Handbook for Predicting Stream Meander Migration and Supporting Software (2004)

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C-1 INSTRUCTIONS FOR INSTALLING DATA LOGGER AND CHANNEL MIGRATION PREDICTOR The following instructions assume that the user has ArcView 3.x installed. The Avenue scripts for the Data Log- ger and Channel Migration Predictor are not supported by ArcGIS 8.x. The procedures for installation of the Data Log- ger extension are as follows: 1. Find the folder named ext32 within the folder in which ArcView is installed. For example, C:\Esri\Arcview\ ext32 would be a fully qualified path name to the folder ext32. Copy the file “DataLogger.avx” from CRP-CD-48 to the folder ext32. On Unix systems, replace the folder ext32 with the directory named ext. 2. Open an ArcView project and select “Extensions . . .” under the “File” menu. 3. Drag the mouse over the check box to the left of the extension named “Data Logger.” Notice the cursor turns into a check mark. 4. Click on the check box for the “Data Logger” exten- sion. When the user clicks on the check box, a check mark will appear. 5. Repeat Steps 3 and 4 for the “CAD Reader” extension. 6. Hit OK and the extension will be loaded. (On the ArcView Extensions Dialog, if the box “Make Default” is checked at this point, then the “Data Logger” will be loaded when any new project is created.) The Channel Migration Predictor is installed using the same procedures. After installing the Channel Migration Pre- dictor extension, make sure the file “frequency.dbf” is in your working directory. (The file “frequency.dbf” is produced by the Data Logger.) CIRCLE-FITTING ALGORITHM The equation for a circle with center at (x, y) = (a,b) and radius R is given by: Let (xi, yi), i = 1, 2, . . . n, be a set of data points in the xy- plane. To determine values of a, b and R, which provide a best least squares fit of a circle to the data points, we seek to minimize: F a b R x a y b Ri i i N , ,( ) = −( ) + −( ) −[ ] = ∑ 2 2 2 2 1 x a y b R−( ) + −( ) =2 2 2 The standard approach to minimizing F calls for setting the partial derivatives of F with respect to a, b, and R equal to zero. Because F is a fourth-degree polynomial in a, b, and R, setting these partial derivatives to zero would lead to a system of three nonlinear equations in a, b, and R. Rather than dealing with a difficult nonlinear system we take the following approach. From: it follows that F can be written as: Next, let c = a2 + b2 − R2 and define a new least squares objective function to be: The values of a, b, and c that minimize G can be found by setting to zero the partial derivatives of G with respect to a, b, and c. This leads to the following system of linear equations: The coefficient matrix for the above system of equations is symmetric (nij = nji), so the system is completely defined by: Gaussian elimination can be used to solve for a, b, and c, and then R can be recovered from .R a b c= + −2 2 n x n x y n x n y n y n N d x y d x y y d x y i i N i i i N i i N i i N i i N i i i N i i i N i i i i N 13 2 1 12 1 13 1 23 1 22 2 1 33 3 2 2 1 2 2 2 1 1 2 2 1 4 4 2 2 4 2 = = = = = = − = +[ ] = +[ ] = +[ ] = = = = = = = = ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ 2xi n n n n n n n n n a b c d d d 11 12 13 21 22 23 31 32 33 1 2 3             =       G a b c x y x a y b ci i i i i N , ,( ) = + − − +[ ] = ∑ 2 2 2 1 2 2 F a b R x x a a y y b b Ri i i i i N , ,( ) = − + + − + −([ ] = ∑ 2 2 2 2 2 2 1 2 2 x a y b R x x a a y y b b R i i i i i i −( ) + −( ) − = − + + − + − 2 2 2 2 2 2 2 2 2 2 APPENDIX C INSTRUCTIONS FOR INSTALLING DATA LOGGER AND CHANNEL MIGRATION PREDICTOR AND DESCRIPTION OF THE CIRCLE-FITTING ALGORITHM