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Contents

1 Introduction 11.1 What is a Model? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2 Reasons for Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1 Scientic Models that Describe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Engineering Models that Predict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Regulatory Models that Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3 The Modeling Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 The Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 The Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 The Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 The Evaluation Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 The Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Analog Models 82.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1.2 Exponents and Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 Process Analogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.1 Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.2 Electricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.3 Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Solutes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3 Scale Analogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Example: Fractal Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Network Models 143.1 Percolation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.1 Frequency Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Discrete Event Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Example: Extreme Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 Linear Systems Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 System Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.3 Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Example: Unit Hydrographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Statistical Models 194.1 Probability and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.1 Sample Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Higher Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Variation of Sample Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.2 Ordinary Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Factor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Model Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 Time Series Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3.1 Autoregressive (AR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Moving Average (MA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 External Inputs (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Differenced (D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3.5 Integrated (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.6 Example: Salt River Pro ject . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Differential Equations 295.1 System Specication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1.1 Types of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Example: Nash Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2 Analytic Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Laplace Transform Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3 Numerical Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Finite Difference Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Boundary Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Analytic Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4 Example: Linear Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Example: Channel Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5.1 General Routing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Nonlinear Outow-Storage Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Muskingum Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Muskingum-Cunge Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Muskingham Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Vectors, Complex Variables, and Quaternions 396.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.1.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.1.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.1.4 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2 Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.3.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.3.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.3.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.3.4 Relationship to Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4 The Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Dual Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Cauchy Integral Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Additional Reading 507.1 Mathematical References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Management Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Systems/Statistical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Watershed/Surface Water Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Groundwater Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Tables

1.1 The Basis of Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2 General Systems Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 Metric System Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Metric System Prexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Physical analogs between processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1 State Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Method for Determining Return Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 Example convolution problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 Statistical Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Variation in Student Heights, cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1 General differentiation rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 General integration rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1 Vector Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Quaternion Multiplication Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Quaternion Hyperbolic Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures

3.1 Triangular Unit Hydrograph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 Venn Diagram of Events A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1 Linear Reservoir Denition Sketch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Cascade of linear reservoirs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Linear reservoir with a constant input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Sketch diagram for Muskingum storage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1 Diagram illustrating boundary ux components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

1.1 What is a Model?From birth, each of us must learn to negotiate the envi-ronment that surrounds us. By experimentation we learnhow to interact with our environment - we learn hot fromcold, up from down. And once we gain condence in ourknowledge, we learn how to control and manipulate oursurroundings. With maturity we learn that certain ap-proaches are more effective than others - we learn rightfrom wrong.

Thus, our world view begins with basic observations -data that we learn and remember. As data becomes moreabundant, we organize these data in ways that help usremember details. The real world is a complex web of en-tities and relationships. Once we have sufficient data, weform relationships between entities. When we are young,we may not understand the reason for these relationships.But with education we begin to understand why these re-lationships exist, and learn how to use these relationshipsto control our environment. Finally, through experienceand training, we gain sufficient wisdom to use this knowl-edge for our own, and perhaps societys, betterment.

A model, then, is a representation of our environment -the observations we collect, the relationships we infer, theknowledge we gain, and the wisdom we draw upon to act.From the very beginning, we are building mental modelsof our environment, trying to make sense of our place onPlanet Earth, and what we can do to better deal with lifescomplexities.

When someone says they wish to build a model , whatthey are really trying to do is capture the rules by whicha system operates. By explicitly stating these rules, theywish to articulate a specic mental construct of their ob-

servations, experiences and knowledge.Many of these rules are absolute, such as the force of gravity, which can have no ambiguity. Such absolute rulescan be quantied mathematically, and are called laws of nature . Other rules are less absolute, such as the patha leaf takes as it falls to the ground. While controlledby gravity, many uncertain variables - such as leaf shape,random gusts of wind, and intervening branches - affectthe path in complicated ways. In these cases a statisticalrepresentation may be more useful in describing how theleaf falls.

Table 1.1: The Basis of Modeling

Data - Observations of our environment, collected by design(hard data: a monitoring program, experimentation) orby circumstance (soft data: ad hoc, incidental)

Information - Establishing relationships between observa-tions by interpretation of data (hard information: sta-tistical tests), or circumstantial relationships (soft infor-mation: anecdotal)

Knowledge - Understanding why relationships exist - leadingto the ability to predict outcomes - based on scienticprinciples (hard knowledge: hypothesis testing), or byexperience (soft knowledge: trail and error)

Wisdom - Using knowledge to successfully manipulate ouenvironment for personal or public good

Regardless of the approach one uses for documentinthe behavior of a system, the process of model buildinprovides the basis for documenting and communicating

the specic approach used to represent a system.In the most general sense, a model is a simplied representation of a system, where a system consists of entitiesand relationships between entities. Each entity is a physi-cally or logically distinct object, such as a person or eventRelationships are the associations between entities, suchas the predator-prey relationships in a food web, or anevent tree in failure analysis.

Each entity and relationship has attributes, such assize, color, intensity, or severity. Parameters are used toquantify attributes, by providing quantitative or qualita-tive values for each entity or relationship. Parameters nor-mally remain constant, while state variables describe at-

tributes that change over time.Inputs are external stimuli that cause changes to thesystem, where the resulting change may be instantaneousor delayed. The stimulus affects state variables as well aoutputs from the system. Outputs are external responsesof the system. If the response is delayed, then the maximum time required for the response to dissipate is calledthe memory of the system.

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CHAPTER 1. INTRODUCTION

Table 1.2: General Systems Terms

System - A set of relationships between entitiesEntity - An object, person, or event

Relationship - An association between objectsAttribute - A quality of an entity or relationship

Parameter - A quantitative measure of the attribute of anentity or a relationship

State Variable - A measurable attribute of an entity or rela-tionship that changes with time or space. To distinguishit from an input and output, state variables do not affectother entities or relationships.

Input - A measurable quantity that behaves as a stimulus anentity or relationship.

Output - A measurable response of an entity or relationshipto a stimulus.

Memory - The length of time that an input affects the output

1.2 Reasons for ModelingHydrologic models are commonly used to explain and pre-dict complex behavior associated with the management of environmental systems. Often, these models are used toevaluate the effectiveness of regulatory controls, such asfor human health [49], pesticide and herbicide registration[7]; waste isolation at proposed, existing, and abandonedwaste disposal sites [12], and to examine the effects of al-ternative regulatory policies [53].

One reason is to learn how a system operates. Theprocess of building a model assists in data interpretation.Data collected in time and space are analyzed for the pur-

pose of establishing relationships. These relationships, inturn, provide information about the system being mod-eled:

Water levels in certain wells respond faster in one areathan another; Floods move faster in one section of a river than another;and Fertilizers applied during one part of the year contami-nate ground water, but not in another part of the year.A second reason is to understand why observed rela-

tionships exist. This understanding is useful for generatingknowledge. We can employ basic physics to explain that:

The recharge rate is dependent upon the amount of rain-fall and the soil moisture content;

The ood wave velocity is a function of the channel cross-section; and Nutrient transport through the unsaturated zone is mod-ied by plant root uptake.Another reason for constructing or using a model is to

identify a cause, or to assign blame. The detection of acontaminant in ground water must be the result of somerelease. Where was the release, who caused it, when didit occur, and why did it occur? A model is often used to

reconstruct a sequence of events, constrained by availabledata, information, and knowledge, to assess the root causeof some observed condtion.

A common reason to model is to predict the consequences of alternative actions. We are often faced withthe situation that we have several options, but are notsure which course of action is best. In these cases we turto models to help explore the possible outcomes of eacalternative action. Which action is most likely to succeed?What are the possible adverse impacts of the alternatives?Given a set of options, which will cause the least amounof damage, or maximize the benets? How can we desiga system that achieves our goals?

Clearly, there are many reasons for modeling. Eacapplication may have a different reason, thus requiring adifferent model - no one model can be used for all potentiaapplications.

Different models and modeling strategies have been developed in response to these multiple modeling purposesWhile scientists generally seek to remove noise and uncertainty during experiments by careful use of controls, aregulator seeks to account for uncertainty by focusing onworst-case scenarios. Engineers, as well, try to account founcertainty by using conservative measures.

Thus, regulatory and engineering models can be readily separated from scientic models. The use of a scientic model that accounts for the myriad forms of physicalchemical and biological interactions may be fascinatingfor a scientist, but not useful for the design of a simplstructure. In the same respect, an engineering model thatgrossly oversimplies a system may provide an adequatbasis for designing a structure, but ignores the componentsthat most interest a scientist.

The regulatory model differs from the engineering modin that the engineer is not able to alter standards; the engi-neer designs a structure to meet a specied standard. Theregulator, on the other hand, has the special obligation ofstandards specication.

The regulator seeks to dene standards that simulta-neously achieve several objectives. One objective is tsafeguard the public interest. This objective often linksa standard to a critical measures of exposure, which canbe addressed using criteria that adequately account foruncertainties and technological capabilities.

On the other hand, standards could be so specic thatonly limited decision-making is given to the engineer, th

result of this which is to restrain efficiency. In an ideenvironment, standards should enable engineers to eval-uate alternatives using measures of cost-effectiveness oreliability, without the standards placing an undo burden.

The role of regulatory modeling, therefore, should be tidentify those measures of system performance that clearlyrepresent the public interest with a minimum of interdic-tion by the state.

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1.2.1 Scientic Models that DescribeOne objective of scientic modeling is to identify a newor unanticipated process, or to identify novel interactionsbetween processes. Keen observation or inference is re-quired to sort through known knowledge about previouslyunknown mechanisms. An additional objective is to char-acterize the magnitude of environmental parameters.

For both objectives, models are used in the inversesense to identify system structure or identify parameters.A forward model is generally used only to form predictionsthat are compared with observed state variables or modeloutputs.

Several hazards arise when scientic models are usedin the regulatory environment. In these cases, model un-certainties may not be explicitly quantied. Parameterdistributions and uncertainties, which are minimized dur-ing a scientic experiment, may not approximate the truediversity of parameter estimates expected under naturalconditions.

Because the scientic estimation problem is inverse,

rather than forward, extension of modeling results fromthe calibration phase to the application phase is not nec-essarily available. That is, a model that has been con-structed using focused laboratory techniques, based onsound scientic method, may never have been used to forma prediction at eld scales in a way that can be veriedand tested. Even when the model may have been testedfor a specic eld-scale experiment, the degree of agree-ment between model prediction and observed results is leftunresolved.

While inverse models may be used in the regulatoryprocess to assign culpability, such as when sources of pol-lution need to be identied for the determination of re-sponsible parties, these inverse techniques generally lackthe designed controls used in scientic studies.

Inference, in this case, is made difficult by the inherentuncertainties in dening the amount and location of con-tamination (i.e., source terms), subsurface material prop-erties, and historical water levels. The exactitude of sci-entic hypothesis testing is further weakened by the legalrequirements of liability, thus placing this type of inverseproblem outside the scientic venue.

1.2.2 Engineering Models that PredictThe design of human structures may not require an ad-vanced theoretical understanding of the components em-bedded within it. Engineers are most concerned aboutthe attributes of the system that are most likely to con-tribute to performance and failure. Thus, those aspects of a design that affect system performance or failure requirestringent characterization.

Estimating the likelihood of failure, as a function of cost and performance, are an important aspect of engi-neering analysis. An engineer may devote resources to-ward estimating a parameter, especially when the value of

the parameter is near a threshold that determines failureor success. An engineer may be willing to accept a margiof error so that a minimal likelihood of failure is possiblAlso, the stability of a specic parameter may be considered when designing materials or components. A morvariable result generates concern, even though the valueof the parameter is otherwise far from the performancethreshold.

Like the scientist, the engineer relies on inverse techniques to estimate system behavior, but unlike the sci-entist, the engineer is satised with statistical certainty,rather physical certainty. Knowing that a parameter isbounded may mean more to an engineer than knowing thenature of the expected value. Because the engineer is seeking a system design for optimal operation, the specicationof an absolute constant is less meaningful than knowingthe worst-case possibility. By containing the behavior todesirable states, one can avoid unwanted outcomes.

The ultimate endeavor of any design effort, therefore, ito provide a system that offers predictable outcomes. Theoutcomes must be desirable, and they must allow for control. By evaluating alternative designs, an optimal designcan be selected based upon a performance measure. Theperformance measure can have various forms; expectedvalue (benets), extreme value (worst case), expected risk(consequences), etc. Because each of these measures arincommensurate in that they generally yield different se-lections, some balance of these measures is used. A standard mechanism is to apply benet-risk analyses, whichbalances the benets to some against the costs to many.

Like scientic models, the use of engineering models bthe regulatory community results in real dangers. Specif-ically, model uncertainties may not conform with regula-

tory intent. An engineering design that minimizes costsgiven a set of environmental standards may, or may not,be better than a model that is free to balance costs againststandards.

An engineering model may yield an optimal solutiofor a specic standard, yet ignores a solution that costsinsignicantly more but reduces risks substantially. Ide-ally, tradeoffs between alternative engineering designs areidentied so that the relative merits of the alternative de-signs can be evaluated. Due to public perceptions of riskthe regulated and regulatory community, along with thegeneral public, should be involved in nalizing the optimadesign.

1.2.3 Regulatory Models that ControlTo better understand the regulatory role of modeling, it isimportant to note that models often serve as the interfacebetween data and decision-making. Few individuals arecapable of making a sensible conclusion by examining largvolumes of data.

Models assist in data interpretation by extracting in-formation, and thence understanding, from data. Onceexperiments generate data, rules are applied via models

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to gain insight into what happened where, when, why andhow. The information and knowledge gained from the in-sight is then used to form a decision. Models are used,therefore, to improve our knowledge of the system.

Many regulatorscommonly employ models to help learnabout processes in the environment. In this sense, themodels serve as true simulators; to teach the user how thesystem behaves. Like pilots who use aircraft simulatorsto improve their performance when learning how to ynew aircraft, most modelers understand that the modelthey use is not an exact replication of the real system.With time, many modelers either move on to newer, morepowerful codes that provide additional insight and under-standing, or they bypass the simulation route by learninghow to interpret data directly.

Veteran regulators acknowledge that simulation mod-els are abstractions of reality. The abstraction is basedon simplied representations of the real world. It is alsorecognized that model predictions are imperfect due touncertainties. At a minimum the regulatory model shouldbe sensitive to uncertainties, in that the performance of a system should be tested under a wide range of possibleinputs, states, and material properties.

The complexity of material properties can be readilyquantied if the project is limited in scale, or the magni-tude of the variation is small. Unfortunately, the effects of environmental change cover a wide area and induce sub-stantial perturbations to the environment in many ways.The complexity of the model structure can also be evalu-ated if the system functions deterministically, but is virtu-ally impossible if the physical, hydraulic, chemical, ther-mal or biological responses are poorly known. Also, thecomplexity due to coupling between processes can be eval-

uated if the coupling are weak and limited in number, butare difficult when the system is strongly coupled, or isnonlinear in the effects of coupling.

From a regulatory viewpoint, simulation is straightfor-ward if there are no uncertainties, but uncertainties tendto introduce ambiguity in reaching a decision. A scienticmodel that articially controls or minimizes uncertaintiesto allow identication of relevant processes and parame-ters may not be useful for a decision-maker who wishesto incorporate uncertainty. The engineering design thatincorporates uncertainties using conservative designs doesnot necessarily allow the optimum balance of risk vs. costto be achieved. Each of these models therefore restrict

the regulator in their ability to incorporate uncertaintiesin the decision-making process.As noted above, a good performance measure should

conform with the regulatory intent of preserving and pro-tecting the public health and welfare. As such, regulationsshould protect public health, while minimizing intrusioninto the design process. The ability of a performance mea-sure to evaluate the suitability of a proposed design, or toassist in the approval of a permit, should reect the abilityof the design to meet the regulatory intent.

A regulatory policy that enhances the efficiency of the

performance measure will require the use of regulatormodels. These models should fully account for model anparameter uncertainties, while providing design exibility,and also incorporate societal goals.

1.3 The Modeling Process

1.3.1 The ObjectiveThe modeling process begins with trying to articulate themodeling objective. Is it a design objective that minimizescost while conforming to regulatory requirements? Is it ascientic objective to understand how a system works? Iit a regulatory objective whose purpose is to establish alimit that is protective of human health and the environ-ment? Ideally, a model should be as robust as possibl(i.e., suitable for use at many locations). Also, a modethat minimizes uncertainty is more useful than one whichgives highly uncertain predictions.

The greatest challenge to modeling results from the

confusion that commonly arises when scientists, engineersand regulators discuss modeling. This can be expectedgiven their disparate ob jectives. Simulation models varydepending upon their purpose. Scientists employ simulation models to formulate an understanding of complexsystems and to identify previously unknown phenomena orprocesses. Engineers use simulation models to test alternate designs and to predict system behavior. Engineeringmodels are generally coupled with prototype designs totest model assumptions. The regulatory community usessimulation models to evaluate the ability of proposed actions or activities to meet regulatory objectives. In manyregulatory applications, the model is used to extract in-

formation about regulatory performance from data so thatinformed decisions can be made.

Clearly, an important rst step is to clearly dene theobjectives of your modeling effort. Examples include:

Developing a Well Field for Water Supplies. Modelsneeded to meet an engineering objective - to providethe most efficient way to develop ground-water resourceswhile minimizing impacts on local wells and wetlandand complying with existing environmental laws.

Establishing Liability at a Superfund Site. Models arto meet an identication objective - to identify the sourceof ground-water contamination so that responsibility forthe contamination can be assigned. The models must bephysically-based because of the likelihood of litigationover any liability that is assigned.

Establishing Nutrient Loading Limits. Models are neeto meet a regulatory objective - to establish an upperlimit that is protective of the aquatic resource and hu-man health. The models may be physically or statis-tically based. Oftentimes, sufficient information is notavailable to form a physically based model, so that sta-tistical relationships are often used.

Approving a Pesticide License Application. Models needed to meet a regulatory objective - to evaluate whethe

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a chemical can be safely applied without undo harm tohuman health and the environment.

1.3.2 The ApproachThe second step is to dene the model approach. Earlyefforts to develop predictive models relied on scale modelsthat reduced the size of the system to a scale that t in

the laboratory. Entire watersheds have been constructedin miniature - with dams, bridges, and miniature obstruc-tions - for the purpose of evaluating alternative designsfor channel structures. Laboratory column experimentsare a type of scale model still commonly used today toevaluate the effects of contaminant transport through thesubsurface.

Another early effort used resistors and capacitors tomimic the permeability and storage properties of an aquifer.These process analog models were based on the similaritybetween uid ow and the ow of electricity. Heat con-ductivity and capacity are also analogs for water ow, andmany heat ow models are commonly employed as surro-gates for uid ow.

Statistical procedures are useful when the behavior of asystem is highly uncertain, such as in rainfall-runoff mod-els. In these cases, the complexities associated with ex-plicitly modeling the physical processes overwhelm the in-formation available to constrain model parameters.

In other cases, the physical processes are known andappropriate parameters are readily available, so that bothanalytic and numeric models can be used to evaluate andpredict behavior. While analytic models are usually pre-ferred, due to their greater simplicity, site conditions maynot be conducive for direct application. Instead, numericalmodels are used to provide greater exibility in modelingthe unique conditions at the site.

The focus of this book is to provide methods and pro-cedures to assist in understanding the relative strengthsand weaknesses these modeling approaches.

Scale Analogs - Models that employ a smaller (or larger)version of the original system as an experimental tool.Examples included model airplanes, Hele-Shaw models,miniature umes, and laboratory tests.

Process Analogs - Models that replace the ow of water withanother physical or chemical process. Examples includeelectric-analogs, temperature-analogs, and solute-transportanalogs.

Statistical Models - Models based upon information, whichuse observed data to identify relationships and to esti-mate model parameters. Examples include regression,convolution, time series, fourier, wavelet, and percola-tion

Analytical Models - Models that provide analytic solutionsfor specic physically-based systems. The solutions areusually based on governing equations that take the formof algebraic, differential, or integral expressions.

Numerical Models - Models that provide solutions when an-alytic solutions for physically-based systems are not pos-

sible. Examples include iterative, Finite Difference, Fi-nite Element, Analytic Element, and other methods.

1.3.3 The StructureOnce a general class of model has been selected, a largnumber of issues still need to be resolved. What processewill be considered? What are the important relationships?

What state variables, inputs, outputs, and parameters canbe measured? How stable are these measurements? Whatrange of application is required?Physical vs. Abstract - A physical model is a scaled repli

of the modeled system, such as a model airplane, meantto perform identically to the original. An abstract modelis a mathematical representation of the modeled system,meant to yield predictions that can be applied to theoriginal.

Natural vs. Devised - A natural model describes the physical environment, while a devised model is one, such alaw, that creates articial rules for human conduct.

Open vs. Closed - A closed system has no input or outputsand is entirely self-contained. An open system interactswith its external environment. Planet Earth is a closedhydrologic system, but is open with respect to energy.

Steady vs. Dynamic - A steady system is one in which theis no temporal variation in inputs, outputs, or state vari-ables, while a dynamic system is one which changes ovetime.

Stable vs. Unstable - A stable system is one which can bcontrolled, while an unstable system is one which is dif-cult or impossible to control. One property of a stablesystem may be that, in the absence of inputs, it returnsto a constant condition, while an unstable system growsor shrinks without stabilizing.

Discrete vs. Continuous - A discrete system has inputs, oputs, or state variables that take on integer values, suchas the sex of a person, while a continuous system is onein which components can vary over an innite number ofconditions, such as a dimmer switch vs. an on-off switch

Deterministic vs. Probabilistic - A deterministic systeis one in which all inputs, outputs, and attributes canbe described with great certainty. A probabilistic modelexplicitly incorporates uncertainties in the model usingassumed distributions and statistical parameters.

Lumped vs. Distributed - Lumped parameters are used wan attribute of a system can be described using a singleparameter or state variable. In some situations, however,a single attribute can not fully describe the variabilitywithin the system, and the entity or relationship mustbe divided into multiple, or distributed, components -each with their own attribute.

Causal vs. Noncausal - A causal system implies that an action at one point or time causes a response at anotherpoint or time. A noncausal system is one in which anyresponse between two points or times is the result of arandom coincidence.

Instantaneous vs. Lagged Response - An instantaneoutem is one in which an input causes an immediate output,

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and there is no memory. A lagged response occurs whenthere is a delay between an input and a response.

Time Variant vs. Time Invariant - A time-variant modelhas parameters that change with time, while a time-invariant model has stable parameters. When faced witha problem where a parameter is time-variant, the modelcan be altered by specifying changing the parameter toa state variable.

Linear vs. Nonlinear - A linear system is one in which themagnitude of the response is a constant multiple of themagnitude of the input. That is, a doubling of the inputcauses a doubling of the output. A nonlinear systemresults when the ratio of the response to the stimulus isnot constant, but varies as a function of the magnitudeof the input.

The denition of the model structure is called the con-ceptual model . For some applications, this conceptual modelmay be uncertain, and alternative conceptual models maybe required. For these cases, additional information mayneed to be collected to evaluate which of the alternativeconceptual models is more appropriate.

1.3.4 The Evaluation StrategyNo phrase seems to induce greater discussion than modelvalidation. Accurate predictions hinge on a correct model[10] [54]. While some argue that models can never bevalidated, due to the inability to accept a hypothesis, oth-ers argue that model validation depends upon the regula-tory context. Efforts to evaluate suitability of model per-formance focus on the key aspects of model verication,which are used to evaluate the accuracy of coded state-ments (software control), and model evaluation, which fo-

cuses on the ability of the theory to replicate observedbehavior. The issue of model validation comprises thecore of performance assessment; risk predictions dependon accurate and precise measurements of anticipated out-comes. The performance assessments are then used fordecision making, establishing regulatory policies, and forevaluating regulatory compliance.

The greatest challenge in modeling is the proper math-ematical specication of the conceptual model. The iden-tication and quantication of relevant processes can beperformed using laboratory and limited-scale eld exper-iments. In general, the mathematical equations that re-sult are greatly simplied from the real world. The quan-

tity, quality and types of data must be specied, both intime and space. Once data are obtained, model resultsare compared with experimental results to evaluate modelassumptions. The conceptual model is iteratively reviseduntil a specied performance measure is achieved. Thesesteps do not guarantee success. The calibration and evalu-ation phases are both limited in areal and temporal extent.Extrapolation beyond the range of experimental testing issubject to great uncertainties, that are generally neglectedwhen running performance assessment simulations.

The performance assessment simulations are used todevelop alternative designs for a wide array of conditionsbeginning with limited-scale testing under relatively con-trolled conditions, to extended-scale testing under openconditions. These design comparisons can be used to provide model results that limit the prototype testing to asmaller set of alternative system designs. The alternativeprototype designs are then evaluated using a variety ofperformance measures, recognizing the uncertainties in-herent in open testing. Finally, the sensitivity of modepredictions to the magnitude of uncertainties is evaluated.

Condence in the validity or appropriateness of an environmental model is enhanced if the model accuratelyand concisely represents the relevant physical, biologicalchemical, hydrologic, geologic, and thermal processes inherent in the system being modeled. Failure to accountin a meaningful way in any of these components macompromise the robustness of model predictions. Estima-tion of model single-phase and coupling parameters mustbe performed under a wide range of ambient conditionsespecially those appropriate to the target environment.And nally, extended prototype testing at multiple spa-tial scales, time horizons, and thermal loadings is requiredto identify unanticipated events. Reliance on natural ana-logues that mimic the long-term effects of environmentadisturbance, such as volcanic and meteorite-impact eventsare a possible mechanism for extending prototype testing.

Validation of model performance lies in the ability toquantify model uncertainties. While some model uncertainties can be quantied, many uncertainties related tomodel structure remain unquantiable. Regulatory poli-cies that rely on model predictions may not provide greatercondence than other approaches. Alternative regulatory

strategies include an increased reliance on prototype test-ing, improved linkages between hydrologic modeling capabilities and standards, and decoupling models developed for scientic purposes from the regulatory processIn some cases, regulatory policies and simulation modecapabilities may be incompatible, leading to the improperuse of simulation models in the regulatory environment.

Engineers routinely establish tradeoffs between modeperformance and standards by incorporating margins ofsafety to account for uncertainties [53]. The degree ocondence reduces the need for margins of safety, and allows for economies in the design and operation of a structure. For a new design, large margins of safety are rec

ommended, which are slowly relaxed as the understand-ing of the system increases. A regulatory policy that acknowledges uncertainties, or develops methods for explicitly linking uncertainties to standards, is one means foraddresses incomplete understanding. Uncertainty magni-tudes are especially large early in the design state, and aresolidied as the prototype testing reduces unknowns.

A regulatory policy that focuses exclusively on risminimization may fail to account for identication of un-known uncertainties. A process of uncertainty characteri-zation and minimization can be monitored using the var-

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ious performance measures addressed above. The processof testing need not be static, however. As new processes orbehaviors are observed, increased surveillance and testingmay be required. To dene, a priori, the magnitude of un-certainty may not be possible. In fact, the magnitude willcertainly be underestimated. Minimization and control of uncertainties leads to reliability of operation, the primaryconcern of the regulator.

1.3.5 The LimitationsA primary inadequacy of environmental models stems fromtheir inability to provide long-term predictions with suf-cient accuracy to meet regulatory requirements [49] [82][84]. Assuring the long-term continuation of sustainablehuman and ecologic conditions for millions of years is diffi-cult, given unforeseeable climate changes, human and an-imal interactions, geologic and astronomic instability, andother, as yet unknown, modications of the global environ-ment. For a regulatory requirement of predictability foronly ten thousand years, many scenarios of climate changeor volcanic activity could adversely affect the integrityof global systems. Incorporating the potential stresses of changes in the external environment is rendered more diffi-cult when observed behaviors are limited by an incompleteunderstanding of potential future states [61].

An additional problem arises when more than a singleinterpretive model can be used to explain observed exper-imental data [37] [23] [80]. In the case of any experimentin which data are incomplete, alternative scenarios canbe devised that may account for the observed distributionof state variables or measured uxes. Thus, alternativehypotheses may not be distinguishable due to incomplete

control of the experiment.Calibration data sets and model structure substan-tively affect model prediction accuracy. For data sets withhigh multicollinearity, the parameter covariances becomelarge, resulting in poor forecasts. Also, extrapolative mod-els in which parameter values that lie far from the meanresults in larger forecasting errors. And nally, the modelshows that the objective of reducing model calibration er-rors can result in a poorly structured parameter covariancematrix, offsetting any reduction in forecast error.

A further confounding factor lies in the situation wherea regulatory policy may not be amenable to quantitativeanalysis [26]. Without being specic, it may be safe to say

that a quantitative denition of safe , is not scienticallypossible. A policy which requires safety requires the speci-cation of a quantiable measure. A quantiable measuremay be the specication of a risk envelope, which is theproduct of the probability of an undesirable state with thelikely consequence of the state, for all possible states [17].The total risk is the sum of risks of all possible adverseevents.

The total risk can be limited to specic subsets, e.g.,the risk from carbon accumulation in the atmosphere. Riskcalculations incorporate all outcomes, with the obvious

bias due to the inability to specify the universe of outcomes. Also, the risk computation should account for risktransfers (e.g., between present and future generations),perceptions of risk (e.g., brief, high exposures may be moracceptable than chronic, low exposures given the same to-tal consequence), and public voice (e.g., poor, minoritypopulations may have less access to the political procesthan established political groups).

While mathematically simple, the use of risk as a measure of safety suffers from the inability to calculate eithethe likely probability of a single hazard, or even the resulting consequence for that hazard [21] [22]. The advantage of risk assessments lies in their ability to incorporateknown uncertainties by coupling them with the magnitudeof the consequence, yet they are biased because they faito identify unknown processes or scenarios, or to quantify unknown uncertainties [60]. The risk measure is relative measure, requiring the balancing of risks betweenalternative decisions [11]. Also, community perceptionof acceptable risks commonly differ from calculated risk[81]. And nally, risk is transferable, allowing individuato benet from the increased exposure of others.

The dilemma arises when a regulatory policy may noadequately incorporate model capabilities and uncertain-ties [60]. A model which supercially treats complex sytems by using expected inputs, states, or material prop-erties, may yield precise estimates of expected outcomesYet this model will undoubtedly fail to accurately repre-sent the wide range of possible outcomes by ignoring thless likely, but still possible, alternatives.

1.4 ProblemsSelect a journal article of your choice that uses a modein the analysis of eld data. For the model you selecidentify the:

1. Modeling objectives (e.g., scientic, engineering, reulatory)

2. Processes described by the model

3. Input information used to construct the model

4. Parameters and state variables within the model

5. Outputs predicted by the model

6. Measures of model performance

7. Construct a visual diagram of the model showinthe entities, relationships, parameters, state vari-ables, inputs, outputs, feedback loops, and calibra-tion checks.

8. Describe alternative modeling methods, and discushow they might be useful for meeting the modelinobjective.

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Chapter 2

Analog Models

An analog model is an articial physical representa-tion of the system that is to be studied. For example,a process analog substitutes a different process, such aselectricity or heat, to represent uid ow. Another classof analog models are scale analogs which use a miniature(or perhaps an expanded) version of the system. Exam-ples of scale analogs include miniature dams and rivers,

or miniature aquifers. These models can be constructedin a laboratory so that experiments can be conducted toevaluate alternative designs.

2.1 ReviewBefore we launch this baby, lets do a bit of review. Imhoping that youve had (at least some of) this before. If not, please give it a try. . .

2.1.1 UnitsThe metric system is the internationally recognized systemof units. All countries in the world (except for Liberia,Burma/Myanmar, and the United States) use the metricsystem as their official way to measure things. MixingEnglish with other units can result in engineering failure,such as the failure of the $125 million Mars Climate Or-biter spacecraft in 1999. In this case, two teams of scien-tists used two different systems of units, which were neverreconciled.

Table 2.1: Metric System Units

Mass kilogram kg

Length meter mVolume liter LTime second sEnergy joule J (kg m 2 )/s 2

Power watt W (kg m 2 )/s 3 J/sForce newton N (kg m)/s 2 J/mPressure pascal P a kg/ (m s 2 ) J/m 3

Table 2.2: Metric System Prexes

d deci 10 1 D deca 10 1

c centi 10 2 h hecto 10 2

m milli 10 3 k kilo 103

micro 10 6 M mega 10 6

n nano 10 9 G giga 109

p pico 10 12

T tera 1012

a femto 10 15 P peta 10 15

2.1.2 Exponents and LogarithmsThe exponent is used to indicate the repeated multiplica-tion of a number:

x3 = x x x (2A negative exponent indicates the reciprocal of the num-ber:

x 3

=1x3 (2

Non-integer exponents are also possible. For example, theexponent one-half indicates the square root of a number:

x1 / 2 = x (2

A general rule for exponents is:

xa / b = xx a (2

There are two standard bases for exponents. One is tbase-10 system, such that any number can be representedusing:

x = 10 b (2

For example, b = 2 represents the number 10 2 = 100.second base is the natural system, represented using e

x = ea (2

where e 2.71828.The logarithm of a number is equal to the exponent:log10 x = log 10 (10b) = b (2

8

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CHAPTER 2. ANALOG MODELS

ln(x) = log e x = log e (ea ) = a (2.8)

There are several basic rules of exponentials and loga-rithms:

x y = 10 b 10c = 10 b+ c (2.9)xy

=10b

10c= 10 b c (2.10)

These relationships also hold for the natural system:x y = eb ec = eb+ c (2.11)xy

=eb

ec= eb c (2.12)

Logarithms also follow similar rules:

log(b c) = log( b) + log( c) (2.13)log(bc) =

log(b)log(c)

(2.14)

for both the base-10 and natural systems. Another loga-rithmic relationship is:

log(ax b) = log( a) + blog(x) (2.15)

2.2 Process AnalogsThe rules that govern uid ow are remarkably similar tothose that govern heat, electrical, and solute ow. All aregoverned by conservation and ux equations, with similarlaws and parameters.

The general conservation equation requires that anyaccumulation or loss in mass, heat, or current be offsetby a corresponding increase or decrease in storage. The

general ux law states that the rate of movement of mass,heat, or current is related to the product of a parameterwith the gradient of a state variable (head, temperature,voltage, or concentration).

The ratio of the ux to the conservation parametersis termed the diffusivity, and always has units of lengthsquared per unit time ( m2 /s ). Thus, the ux parameterunits must be paired with the conservation variable pa-rameter so that concomitant units are used.

2.2.1 Fluid FlowThe steady and unsteady ow equations for water in the

subsurface are: q = K h (2.16)

q = S sht

(2.17)

D2h =

ht

(2.18)

where q is the ux vector, K is the hydraulic conductivityparameter, h is the gradient in the hydraulic head, D =K/S s is the hydraulic diffusivity, is the volumetric watercontent, and S s = d/dH is the specic storage coefficient.

2.2.2 ElectricityThe corresponding equations for electricity are:

i = V

R(2.

i = C V t

(2.

D2V = V

t(2.

where i is the electrical ux, or amperage, vector, R is electrical resistivity parameter, V is the gradient in thelectrical potential, or voltage, D = 1 /RC is the electridiffusivity, and C is the capacitance.

In this case, the ux of electrons replaces the ux of water, the reciprocal of the resistivity replaces the hydraulicconductivity, the voltage replaces the hydraulic head, andthe capacitance replaces the specic storage coefficient.

As an example, one can construct a ground-water owmodel using a network of resistors. Resistors are inverselrelated to the hydraulic conductivity, so a low permeabil-ity aquifer would be constructed using resistors with highresistances. A voltage can be applied at any point withinthe network to simulate the addition or removal of water.A ground is applied at either a point, or along a line, oat multiple points to simulate the return of the water tothe system.

Another means for studying uid ow is to use electrcally conductive paper. The ow domain is drawn on thepaper and then the non-conducting part of the domain iscut off. An electrode is placed at one or more locationon the paper, while a second electrode is placed elsewhereThe voltage on the paper is used to simulate the resultingow eld. This analog is limited to uniform conditionunless papers with different electrical conductivities areused.

2.2.3 HeatThe ux equation of heat is:

j = K T (2.

j = C pT t

(2.

D

2

T =T t (2.

where j is the heat ux vector, K is the thermal condutivity parameter, T is the gradient in the temperatureD = K/C p is the thermal diffusivity, and C p is the hecapacity.

A similar analog using heat can also be constructed. Aheat-conducting surface is constructed in the shape of thesystem to be studied. A constant heat source is placedat one or more locations, and the resulting temperatureis measured. Materials with variations in conductivity

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Table 2.3: Physical analogs between processes

Process Conservation Variable (parameter) Flux Law (parameter)Water Mass, Volume (storage coefficient) Darcys (hydraulic conductivity)Heat Calories, Joules (heat capacity) Fouriers (thermal conductivity)Electricity Amps, Electrons (capacitance) Ohms (resistance)Solutes Mass, Moles Ficks

can be used to reproduce variations in water transmittingproperties.

If steady ow is desired, then only the conductance,or ux, component is needed. However, the storage mustbe included when modeling dynamic conditions, i.e., whenthere are temporal changes in the system. For the electri-cal analogs, capacitors must be placed alongside the resis-tors. For inertial components, inductors (or coils) must beused. For thermal analogs, the heat capacity of the sur-face should match the storage coefficient of the hydraulicsystem.

2.2.4 SolutesThe ux equation of solutes due to a concentration gradi-ent is:

J = DC (2.25)

J = C t

(2.26)

D2C =

C t

(2.27)

where J is the solute ux vector, D is the solute diffusivity,

C is the gradient in the solute concentration.

2.3 Scale AnalogsA scale model is a common tool for studying large sys-tems. Like a map, the original system is reduced in size sothat it ts within a laboratory or other reasonably sizedstructure. For a system with an original length of Lo , themodeled scale, Lm , is:

Lm = L o (2.28)

where is the model scale. Note that < 1, meaning

that a smaller scale involves a larger reduction - a 1:1000model is a smaller scale than a 1:100 model.

For example, a river with a width of 1 km, a depthof 10 m, and a length of 1000 km can be scaled downto a manageable size using a = 1:10,000 scale, so thatthe model river is 10 cm wide, 1 mm deep, and 100 mlong. While this is still the length of a football eld, modelsimulations might be performed within an indoor arena of sufficient size.

The features of the owing river would also have to bereproduced. If the original bed material within the river is

a sand with a diameter of 0.3 mm, then the bed materialof the scale model would have to have a diameter of 30 nmor 300 A, which is a ne clay particle. Also, a velocity 3 m/s in a natural setting would correspond to a velocityof 0.3 mm/s, or about 1 m/hr.

The viscosity of the uid would also have to change because the shear stress of uids is a function of the lengthscale used. As the depth of water, or any uid, gets thin-ner, then the effects of viscosity increase. The Reynolnumber, R, describes the ratio of inertial to viscous force

R =vL

=vL (2.

where v is the uid velocity, L is the length scale, is uid density, is the dynamic viscosity, and = /the kinematic viscosity. For water at 4 C, = 1 kg/ = 1 g/m/s, and = 10 6 kg/m 2 /s. We know thwater ows is in a laminar manner if R < 10, and owsa turbulent manner when R > 100.

The viscosity of water is a measure of its ability to ow- uids with high viscosity ow more slowly than uidwith lower viscosity. A Newtonian uid is one which obethe relationship that:

= vxy (2.where (Pa s), is the dynamic viscosity of water, and is the shear stress within the uid, and vx /y is the rof change in the y-direction of the uid velocity in thedirection. The shear stress is a measure of the forces othe uid, and turbulent ow occurs if the forces exceed threshold value.

Thus, in order to maintain the proper ow conditions,laminar or turbulent, then the viscosity of the uid mustchange at a rate equal to the product of the velocity andlength scales.

2.3.1 Dimensional AnalysisWhenever you use an equation, pay particular attentionto the units - they must balance each other on each sideof the equation. For example:

v (m/s ) =dx (m)dt (s)

(2.

where v is the velocity, in units of meters per second,is the change in distance, in units of meters, and dt

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the change in time, in units of seconds. Note how theunits on the left-hand side perfectly balance those on theright. One can check the accuracy of the equation, or yourcalculations, by examining whether the units are correct.

When modeling a new system, one can develop a func-tional relationship just based on the units. For example, if we know that the velocity is based on distance and time,there is only one combination that yields a balanced set of units. So, rather than having to remember the equation,one only needs to remember the units.

Taking this one step further, the Buckingham- theo-rem states that one can establish a set of possible phys-ical relationships if the number of independent variablesare known. For example, if we know that the velocity of falling water is affected by distance and time, as well asgravitational acceleration, then, by dimensional analysis,we arrive at the following possible combinations of vari-ables:

v (m/s ) = f {g (m/s 2), dt (s)} (2.32)v (m/s ) = f {g (m/s 2), dx (m)} (2.33)g (m/s 2) = f {v (m/s ), dt (s)} (2.34)

where g is the gravitational acceleration in units of m/s 2 .Using a bit of math, we can show that these are equivalentto:

v = g dt (2.35)

v2 = 2 g dx (2.36)

g =dvdx

(2.37)

2.3.2 Example: Fractal ScalingSome systems display self-similar behavior, meaning thatit looks the same regardless of the scale of measurement.Examples include:

Shorelines: Length of land-ocean boundary increases asruler length decreases

River densities: Number and length of waterways in-creases as map scale becomes ner

Soil Physics: Scaling of particles shifts soil-moisture char-acteristic curves to common shape

Geophysical Measurements: Bulk resistivity is not justproduct of resistivity and porosity

Fractured Media: Fracture density changes as the scaleof measurement changes

We will examine tortuosity which is a commonly ob-served property of environmental systems. The denitionof tortuosity used here (recognizing that there are severaldifferent denitions in the literature) is:

Tortuosity, : The ratio of the path length s, to truler (straight-line) length, x:

= s x

(2.

The path length along a streamline between two points isgenerally unknown, however. In general, the parameterswhich may be measured using experimental tests are:

Total Head Difference, h: The change in head btween two points:

h = hb ha (2.Hydraulic Gradient, i: The change in head, h, unit distance, x:

i = h x

(2.

Hydraulic Conductivity, K : The ability of a geolomedium to transmit water, calculated using:

K =q i

(2.

Travel Time, t t : The time required for a particle owater to move from one point to another, equal to thedistance, x, divided by the uid velocity, v:

t t = xv

(2.

Fluid Velocity, v: The rate at which water moves throu

the aquifer, equal to the ux, q , divided by the effectiporosity, n :

v =q n

(2.

Effective Porosity, n : The volume of voids, V V , punit volume of geologic medium, V T :

n =V V V T

(2.

The hydraulic gradient can be calculated at two scales,as a straightline gradient along x, or along the cur

associated with the true path described by s

ix = h x

and i s = h s

(2.

It is easy to see that

ix = h x

= h s

s x

= is (2.

or

is =ix

(2.

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The hydraulic conductivity at the experimental scalecan also be related to the value at the streamline scale:

K x =q ix

=q

i s=

K s

(2.48)

K s =q is

= q ix

= K x (2.49)

For extrapolating tests from one scale, say at a eld orlaboratory scale of size, x1 , to a different scale, x2 , thefollowing relationship can be used:

K 1K 2

=i2i1

= 2 1

(2.50)

Similar to the gradient and the hydraulic conductivity,the calculated travel time may also be affected by the scaleof measurement. The travel time is dened here as theintegral of the inverse velocity along a one-dimensionalstreamline:

t t =

s b

s av 1ds (2.51)

where t t is the uid travel time, v is the uid velocity alongstreamline, s is the distance along streamline, and sa andsb are the particle starting and ending positions, respec-tively. The uid velocity is the volumetric ow rate perunit area (i.e., the darcian ux) divided by the porosity,or:

v =q n

=Q

nA=

K s isn

(2.52)

where q = K s is is the uid ux, n is the porosity, Q isthe total ow, A is the cross-sectional area, K s is the localhydraulic conductivity, and is is the local hydraulic gradi-ent. By assuming constant velocity along the streamline,we obtain:

t t = sv

=n s

q =

n sKi

= n s2

K s h(2.53)

where i = h/ s, and where s = sb sa is the distancealong the streamline, and h = hb ha is the head dropalong the streamline. Switching to ruler lengths, we have:t t =

n 2 x2

K x h=

n x2

K x h(2.54)

If the concept of fractal scaling is employed, then arelationship between the tortuosity at one scale can berelated to the tortuosity at a different scale:

1 = x 1o (2.55)

and

2 = x 2o (2.56)

where is a fractal scaling parameter and o is a dimen-sionless fractal tortuosity parameter.

Summary. This example demonstrates the effect of tortuosity and the scale of measurement on the hydraulicconductivity, hydraulic gradient, and travel time. A crit-ical parameter in this analysis is the tortuosity, a resultof the geometry of the ow regime. Additional researcis required related to the effects of spatial variability ontortuosity, and the relationship between scale and the es-timated tortuosity.

The spatial variability of tortuosity along the stream-line may or may not have signicant effects on the estimated travel time. It may be possible that local uctu-ations in this parameter may not signicantly affect re-gional travel times. Also, if the tortuosity is scale invariant, then laboratory and eld parameter estimates can bedirectly applied to regional-scale models without the needfor incorporating scale effects.

2.4 Problems1. Exponents and Logarithms

(a) What is the range of ex if the range of x

< x < ?(b) What is the range of x if the range of log( x

< log(x) < ?2. Process Analogs Using the provided electrical co

ducting paper:

(a) Construct a uid ow model using a simple gometry, like a stream with pools and narrowsor an aquifer with varying shape.

(b) Attach a low-voltage electrical source to the pa

per - such as a 9-V battery - so that the negative terminal is connected to one end of thpaper and the positive terminal is attached tothe opposite end.

(c) Map lines of constant voltage on the paper.(d) Discuss how one could construct a heat analo

model using various metals.(e) Discuss the advantages and disadvantages of as-

tronauts using swimming pools to simulate theffects of zero gravity.

3. Scale Analogs Consider a miniature aquifer mea

suring 1-m long, 10-cm wide, and 1-cm thick, which food-coloring is added to show contaminantransport:

(a) Discuss the effects on model results if water ana sand medium is used.

(b) Discuss how these effects might be reduced oeliminated by using alternate materials.

4. Tortuosity Consider an aquifer with a tortuosity o2.

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(a) Discuss the impacts of a travel time predictionif the hydraulic conductivity was estimated us-ing ow between two wells.

(b) Explain how the tortuosity (sometimes calledthe formation factor) might be estimated usingelectrical methods.

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Chapter 3

Network Models

We are often faced with the challenge of organizingdata. We start by grouping like quantities together; putall lakes in one group, all rivers in another, and all ground-water in a third. These groupings could be considered tobe entities. We then try to nd relationships betweenthese entities; surface and ground water interact, riversow into and out of reservoirs.

This organization is called a network, in that each en-tity is qualitatively different from the others, and there areunique relationships between each entity. As the modeldevelops, one may begin to distinguish between specicelements within each entity; small lakes behave differentlythan large lakes, surcial aquifers behave differently fromconned aquifers. We then create new entities and rela-tionships between these entities.

This chapter focuses on network models, and how theycan be used to represent environmental systems.

3.1 Percolation Models

Percolation models use nodes to describe locations thathave discrete values, such as occupied or vacant, on or off.They also uses bonds that connect the nodes, which canbe open or closed, and can have distinct attributes. In thissection, we describe several types of percolation models,such as:

Water percolating through a soil column Contaminants percolating a fractured rock network Wells connected through a layered aquifer systemThese models are a function of scale and dimension:

As distance decreases between boundaries, connectednetwork probability changes For 1-d ow, only one bond needs to be broken toshut down the network, fewer bonds are needed to

complete a network in higher dimension

Fracture length strongly affects network connectiv-ity, long fractures form a backbone

3.1.1 Frequency DistributionsThere are two classes of distributions - discrete and contin-uous. A discrete distribution is used when only countablenumber of outcomes are possible, such as the tosses of coin, or the number of students in a class. Continuous dis-tributions are used for describing outcomes can have anyfractional value, such as the monthly or annual rainfall

depth.

Discrete Distributions

Examples of discrete distributions that can be used tomodel these problems include the uniform, binomial, andgeometric distributions.

The discrete uniform distribution, as the name implieshas a constant probability for all outcomes between x1 ax2 . The corresponding probability for each outcome, x,

P (x) =1n

(3

where n are the number of outcomes.Another discrete distribution is the binomial , wh

we used earlier to predict the probability of the number ofheads when a coin is repeatedly tossed n times:

P (x) =nx

px (1 p)n x (3where

nx

=n!

x! (n x)!(3

is the combinatorial operator that accounts for the numberof opportunities for getting the same outcome.

Another common discrete distribution is the geometrwhich is used to calculate the probability of a failure afterx attempts:

P (x) = p (1 p)x (3where p is the likelihood of failure on each attempt.

14

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CHAPTER 3. NETWORK MODELS

Continuous Distributions

Examples of continuous distributions include the uniform,normal, exponential, gamma, and Gumbel extreme value.One can increase the number of distributions by takingthe logarithm of the random variable, resulting in distri-butions such as the log-normal, log-gamma, etc.

The continuous uniform distribution takes the form:

P (x) = 1ba

(3.5)

when a x b. This is a two-parameter distribution,meaning that two numbers, a and b, are sufficient to de-scribe the distribution.

The normal distribution is another two-parameter dis-tribution, requiring the mean and standard deviation, xand sx , respectively. The distribution is unbounded be-low and above, dened using:

P (x) =1

2 x ez 2 / 2 (3.6)

and where

z =x x

x(3.7)

is the standard normal variable with mean zero, z = 0,and unit standard deviation, z = 1.

The log-normal distribution is a two-parameter distri-bution obtained by setting y = ln( x) so that x = ey islog-normally distributed. The log-normal distribution isunbounded above, but bounded by zero on the left.

P (x) =1

2 ln x ez 2 / 2 (3.8)

z =ln x

ln x

ln x (3.9)The exponential distribution has just one parameter,

x , and takes the form:

P (x) =1x

ex/ x (3.10)

Note that this distribution is bounded by 0+ on the left,and on the right.The gamma distribution, is a two-parameter distribu-tion that represents the sum of exponentials:

P (x) = x 1 e x/

()(3.11)

where = ( x/ x)2 , = x2/ x, and ( ) = ( 1)! is thegamma function. The gamma function is the same as the 2 distribution when = / 2 and = 2.

The log-gamma function is also known as the Log-Pierson Type III distribution which is widely used forextreme value problems. Another common extreme-valuedistribution is the Gumbel :

P (x) =1x

ez e ez

(3.12)

where z = ( x x)/ x is the standard normal variable.

3.1.2 Markov ChainsA Markov Chain model has discrete (or nite) states, suchas rainy and dry days. We can use the concept of probability to say that the sum of all states has to equal oneThat is, if p is the probability that a day is rainy, andthe probability that it is dry, then p + q = 1.

Once the states are dened, we need to dene a st

transition matrix , which is simply the function that indicates the probability of moving from one state to another.

Table 3.1: State Transition Matrix

TodayRainy Dry

Tomorrow Rainy P ww P dwTomorrow Dry P wd P dd

where P ww is the probability of a rainy day followed brainy day, etc.

It is easy now to estimate the probability of differentypes of events:

Two dry days followed by a rainy day:P (d,d,w) = P dd P dw (3.

n dry days followed by a rainy day:P (d,d,...,d,w ) = ( P dd )n 1 P dw (3.

n wet days followed by a dry day:P (w,w,.. . ,w,d ) = ( P ww )n 1

P wd (3.

Note also, the following properties:

P (w) = P (w|w)P (w) + P (w|d)P (d) P (w) = P (w|d)/ [1P (w|w) + P (w|d)]

which are forms of Bayes Theorem . Also note that:

The transition probability decreases for increasingprecipitation depths (0.01, 0.10, 1.00) because thelikelihood of two days in a row with heavy rain much less likely than days with light rain.

P (w|w) = f (z, t ), is relatively stable geographicaland seasonally. Note that P (w|w) is the storm psistence probability, and P (d|w) = 1 P (w|w)the storm departure probability. If the probabilitis constant, this means that storm persistence is rel-atively uniform in space and time.

P (d|d) = f (z, t ), varies geographically and seasoally. Again, this is the drought persistence probbility and P (w|d) = 1 P (d|d) is the storm arrivprobability. It is clear that desert climates have a

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much lower chance of storms arriving than in a wet-ter climate. Also, areas with strong seasonal varia-tion in weather, such as Monsoonal and the Mediter-ranean climates, will have different rates of stormarrivals over the course of a year.

Problem of persistence: extremely dry and wet weathermay reinforce themselves. The solution is to add ad-ditional states that represent drought and ood con-ditions. This results in additional transition proba-bilities.

3.1.3 Discrete Event ModelingSome things happen as distinct events - heads vs. tails,on vs. off, boy vs. girl, day vs. night. While otherthings happen along a gradient - shades of gray, moisturecontent, income. Models that describe the discrete natureof things are often easier to understand and build thanones that handle continuously changing variables.

The roadway over a river will be ooded or dry.

The water level in a well will drop below the bottomof the well.3.1.4 Example: Extreme ValuesThe objective is to t observed frequencies to a probabilitymodel. Extreme value distributions are commonly appliedto maximum or minimum annual discharge. In this casethe largest or smallest event in each year is identied. Al-ternatively, a data set could be constructed using all dailyobservations above, or below, a threshold discharge. Thisalternative technique is called a partial duration series.

3.2 Linear Systems ModelsA system is composed of inputs, outputs, state variables,and parameters. One can draw a box with an input ar-row, an output arrow, parameter boxes, and state-variabledials.

A linear system is one in which the value of the outputvaries linearly with the value of the input. For example:

y = f (x) = 2 x (3.16)

is a linear system, because doubling the input, x = 2 x,

results in a doubling of the output y

= 2 x

). Yet,y = f (x, b) = 2 x + b (3.17)

is not a linear system because 2 y= f (2x, b):

y= f (x, b) = 2 x+ b = 2( x+ b) = 2 x+ 2 b (3.18)

One can convert the non-linear system into a linear oneby removing the constant, b, so that y = y b:

y = y b = f (x) = 2 x (3.19)which is now a linear system in y .

Table 3.2: Method for Determining Return Periods

1. For discharge, and other heteroscedastic variablestransform the observations, x, using the logarith(base-10), y = log x to make them homoscedastic.

2. Rank each observation from largest to smallest, m1, 2,

, n , where n is the number of events

3. Calculate the exceedence probability of each observation, P i , using one of the following ranking statitics:

Method Exceedence Probability

Weibull m / (n + 1)

Hazen ( m 0.5) / nCunnane ( m 0.4) / (n + 0 .2)

4. Plot the exceedence probability, P i , against the o

servations, x i

5. Calculate the recurrence interval, T i = 1 /P i .

6. Plot the recurrence interval against the observations.

7. Specify a probability model (log-normal, log-gammaetc.)

8. Estimate the sample moments and use these to esti-mate the model parameters

9. Show the calculated probabilities on the above plots

3.2.1 System InputsWhile many systems have random inputs, it is often help-ful to examine the response to specic types of inputssuch as:

Dirac Delta

(to) = t = to0 t = to continuous time (3.

Kronecker Delta

(to) =1 t = to0 t = to

discrete time (3.

Heaviside (step)

H (to) =

(to) dt

= 0 t < t o1 t to(3.

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Pulse

P (ta , tb) = H (ta ) H (tb)

=0 t < t a1 ta t < t b0 tb t

(3.23)

Ramp

R(ta , tb) =1

tb ta t b

t aP (ta , tb) dt

=0 t < t a

t t at b t a ta t < t b

1 tb t(3.24)

3.2.2 ConvolutionConvolution is a type of mathematical operator, denedusing y = hx whereis the convolution operator and h is

the unit response function. For the case where the inputis a Delta function (either continuous or discrete), theny = h. This means that a spike input causes an outputequal to the unit response function.

Applications include:

Rainfall-Runoff Soil Mo

8740 notes

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