Transcript Subsurface Hydrology - Illinois State University

The Calibration Process
• Calibration of a flow model refers to a
demonstration that the model is capable of
producing field-measured heads and flows which
are the calibration values.
• Calibration is accomplished by finding a set of
parameters, boundary conditions, and stresses that
produce simulated heads and fluxes that match
field-measured values within a pre-established
range of error.
Targets in Model Calibration
• Head measured in an observation well is known
as a target. Baseflow measurements or other fluxes
are also used as targets during calibration.
• The simulated head at a node representing an
observation well is compared with the measured
head in the well. (Similarly for flux targets…)
Residual error = observed - simulated
• During model calibration, parameter values (e.g., R
and T) are adjusted until the simulated head matches
the observed value within some acceptable range of
error. Hence, model calibration solves the inverse
problem.
Target Values
Inverse Problem
• Objective is to determine values of the parameters
and hydrologic stresses from information about
heads, whereas in the forward problem system
parameters such as hydraulic conductivity, specific
storage, and hydrologic stresses such as recharge
rate are specified and the model calculates heads.
• The inverse problem is an estimation of boundary
conditions, hydrologic stresses, and the spatial
distribution of parameters by methods that do not
involve consideration of heads.
• Calibration can
be performed:
• steady-state
– Requires some
flux input to
the system
• transient data
sets.
Information needed for calibration:
• head values and fluxes or other calibration
data (called sample information),
• parameter estimates (called prior
information) that will be used during the
calibration process.
Sample Information
• Heads
– Sources of error
• transient effects that are not represented in the model.
• measurement error associated with the accuracy of the water
level measuring device
• Interpolation Error
– Calibration values ideally should coincide with nodes, but in
practice this will seldom be possible. This introduces
interpolation errors caused by estimating nodal head values. This
type of error may be 10 feet or more in regional models. The
points for which calibration values are available should be shown
on a map to illustrate the locations of the calibration points
relative to the nodes. Ideally, heads and fluxes should be
measured at a large number of locations, uniformly distributed
over the modeled region.
Examples of Sources of Error
•
•
•
•
•
•
Surveying errors
Errors in measuring water levels
Interpolation error
Transient effects
Scaling effects
Unmodeled heterogeneities
Sample Information
• Fluxes
– Field-measured fluxes, such as baseflow,
springflow, infiltration from a losing stream, or
evapotranspiration from the water table may
also be selected as calibration values.
– associated errors for flux are usually larger than
errors associated with head measurements.
– Calibration to flows gives an independent check
on hydraulic conductivity values.
Prior Information
• Calibration is difficult because values for aquifer parameters
and hydrologic stresses are typically known at only a few nodes
and, even then, estimates are influenced by uncertainty.
• Prior information on hydraulic conductivity and/or
transmissivity and storage parameters is usually derived from
aquifer tests.
• Prior information on discharge from the aquifer may be
available from field measurements of springflow or baseflow.
• Direct field measurements of recharge are usually not available
but it may be possible to identify a plausible range of values.
• Uncertainty associated with estimates of aquifer parameters and
boundary conditions must also be evaluated.
Calibration Techniques
• two ways of finding model parameters to
achieve calibration
– (1) manual trial-and-error adjustment of
parameters
– (2) automated parameter estimation.
• Manual trial-and-error calibration was the
first technique to be used and is still the
technique preferred by most practitioners.
Calibration parameters are parameters whose values
are uncertain. Values for these parameters
are adjusted during model calibration.
Typical calibration parameters include
hydraulic conductivity and recharge rate.
Parameter values can be adjusted manually by trial and error.
This requires the user to do multiple runs of the model.
…or parameter adjustment can be done with the help
of an inverse code.
Trial-and-Error Calibration
• Parameter values assigned to each node or element in the grid.
• The values are adjusted in sequential model runs to match
simulated heads and flows to the calibration targets.
• For each parameter an uncertainty value is quantified. Some
parameters may be known with a high degree of certainty and
therefore should be modified only slightly or not at all during
calibration.
• The results of each model execution are compared to the
calibration targets; adjustments are made to all or selected
parameters and/or boundary conditions, and another trial
calibration is initiated.
• 10s to 100s of model runs may be needed to achieve calibration.
• No information on the degree of uncertainty in the final parameter
selection
• Does not guarantee the statistically best solution, may produce
nonunique solutions when different combinations of parameters
yield essentially the same head distribution.
Trial and Error Process
Automated Calibration
• Automated inverse modeling is performed using specially developed
codes
– Example PEST – Parameter ESTimation,
– Direct solution - unknown parameters are treated as dependent variables in
the governing equation and heads are treated as independent variables.
• The direct approach is similar to the trial-and-error calibration in that the
forward problem is solved repeatedly. However, the code automatically
checks and updates the parameters to obtain the best solution.
• The inverse code will automatically find a set of parameters that
matches the observed head values.
• An automated statistically based solution quantifies the uncertainty in
parameter estimates and gives the statistically most appropriate
solution for the given input parameters provided it is based on an
appropriate statistical model of errors.
Evaluating the Calibration
• The results of the calibration should be evaluated both
qualitatively and quantitatively. Even in a quantitative
evaluation, however, the judgment of when the fit between
model and reality is good enough is a subjective one.
• There is no standard protocol for evaluating the calibration
process.
• Traditional measures of calibration
– Comparison between contour maps of measured and simulated
heads
– A scatterplot of measured against simulated heads is another way
of showing the calibrated fit. Deviation of points from the straight
line should be randomly distributed.
Basecase simulation for the Final Project
Perfect fit
Residual = observed - simulated
Tabular Data
Expressing differences between
simulated and measured heads
• The mean error (ME) is the mean difference between
measured heads (hm) and simulated heads (hs).
n
ME  1 / n (hm  hs ) i
i 1
• where n is the number of calibration values.
• Simple to calculate
• Both negative and positive differences are
incorporated in the mean and may cancel out the
error.
• Hence, a small mean error may not indicate a good
calibration.
Example of Mean Error
Tabular Data
Expressing differences between
simulated and measured heads
• The mean absolute error (MAE) is the mean of the
absolute value of the differences in measured and
simulated heads.
n
MAE  1 / n (hm  hs ) i
i 1
• All errors are positive.
• Hence, a small mean error may would indicate a
good calibration.
Expressing differences between
simulated and measured heads
• The root mean squared (RMS) error or the
standard deviation is the average of the
squared differences in measured and
0.5
simulated heads.
n

2
RMS  1 / n (hm  hs ) i 
•
i 1


• As with MAE, all errors are positive.
• Hence, a small mean error may would
indicate a good calibration.
Example of Root Mean Squared
Error
• The RMS is
usually thought
to be the best
measure of error
if errors are
normally
distributed.
However, ME
and MAE may
provide better
error measures
(Figure 32).
Sensitivity Analysis
• Purpose: to quantify the uncertainty in the calibrated model caused by
uncertainty in the estimates of aquifer parameters, stresses, and
boundary conditions.
• Process: Calibrated values for hydraulic conductivity, storage
parameters, recharge, and boundary conditions are systematically
changed within the previously established plausible range. The
magnitude of change in heads from the calibrated solution is a measure
of the sensitivity of the solution to that particular parameter.
• Sensitivity analysis is typically performed by changing one parameter
value at a time.
• A sensitivity analysis may also test the effect of changes in parameter
values on something other than head, such as discharge or leakage