Approximate Analytical Solutions to the Groundwater Flow Problem CWR 6536 Stochastic Subsurface Hydrology.

Slides:

Advertisements

Similar presentations

Autocorrelation Functions and ARIMA Modelling

Advertisements

The Spectral Representation of Stationary Time Series.

Approximate Analytical/Numerical Solutions to the Groundwater Transport Problem CWR 6536 Stochastic Subsurface Hydrology.

ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Response to a Sinusoidal Input Frequency Analysis of an RC Circuit.

Outline input analysis input analyzer of ARENA parameter estimation

What is the language of single cells? What are the elementary symbols of the code? Most typically, we think about the response as a firing rate, r(t),

Total Recall Math, Part 2 Ordinary diff. equations First order ODE, one boundary/initial condition: Second order ODE.

Stochastic processes Lecture 8 Ergodicty.

Statistical analysis and modeling of neural data Lecture 6 Bijan Pesaran 24 Sept, 2007.

Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first.

Review of Probability and Random Processes

بيانات الباحث الإ سم : عبدالله حسين زكي الدرجات العلمية : 1- بكالوريوس هندسة الطيران – جامعة القاهرة - بتقدير جيد جداً مع مرتبة الشرف 2- ماجستير في الرياضيات.

Part 2.  Review…  Solve the following system by elimination:  x + 2y = 1 5x – 4y = -23  (2)x + (2)2y = 2(1)  2x + 4y = 2 5x – 4y = -23  7x = -21.

Review of Probability.

Algebra Distributive Property. Vocabulary Equivalent expressions: two expressions that have the same output value for every input value Distributive.

H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure.

Random Process The concept of random variable was defined previously as mapping from the Sample Space S to the real line as shown below.

Graphing Systems of Equations Graph of a System Intersecting lines- intersect at one point One solution Same Line- always are on top of each other,

Bringing Inverse Modeling to the Scientific Community Hydrologic Data and the Method of Anchored Distributions (MAD) Matthew Over 1, Daniel P. Ames 2,

Monte Carlo Simulation CWR 6536 Stochastic Subsurface Hydrology.

Week 2ELE Adaptive Signal Processing 1 STOCHASTIC PROCESSES AND MODELS.

1 Part 5 Response of Linear Systems 6.Linear Filtering of a Random Signals 7.Power Spectrum Analysis 8.Linear Estimation and Prediction Filters 9.Mean-Square.

Formulation of the Problem of Upscaling of Solute Transport in Highly Heterogeneous Formations A. FIORI 1, I. JANKOVIC 2, G. DAGAN 3 1Dept. of Civil Engineering,

Prepared by Mrs. Azduwin Binti Khasri

Use the Distributive Property to: 1) simplify expressions 2) Solve equations.

LECTURE 3: ANALYSIS OF EXPERIMENTAL DATA

Silesian University of Technology in Gliwice Inverse approach for identification of the shrinkage gap thermal resistance in continuous casting of metals.

Review of Random Process Theory CWR 6536 Stochastic Subsurface Hydrology.

Robotics Research Laboratory 1 Chapter 7 Multivariable and Optimal Control.

Chapter 1 Random Process

ENEE631 Digital Image Processing (Spring'04) Basics on 2-D Random Signal Spring ’04 Instructor: Min Wu ECE Department, Univ. of Maryland, College Park.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 1/45 GEOSTATISTICS INTRODUCTION.

Discrete-time Random Signals

Stochastic Analysis of Groundwater Flow Processes CWR 6536 Stochastic Subsurface Hydrology.

Surveying II. Lecture 1.. Types of errors There are several types of error that can occur, with different characteristics. Mistakes Such as miscounting.

Holt McDougal Algebra Multiplying and Dividing Rational Expressions Simplify rational expressions. Multiply and divide rational expressions. Objectives.

Multi Step Equations. Algebra Examples 3/8 – 1/4x = 1/2x – 3/4 3/8 – 1/4x = 1/2x – 3/4 8(3/8 – 1/4x) = 8(1/2x – 3/4) (Multiply both sides by 8) 8(3/8.

CWR 6536 Stochastic Subsurface Hydrology

1 Review of Probability and Random Processes. 2 Importance of Random Processes Random variables and processes talk about quantities and signals which.

Solve a two-step equation by combining like terms EXAMPLE 2 Solve 7x – 4x = 21 7x – 4x = 21 Write original equation. 3x = 21 Combine like terms. Divide.

Approximate Analytical Solutions to the Groundwater Flow Problem CWR 6536 Stochastic Subsurface Hydrology.

Sec Math II 1.3.

Solving Equations. An equation links an algebraic expression and a number, or two algebraic expressions with an equals sign. For example: x + 7 = 13 is.

3.2 Solve Linear Systems Algebraically Algebra II.

DYNAMIC BEHAVIOR OF PROCESSES :

Linear Algebra: What are we going to learn? 李宏毅 Hung-yi Lee.

Chapter 6 Random Processes

Colorado Center for Astrodynamics Research The University of Colorado 1 STATISTICAL ORBIT DETERMINATION Statistical Interpretation of Least Squares ASEN.

CWR 6536 Stochastic Subsurface Hydrology Optimal Estimation of Hydrologic Parameters.

Direct Variation If two quantities vary directly, their relationship can be described as: y = kx where x and y are the two quantities and k is the constant.

Multiple Random Variables and Joint Distributions

Linear Constant-Coefficient Difference Equations

STATISTICAL ORBIT DETERMINATION Kalman (sequential) filter

Why Stochastic Hydrology ?

Laplace Transforms Chapter 3 Standard notation in dynamics and control

Objective I CAN solve systems of equations using elimination with multiplication.

3.1 Introduction Why do we need also a frequency domain analysis (also we need time domain convolution):- 1) Sinusoidal and exponential signals occur.

STATISTICS Joint and Conditional Distributions

Random Process The concept of random variable was defined previously as mapping from the Sample Space S to the real line as shown below.

Class Notes 18: Numerical Methods (1/2)

Test 1 Review Chapter 1, Hydrologic cycle and the water balance

B.Sc. II Year Mr. Shrimangale G.W.

Multiplying and Factoring

Solving Systems of Equation by Substitution

Solution of ODEs by Laplace Transforms

Professor Ke-sheng Cheng

Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first.

Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first.

Presentation transcript:

Approximate Analytical Solutions to the Groundwater Flow Problem CWR 6536 Stochastic Subsurface Hydrology

3-D Steady Saturated Groundwater Flow K(x,y,z) random hydraulic conductivity field  (x,y,z) random hydraulic head field want approximate analytical solutions to the 1st and 2nd ensemble moments of the head field

System of Approximate Moment Eqns to order  2 Use  0 (x), as best estimate of  (x) Use   2 =P  (x,x) as measure of uncertainty Use P  (x,x’) and P f  (x,x’) for cokriging to optimally estimate f or  based on field observations

Possible Solution Techniques Fourier Transform Methods (Gelhar et al.) Greens Function Methods (Dagan et al.) Numerical Techniques (McLaughlin and Wood, James and Graham)

Fourier Transform Methods Require A solution that applies over an infinite domain Coefficients in equations that are constant, or can be approximated as constants or simple functions Stationarity of the input and output covariance functions (guaranteed for constant coefficients) All Gelhar solutions use a special form of Fourier transform called the Fourier-Stieltjes transform.

Recall Properties of the Fourier Transform In N-Dimensions (where N=1,3): Important properties:

Recall properties of the spectral density function Spectral density function describes the distribution of the variation in the process over all frequencies: Eg

Look at equation for P f  (x,x’) Are coefficients constant? Can input statistics be assumed stationary? If so output statistics will be stationary. Assume ; substitute

Solve equation for P f  (x,x’) Expand equation Take Fourier Transform

Solve equation for P f  (x,x’) Rearrange Align axes with mean flow direction and let

Look at equation for P  (x,x’) Are coefficients constant? Can input statistics be assumed stationary? If so output statistics will be stationary. Assume ; substitute

Solve equation for P  (x,x’) Expand equation Take Fourier Transform

Solve equation for P  (x,x’) Rearrange Align axes with mean flow direction and let

Procedure Given P ff (  ) Fourier transform to get S ff (k) Use algebraic relationships to get S f   (k) and S     (k) Inverse Fourier transform to get P f   (  ) and P     (  ) Then multiply each by  2 =  lnK 2 to get P f  (  ) and P  (  )

Results Head Variance: Head Covariance