness, and so on. While knowledge of this classification is not necessary to use the program, it is helpful in deciding which of the models to use for various applications. For example, if the goal is to create a model for predicting runoff from an ungaged watershed, the fitted-parameter models included in the program that require unavailable data are a poor choice. For long-term runoff forecasting, use a continuous model, rather than a single-event model; the former will account for system changes between rainfall events, while the latter will not.

Event or Continuous

This distinction applies primarily to models of infiltration, surface runoff, and baseflow. An event model simulates a single storm. The duration of the storm may range from a few hours to a few days. The key identifying feature is that the model is only capable of representing watershed response during and immediately after a storm. Event infiltration models do not include redistribution of the wetting front between storms and do not account for drying of the soil through evaporation and transpiration. Often but not always, event infiltration models use a function of cumulative loss to compute infiltration capacity. Unit hydrograph models of surface runoff are all classified as event models because the respond to excess precipitation. Excess precipitation is the actual precipitation minus any infiltrated loss. By definition, excess precipitation can only occur during a storm so these are also event models. Some baseflow models are classified as event because they compute receding flow based on the peak flow rate computed during a storm event. These same methods must include a capability to reset between storms to begin the next recession; nevertheless they remain event models.

A continuous model simulates a longer period, ranging from several days to many years. In order to do so, it must be capable of predicting watershed response both during and between precipitation events. For infiltration models, this requires consideration of the drying processes that occur in the soil between precipitation events. Surface runoff models must be able to account for dry surface conditions with no runoff, wet surface conditions that produce runoff during and after a storm, and the transition between the two states. Baseflow methods become increasing important in continuous simulation because the vast majority of the hydrograph is defined by inter-storm flow characteristics. Most of the models included in HEC-HMS are event models.

Spatially-Averaged or Distributed

This distinction applies mostly to models of infiltration and surface runoff. A distributed model is one in which the spatial (geographic) variations of characteristics and processes are considered explicitly, while in a spatially-averaged model, these spatial variations are averaged or ignored. While not always true, it is often the case that distributed models represent the watershed as a set of grid cells. Calculations are carried out separately for each grid cell. Depending on the complexity of the model, a grid cell may interact with its neighbor cells by exchanging water either above or below the ground surface.
It is important that note that even distributed models perform spatial averaging. As we will see later in detail, most of the models included in HEC-HMS are based on differential equations. These equations are written at the so-called point scale. By point scale we mean that the equation applies over a length ∆x that is very small (differential) compared to the size of the watershed. In a spatially-averaged model, the equation is assumed to apply at the scale of a subbasin. Conversely, in a distributed model the equation is typically assumed to apply at the scale of a grid cell. Therefore it is accurate to say that distributed models also perform spatial averaging but generally do so over a much smaller scale than typical spatially-averaged models. HEC-HMS includes primarily spatially-averaged models.

Empirical or Conceptual

This distinction focuses on the knowledge base upon which the mathematical models are built. A conceptual model is built upon a base of knowledge of the pertinent physical, chemical, and biological processes that act on the input to produce the output. Many conceptual models are said to be based on "first principles." This usually means that a control volume is established and equations for the conservation of mass and either momentum or energy are written for the control volume. Conservation is a basic principle of physics that cannot be broken. Through the writing of the equations, a model of the process will emerge. In other cases, conceptual models are developed through a mechanistic view instead of first principles. A mechanistic view attempts to represent the dynamics of a process explicitly. For example, water has been observed to move through soil in very predictable ways. A mechanistic view attempts to determine what processes cause water to move as it is observed. If the processes can be described by one or more mathematical equations, then a model can be developed to directly describe the observed behavior.

An empirical model, on the other hand, is built upon observation of input and output, without seeking to represent explicitly the process of conversion. These types of models are sometimes called "black box" models because they convert input to output without any details of the actual physical process involved. A common way to develop empirical models is to collect field data with observations of input and resulting output. The data is analyzed statistically and a mathematical relationship is sought between input and output. Once the relationship is established, output can be predicted for an observed input. For example, observations of inflow to a river reach and resulting flow at a downstream location could be used to develop a relationship for travel time and attenuation of a flood peak through the reach. These empirical models can be very effective so long as they are applied under the same conditions for which they were originally developed. HEC-HMS includes both empirical and conceptual models.

Deterministic or Stochastic

A deterministic model assumes that the input is exactly known. Further, it assumes that the process described by the model is free from random variation. In reality there is always some variation. For example, you could collect a large sample of soil in the field and take it into a laboratory. Next you could divide the large sample into 10 equal small samples and estimate the porosity of each one. You would find a slightly different value for the porosity of each small sample even though the large sample was collected from a single hole dug in the field. This is one example of natural variation in model input. Process variation is somewhat different. Suppose a flood with a specific peak flow enters a section of river. The flood will move down through the reach and the resulting outflow hydrograph will show evidence of translation and attenuation. However, the bed of the river is constantly moving in response to both floods and inter-flood channel flows. The movement of the bed means that the exact same flood with the same specific peak flow could happen again, but the outflow hydrograph would be slightly different. While you might try to describe the reach carefully enough to eliminate the natural variation in the process, it is not practically possible to do so.

Deterministic models essentially ignore variation in input by assuming fixed input. The input may be changed for different scenarios or historical periods, but the input still takes on a single value. Such an assumption may seem too significant for the resulting model to produce meaningful results. However, deterministic models nevertheless are valuable tools because of the difficulty of characterizing watersheds and the hydrologic environment in the first place. Stochastic models, on the other hand, embrace random variation by attempting to explicitly describe it. For example, many floods in a particular river reach may be examined to determine the bed slope during each flood. Given enough floods to examine, you could estimate the mean bed slope, its standard deviation, and perhaps infer a complete probability distribution. Instead of using a single input like deterministic models, stochastic models include the statistics of variation both of the input and process. All models included in HEC-HMS are deterministic.

Measured-Parameter or Fitted-Parameter

This distinction between measured and fitted parameters is critical in selecting models for application when observations of input and output are unavailable. A measured-parameter model is one in which model parameters can be determined from system properties, either by direct measurement or by indirect methods that are based upon the measurements. The Green and Ampt infiltration model is an example of a measured parameter model. It includes hydraulic conductivity and wetting front suction as parameters. Both parameters can be measured directly using appropriate instruments imbedded in the soil during a wetting-drying cycle. Many other parameters used in infiltration models can be reliably estimated if the soil texture is known; texture can be determined by direct visual examination of the soil.

A fitted-parameter model, on the other hand, includes parameters that cannot be measured. Instead, the parameters must be found by fitting the model with observed values of the input and the output. The Muskingum routing model is an example of a fitted parameter model. The K parameter can be directly estimated as the travel time of the reach. However, the X parameter is a qualitative estimate of the amount of attenuation in the reach. Low values of X indicate significant attenuation while high values indicate pure translation. The only way to estimate the value of X for a particular reach is to examine the upstream hydrograph and the resulting outflow hydrograph. HEC-HMS includes both measured-parameter models and fitted-parameter models.