DMSTA Sensitivity Analysis D R A F T
DMSTA includes a routine for testing the sensitivity of results to each input variable.  
Each non-zero entry in the parameter input table is modified by a specified percentage & the simulation is re-run.
The input variables are ranked in order of decreasing sensitivity.
Because the user-specified percentage change is arbitrary, results should not be interpreted as measures of uncertainty.
A separate uncertainty analysis procedure is decribed below.
Results are tabulated for the following output variables describing the combined outflow from the entire treatment area:
1 Flow-weighted-Mean Outflow Conc - Including Bypass
2 Flow-weighted-Mean Outflow Conc - Excluding Bypass
3 Geometric Mean Outflow Conc - Composite Samples - Excluding Bypass
4 Outflow Load - Including Bypass
For each output variable, results are expressed as follows:
1 Simulated value with a positive change in the input parameter
2 Simulated value with a negative change in the input parameter
3 Average absolute change relative to the base value =  [ ( YH - YB ) + (YB - YL ) ] /  2   =  ( YH - YL ) / 2
Negative values indicate that the output value decreases when the input value increases & vice-versa.
The positive & negative changes differ to the extent that the response is non-linear
4 Average absolute change as a percentage of the base value
User inputs include the following:
User Input Value Description
Sensitivity Factor P Modify each input variable by +/- this percentage 
High Value = X  ( 1  +  P )
Low Value = X / ( 1 + P ) <--- prevents negative values
Input Parameter Set 1 Test all inputs (except initial conditions & discharge exponent)
Results will be insensitive to initial conditions if sufficient iterations are performed.
The discharge exponent is not tested because it is highly correlated with the discharge intercept.
Testing the discharge intercept provides a sufficient indication of sensitivity to outflow hydraulics
2 Test P cycling parameters only 
3 Test design features only (all inputs except P cycling parameters)
Test Option 1 Test high results only (increase input variable by specified percentage)
This will speed computation, but with some loss of accuracy if the output response is nonlinear
2 Test high & low results
A button is provided to sort the results in order of decreasing sensitivity.   
The output variable used for sorting is the one selected on the 'UncertaintyAnalysis' sheet.
DMSTA Uncertainty Analysis D R A F T
A routine is provided to derive approximate estimates the uncertainty in each of the output variables, based upon the following factors:
Inherent model error from testing of p-cycling model calibrations against independent datasets
Uncertainty in input variables from calibration of p-cycling model
Uncertainty in design input variables user-specified
Sensitivity of the output variable to each input from sensitivity analysis procedure
Results are based upon a first-order error analysis (Walker, 1982).  The following key assumptions are made:
Uncertainty in each input value is known with reasonable accuracy (expressed as a coefficient of variation or % standard deviation) 
Errors in inputs are statistically independent.
The model response to each input is approximately linear of the range of the uncertainty in that input variable.
First-order analyses have been shown to give results that are similar to Monte-Carlo analyses in simple lake eutrophication models (Walker, 1982).
While possibly more accurate, the time required for a Monte-Carlo analysis would be prohibitive using the Excel platform.
Because the assumptions are not strictly met & the estimates of input variable uncertainty are themselves uncertain, results of the uncertainty analysis procedure yields approximate results.   Results are valid only if the recommended parameter sets are assigned to each vegetation type.
The relevant equations for the first-order analysis are:
Var (Y)  =   Sum [  EJ ]    for  J  =  1 to N
EJ  =  [  d Y / d X J  x  V ( XJ )  ] 2
where,
Var (Y) =  uncertainty  (variance) in estimate of output variable
Var ( XJ ) =  uncertainty in input variable J  ( from calibration & user input  )
N = number of input variables
EJ   = contribution of input variable J to total variance in output variable
d Y / d X J  = first derivate of output with respect to input (from sensitivity analysis)
User-defined estimates of uncertainty in the design variables are entered in column B of the Uncertainty Analysis sheet,
expressed as cofficients of variation (standard error / mean )
Derivatives are displayed on the Uncertainty Analysis sheet, expressed as 'Sensitivity Coefficients'
Sensitivity Coef.  =  % change in output / % change in input;    e.g,. 1.0  means that the output is proportional to the input.
For the user-specified output variable, results are summarized as follows:
Coefficient of variation (CV = standard error as percent of the predicted value)
Approximate 80% confidence interval (10th to 90th percentile range), assuming a normal distribution
To provide perspectives on important sources of uncertainty, CV's are listed for the following cases:
Input Value Error attributed to model inputs other than p cycling parameters
Parameter Error attributed to p cycling parameters
Model Error attributed to inherent model error
Total Error attributed to all of the above  ( root mean sum of squares of above )
Uncertainty analysis results may vary depending on the fixed percentage change used in the sensitivity analysis.
This variation reflects nonlinearity in the model & can be explored by testing alternative percentage values.  
A percentage range of 10 - 25% is recommended.
Approximate estimates of errors in p cycling parameters & model error are derived from model calibration & testing.
These estimates are still being developed.  Preliminary values are specified on the calibration sheet.
Reference: Walker, "A Sensitivity & Error Analysis Framework for Lake Eutrophication Modeling", Wtr Res Bul, Feb 1982
6/8/2002