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DRILL GUIDE - environmental site investigation and assessment software - kriging - geostatistics
DRILL GUIDE Categories: exposure assessment, geostatistical analysis, soil, solid waste management, visualization software, water quality, civil engineering, air pollution
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Geostatistics in DRILL GUIDE
The following provides an understanding of the philosophy and implementation of parameter estimation and geostatistics in DRILL GUIDE. DRILL GUIDE's
capabilities to grid and interpolate sparse measured data in three dimensions are unparalleled. DRILL GUIDE performs all interpolation using an accurate and geostatistically-defensible process called Kriging.
Kriging is a mathematical process recognized by the U.S. Environmental Protection Agency (EPA) as the best and standard means for interpolation and extrapolation of measured data. DRILL GUIDE provides a
user-friendly expert system to drive its Kriging modules, lifting the burden of determining optimal variogram parameters from the user. With DRILL GUIDE, the user can rely on expert system-calculated default values
to provide quality answers in minimal time.
Kriging is the only data estimation method that also provides statistical measures of goodness. When data is analyzed in DRILL GUIDE, the user is
also provided statistical confidence and an additional parameter that we call uncertainty. These additional statistical measures are extremely useful in guiding future site investigation efforts. Our experience is
that by using maximum uncertainty to guide site investigations, a 30% reduction in sampling locations can be realized for an equivalent quality of assessment.
As a user-defined alternative to confidence and uncertainty, DRILL GUIDE will provide statistical bounds on the parameter estimate. In other words,
DRILL GUIDE will determine the nominal, minimum, and maximum estimated distribution based on a user-specified confidence level. With DRILL GUIDE, you can now directly answer the question: With my limited measured
data, to an 80% confidence level, what is the largest and smallest plume I can expect?
DRILL GUIDE Kriging
Kriging is a weighted moving average interpolation (extrapolation) method that minimizes the estimated variance of a predicted point (node) with the
weighted average of its neighbors. The weighting factors and the variance are calculated using a variogram model that describes the differences versus distance for pairs of samples in the input dataset. Determining
an optimal variogram model is the important first step in producing a defensible kriged estimate. With other software, the burden of fitting the variogram is left to the user. In DRILL GUIDE, this task is automated
with an expert system.
DRILL GUIDE's kriging is based on sound mathematical and statistical concepts. The U.S. EPA developed a two-dimensional kriging software package
called Geo-EAS. Geo-EAS version 1.1 was released in September 1988. DRILL GUIDE offers an important enhancement to the 2-D kriging methods used in Geo-EAS (variogram asymmetry), but if this is turned-off, DRILL
GUIDE and Geo-EAS will calculate the same results.
DRILL GUIDE gridding options for two-dimensional estimation include: rectilinear grids with uniform spacing in the x and y directions; convex hull
bounded gridding; and adaptive gridding which automatically refines gridding in the cell(s) surrounding measured samples to ensure that the interpolated results and contours accurately honor measured sample data.
DRILL GUIDE Variograms
The process of determining an appropriate variogram stands out as one of the most significant discriminators between DRILL GUIDE and all other
analysis and visualization programs. Anyone who has ever struggled with trying to fit an appropriate variogram to real world data understands how tedious and difficult it is. DRILL GUIDE applies the experience and
knowledge of a team of experts to that task in the form of expert system algorithms that assess the data and assure the appropriateness of the results. DRILL GUIDE also offers a technological breakthrough by
extending the concept of the variogram. A traditional variogram considers the differences between samples versus their distance from one another. DRILL GUIDE extends this concept by also considering the direction
(vector distance).
Symmetry
Before beginning the process of best-fitting the variogram, the user can set an allowable variogram symmetry. If that symmetry is set to one (1.00),
the direction or vector component of distance is ignored during variogram fitting. If, however, the symmetry is reduced, DRILL GUIDE will analyze trends in the data pairs and determine a "major axis" orientation.
Instead of a variogram that can be represented by a line in two-space, DRILL GUIDE produces a variogram that must be described as a surface in three-space. Major and minor axis variogram parameters are determined
and these are blended together in the regions between. The resulting multidimensional variogram is capable of better representing anisotropies or trends in the data such as a ground-water plume being driven by a
distinct flow direction.
The symmetry is defined as the allowable quotient between the minor axis range and sill and the major axis range and sill.
DRILL GUIDE's Geostatistical Confidence
Start DRILL GUIDE and create a simple network with Krig_Z (red port) connected to the Viewer. Select css_0.csv as the data file and modify the
Kriging Parameters to choose Maximum as the Extract Method. Select Accept All Current Values on Krig_Z and when it runs, your Kriging Parameters should match:
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Note that the Conf(idence) Bound value is set to 10. That value will be used to determine the confidence calculations (which will be discussed
below).
Now close the Kriging Parameters and bring up the Postprocessing panel. Let's set the Data Scaling to 20.0.
Let's also bring up the Az-El panel and set the view so that Scale = 0.8; Elevation = 45; and Azimuth = 210. Your view should now be:
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This surface is colored according to concentration, and also has surface elevations (topology) which is the log of concentration times the data
scaling value of 20.0.
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We will now select Confidence as our Data Component. This will recover the surface but leave the elevations unchanged. From a single picture, we
will be able to see confidence levels (by color) and concentration levels (by elevation).
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Note that the Min(imum) confidence is 26.4. Since our default color scale runs from dark blue to red, the picture below has no dark blue because our
minimum is 26 and the color range has been set to have dark blue map to a confidence of zero (0.0).
Notice that the picture below has a lot of red spots. Each of these spots corresponds to a measured sample location. At every measured sample, the
confidence in the estimate is 100%.
But what do we mean by Confidence? In DRILL GUIDE, the confidence is calculated in response to a question. For this model, the question is: What
is the confidence that the predicted concentration will be within a factor of 10 of the actual concentration?
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Why a "factor of ten"? Because we are working on a log10 scale, DRILL GUIDE took the log10 of the Conf Bound value (the value was 10, so the log is
1.0). It then compares the log concentration values and a corresponding standard deviation that was calculated for every node in our domain. For log concentrations, one unit is a factor of ten; therefore, we are
asking what is the probability that we will be within one unit. If you had changed the Conf Bound to 2.0, the question would have been: What is the confidence that the predicted concentration will be within a factor
of 2 of the actual concentration?
The actual calculation to determine confidence requires the standard deviation of the estimate at a node and the Conf Bound value. The figure below
shows the confidence (as the shaded area under the "bell" curve) for a Conf Bound of 10 at a node where the predicted concentration was 10 ppm (1.0 log concentration) and the standard deviation for this point was
1.1 (in log10 units). For this example, the confidence would be ~64%, which means that 64% of the time, the value would lie in the shaded region.
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Uncertainty
At first glance, confidence seems to be a reasonable measure of site assessment quality. We know that if the confidence is high, we can be assured
of the reasonableness of the predicted values. You might be tempted to collect samples everywhere that the confidence was low, and if you did, your site would be well characterized.
But, there is a better, more cost-effective way. Instead of focusing on every place where confidence was low, we could focus on only those locations
where there was low confidence and where the predicted concentration was reasonably high. We could try to see these locations by looking at the confidence colored plot for high elevation areas.... or, we could use
the DRILL GUIDE measure called uncertainty. In DRILL GUIDE, uncertainty is high where concentrations are predicted to be relatively high (above the Clip Min), but the confidence in that prediction is low. If the
goal is to find the contamination, using uncertainty will allow for more rapid, cost-effective site assessment.
Let's look at the Uncertainty plot. Choose Uncertainty as the Data Component. Note that it ranges from 0.00 to 21.45. When we ran this
calculation, the DRILL GUIDE Message Console reported the precise coordinate of the maximum uncertainty. That location is a very appropriate next sampling site.
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Max uncertainty#0= 21.46 at x= 11492.93 y= 12935.60:
Predicted Concentration = 1.6443
Note the bright red spot (dark spot in black and white) on the left side of the domain. This area is far from any sample locations (regions of high
confidence) and is on the side of our topological hill (meaning that the concentration is well above our minimum. To make the region more obvious for this document, we have added labeled isolines at uncertainty
values of 5, 10, 15, and 20.
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Uncertainty does not have physical units, rather it is a dimensionless parameter for which higher values are bad and lower values are good. High
uncertainty means a high probability of poorly characterized contamination. Low uncertainty means a low probability of uncharacterized contamination. The function used to compute uncertainty has been optimized to
minimize the number of new sampling locations required to lower the maximum uncertainty anywhere in the model domain to a specific low value. The confidence levels and corresponding uncertainties are data dependent.
If the measured data has regions of high concentration gradient, nearby areas away from sample locations will have lower confidence and higher uncertainty.
The absolute magnitude of uncertainty values is also dependent on three key user-specified values; the confidence interval, and the display clipping
minimum and maximum. The "clip min" establishes the floor for concentration values. Values below this level are considered unimportant (or more specifically, they are set to that level). The "clip
max" establishes the ceiling for concentration values. Values above this level are clipped to this value. Setting the clip max at the concentration value of most interest, such as the toxic level for a plume
boundary, will cause the selection of new sampling locations to converge most rapidly on defining the plume boundary. This will, however, sacrifice the accurate characterization of the most contaminated regions in
the domain. This limitation is generally acceptable if the primary goal is determination of the extent (rather than specific distribution) of contamination.
The next figure shows the calculated uncertainty as a function of predicted concentration for 5 different confidence levels. This plot assumed a
clip min of .001, and clip max of 1000, and a confidence interval of 10. During the early stages of site characterization, uncertainty levels can exceed 100.
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Min-Max Plume
Sometimes confidence and uncertainty just won't answer your (or your customer's) question. Just how big (or how small) might the contaminant plume
be? DRILL GUIDE displays concentration, but what are the limits? The Min-Max Plume option directly answers those questions.
Open up the Kriging Parameters and choose Maximum for the Extract Method and Min-Max Plume for the Statistics Options. Note that the Confidence
parameter is set to 60%.
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Choose css_0.csv again as the data file and after it has run, close Kriging Parameters and open up Postprocessing.
We want to set a clamped range for outputting the min, nominal and max plumes. The limits of all three will be slightly different so we should
choose values which will be compatible with all. We will set the Clamp toggle on and the Min value to -2 (equiv. to 0.01 ppm) and the Max value to 4.0 (equiv. to 10,000 ppm). Also set the Data Scaling to 20.
Your parameters should match:
Bring up the Az-El panel and set the view so that Scale = 0.8; Elevation = 45; and Azimuth = 210. Your view should now be:
The white outline is the nominal plume at 0.1 ppm (log value of -1.0). What does nominal mean?
Nominal is the distribution where 50% of the time the predicted values are higher and 50% of the time they are lower. That means that our confidence in this plume size is never lower than 50%. Be careful, this confidence has a different meaning than the confidence discussed in the early section.
NOMINAL PLUME
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Now let's see what the minimum size plume is for our 60% confidence level. To do this we need to change both the Data Component in Krig_Z:
Postprocessing and the Iso Component in isolines to Conc_Min0. Note a marked reduction in the 0.1 ppm plume versus the nominal plume.
MINIMUM PLUME
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Similarly, the maximum plume (for a 60% confidence) is shown below.
What would have happened if we had set the Confidence level higher (or lower)? The Confidence parameter cannot be set lower that 50%, because by
default, the nominal distribution has a 50% confidence. If it is set at 50%, the Min Plume will be the same as the Max Plume. Values below 50% don't make sense.
Similarly, a value of 100% is impossible. We can never be 100% sure of estimates. The maximum allowable value is 99.99% and we don't recommend it.
Unless your site is extremely well characterized, there will be significant variation between min and max plumes at Confidence levels of 80 or 90%.
Everything we have covered on geostatistics has used two-dimensional kriging. Is that all DRILL GUIDE can do?..........NO!
MAXIMUM PLUME
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Min-Max Algorithm
DRILL GUIDE determines the Min and Max plumes by first calculating the nominal value and associated standard deviation at every node in the model.
For the case of Max Plume and 80% confidence, at each node, a maximum value is determined such that 80% of the time, the actual values will fall below the maximum value (for that nominal concentration and standard
deviation). This process is shown below pictorially for the case of a nominal value of 10 ppm with a standard deviation of 1.1 (log units). For this case, the maximum value at that node would be ~84 ppm. This
process is repeated for every node (tens or hundreds of thousands) in the model.
Note that for this plot, the entire left portion of the bell curve is shaded. If we were assessing the minimum value, it would be the right side.
Statistically, we are asking a different type of question than when we calculate confidence for our nominal concentrations.
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DRILL GUIDE Conclusion
DRILL GUIDE combines ease of use with the tools and speed to get your jobs done on-time and on-cost. Its strong geostatistical foundation guarantees
that your work will be defensible and that work can confidently be used to guide a cost-effective remediation program. The use of DRILL GUIDE technology to analytically guide environmental site assessments offers
significant cost benefits. A 30%–80% reduction in sampling locations is achieved when the sampling locations are geostatistically determined versus the use of a regular grid.
DRILL GUIDE will pay for itself on your first job and save you and your customers a great deal of time and money.
DRILL GUIDE Requirements: PC running Windows 95/98/2000/NT with 24 MB RAM.
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Global Enviro Software 1204 W South Jordan Prkwy Ste B South Jordan, Utah 84095
Phone (801) 208-3011 Toll Free (U.S.) 1-866-620-9214 Fax (801) 302-1160
E-mail info@scisoftware.com
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