Prof. Subhankar KarmakarIIT Bombay

Philosophy of Mathematical Models of Watershed Hydrology

Module 22 Lectures

Philosophy of mathematical models of watershed hydrology

Lecture 1

Objectives of this module is to introduce the terms andconcepts in mathematical modelling which will form as a toolfor effective and efficient watershed management throughwatershed modelling

Module 2

Topics to be covered

Concept of mathematical modeling

Watershed - Systems Concept

Classification of Mathematical Models

Different Components in Mathematical Modelling

Module 2

A model is a representation of reality in simple form based on hypotheses and equations:

There are two types of models Conceptual Mathematical

Modeling Philosophy

Experiment

Computation

Theory

Module 2

Conceptual Models

Qualitative, usually based on graphs

Represent important system:

components

processes

linkages

Interactions

Conceptual Models can be used:

As an initial step

For hypothesis testing

For mathematical model development

As a framework

For future monitoring, research, and management actions at a site

Modeling = The use of mathematics as a tool to explain and make predictions of natural phenomena (Cliff Taubes, 2001)

Mathematical modelling may involve words, diagrams, mathematical notation and physical structure

This aims to gain an understanding of science through the use of mathematical models on high performance computers

Science

MathematicsComputer Science

Module 2

Mathematical Models

Mathematical modeling of watershed can address a wide range of environmental

and water resources problems.

Planning, designing and managing water resources systems involve impact

prediction which requires modelling.

Developing a model is an art which requires knowledgeof the system being modeled, the user’s

objectives, goals and information needs, and some analytical and programming skills.

(UNESCO, 2005)

Module 2

Mathematical Models Contd…

Mathematical Modeling Process

Working Model

Mathematical Model

Computational Model

Results/

Conclusions

Real World Problem

Simplify Represent

Translate

Simulate

Interpret

Module 2

Mean – average or expected value

Variance – average of squared deviations from the mean value

Reliability – Probability (satisfactory state)

Resilience – Probability (satisfactory state following unsatisfactory state)

Robustness – adaptability to other than design input conditions

Vulnerability – expected magnitude or extent of failure when

unsatisfactory state occurs

Consistency- Reliability or uniformity of successive results or events

Module 2

Overall measures of system performance

Watershed - Systems Concept

Input Output(Eg. Rainfall, Snow etc.)

(Eg. Discharge)

http://www.desalresponsegroup.org/alt_watershedmgmt.html

Module 2

The Modeling Process

Model World

Mathematical Model(Equations)

Real World

Input parameters

Interpret and Test(Validate) Formulate

Model World Problem

Model Results

Mathematical Analysis

Solutions,Numericals

Module 2

Model:

A mathematical description of the watershed system.

Model Components:

Variables, parameters, functions, inputs, outputs of the watershed.

Model Solution Algorithm:

A mathematical / computational procedure for performing operations on the model for getting outputs from inputs of a watershed.

Types of Models Descriptive (Simulation)

Prescriptive (Optimization)

Deterministic

Probabilistic or Stochastic

Static

Dynamic

Discrete

Continuous

Deductive, inductive, or floating

Basic Concepts

Module 2

Categories of Mathematical Models

TypeEmpirical

Based on data analysisMechanistic

Mathematical descriptions based on theory

Time FactorStatic or steady-state

Time-independentDynamic

Describe or predict system behavior over time

Treatment of Data Uncertainty and VariabilityDeterministic

Do not address data variabilityStochastic

Address variability/uncertainty

Module 2

Classification of Watershed Models

Based on nature of the algorithms

Empirical

Conceptual

Physically based

Based on nature of input and uncertainty

Deterministic

Stochastic

Based on nature of spatial representation

Lumped

Distributed

Black-box

Module 2

Based on type of storm event

Single event

Continuous event

It can also be classified as:

Physical models

Hydrologic models of watersheds;

Scaled models of ships

Conceptual

Differential equations,

Optimization

Simulation modelsModule 2

Classification of Watershed Models Contd…

Descriptive:

That depicts or describes how things actually work, and answers the

question, "What is this?“

Prescriptive:

suggest what ought to be done (how things should work) according to an

assumption or standard.

Deterministic:

Here, every set of variable states is uniquely determined by parameters in the

model and by sets of previous states of these variables. Therefore, deterministic

models perform the same way for a given set of initial conditions.

Module 2

Classification of Watershed Models Contd…

Probabilistic (stochastic):In a stochastic model, randomness is present, and variable states are not describedby unique values, but rather by probability distributions.

Static:A static model does not account for the element of time, while a dynamic modeldoes.

Dynamic:Dynamic models typically are represented with difference equations or differentialequations.

Discrete:A discrete model does not take into account the function of time and usually usestime-advance methods, while a Continuous model does.

Module 2

Classification of Watershed Models Contd…

Deductive, inductive, or floating: A deductive model is a logical structure based on

a theory. An inductive model arises from empirical findings and generalization from

them. The floating model rests on neither theory nor observation, but is merely the

invocation of expected structure.

Single event model:

Single event model are designed to simulate individual storm events and have no

capabilities for replenishing soil infiltration capacity and other watershed abstraction.

Continuous:

Continuous models typically are represented with f(t) and the changes are reflected

over continuous time intervals.

Module 2

Classification of Watershed Models Contd…

Black Box Models:

These models describe mathematically the relation between rainfall and surface

runoff without describing the physical process by which they are related. e.g. Unit

Hydrograph approach

Lumped models:

These models occupy an intermediate position between the distributed models and

Black Box Models. e.g. Stanford Watershed Model

Distributed Models:

These models are based on complex physical theory, i.e. based on the solution of

unsteady flow equations.Module 2

Classification of Watershed Models Contd…

Watershed Modelling Terminology

Input variablesspace-time fields of precipitation, temperature, etc.

Parameters Size Shape Physiography Climate Hydrogeology Socioeconomics

State variablesspace-time fields of soil moisture, etc.

Drainage Land use Vegetation Geology and Soils Hydrology

Module 2

Equations variables

Independent variablesspace x

time t

Dependent variablesdischarge Q

water level h

All other variables are function of the independent or dependent

variables

Module 2

Watershed Modelling Terminology Contd…

Goals & Objectives

Both goals and objectives are very important to accomplish a project. Goals without

objectives can never be accomplished while objectives without goals will never take

you to where you want to be.

Goals Objectives

Vague, less structured Very concrete, specific and measurable

High level statements that provide

overall context of what the project is

trying to accomplish

Attainable, realistic and low level

statements that describe what the project

will deliver.

Tangible

Intangible

Long term

Short termGoals

Module 2