1. Using General Inverse Theory to Calibrate the Island Recharge Problem in EXCEL

2. What I hope to do with this talk… Provide enough general theory so folks will be better prepared to tackle manuals for automated calibration programs such as PEST Demonstrate a simple tool to help illustrate some of these ideas

3. Why is it called inverse modeling? The forward model –solve for heads given R and K: A is called the sensitivity matrix (m x n) x is the parameter (or upgrade) vector (m x 1) b is the observation (or residual) vector (n x 1) m is the number of observations n is the number of parameters

4. Why is it called inverse modeling? The inverse model –Solve for K and R given a set of measured heads

5. So what is the sensitivity matrix? Sensitivities are the derivatives of the groundwater flow equation with respect to parameters PEST and UCODE approximate these derivatives using finite differences –No different than how we have been approximating the groundwater flow equation –Called perturbation sensitivities

6. So what is the sensitivity matrix? Both PEST and UCODE approximate the derivatives using forward and central differences PEST provides a few more options including a best-fit or the slope of a polynomial fit to the three points p(1+derinc)pp(1-derinc) calculated head

7. So what is the sensitivity matrix? One row for each observation One column for each parameter

8. So how do we invert A?

9. UCODE: –Modified Gauss-Newton (small problems) –Parameter Parsimony (Arbitrary zonation???) PEST: –Modified Gauss-Newton –SVD (medium) –LSQR! (large) Both programs can make use of: Damping (Marquardt parameter)

10. So how do we invert A? Modified Gauss-Newton: –Can only invert square matrices –Called the normal equation; A T A is a square n x n matrix –As the number of parameters increases the likelihood of A T A being singular (not invertible) increases

11. So how do we invert A? Singular-Value Decomposition (SVD): –Decomposes a matrix according to: –Finds the pseudo-inverse of any matrix (square or not) according to: –Can solve for several thousands of parameters given thousands of observations before it just takes too long to be practical

12. Observation and sensitivity weights Can be used to scale observations or sensitivities so they are all within the same order of magnitude The normal equation becomes:

13. Marquardt-Lambda I am not going to get into this as I don’t fully understand it yet myself; I just know its needed for some problems to work (even 2x2 problems) The normal equation becomes: This is the form of the equation the EXCEL file solves for a simple 1x2 or 2x2 problem

14. Weighted Least-Squares: or And finally the objective function

15. EXCEL Demo 1.Estimate R and K given 1-head observation 2.Estimate R and K given 1-head and 1-flux observation i)Without observation weights (Q=1) ii)With observation weights 3.Estimate R and K given 1-head observation with scaling of the sensitivity matrix