RCRA GROUND-WATER

MONITORING

DECISION PROCEDURES V I E W E D AS Q U A L I T Y C O N T R O L S C H E M E S THOMAS H. STARKS Environmental Research Center, University of Nevada, Las Vegas, N V 89154, U.S.A,

and GEORGE T. FLATMAN U.S. Environmental Protection Agency, Environmental Monitoring Systems Laboratory, P.O. Box 93478, Las Vegas, N V 89193-3478, U.S.A.

(Received February 1989)

Abstract. The problems of developingand comparing statistical procedures appropriate to the monitoring of ground water at hazardous waste sites are discussed. It is suggestedthat these decision procedures should be viewedas quality control schemesand compared in the same way that industrial quality control schemes are compared. The results of a Monte Carlo simulation study of run-length distribution of a combined Shewhart-CUSUM quality control scheme are reported.

1. Introduction Under the Resource Conservation and Recovery Act of 1976 (RCRA), the U.S. Environmental Protection Agency (EPA) has developed regulations for landfills, surface impoundments, waste piles, and land treatment units that are used to treat, store, or dispose o f hazardous wastes. These regulations include requirements for the monitoring of ground water in the uppermost aquifer below the hazardous waste site (HWS). This monitoring involves the drilling of wells into the uppermost aquifer that are at appropriate locations and depths to yield ground water samples that represent the quality of background ground water and the quality of ground water passing the point of compliance. The sampling and analysis of monitoring-well water are conducted at regular time intervals to help determine whether a release f r o m the H W S has entered the aquifer. There are several areas of methodology in this monitoring p r o g r a m that appear to need further development. One of these areas of methodology is that of decision rules. H o w does one develop or choose a good decision rule, based on measurements of water samples drawn from wells near the HWS, for determining when additional regulatory action may be required? In this paper, it is argued that all decision rules that might be applied to a R C R A site are actually quality control schemes, and that, for this reason, they should be compared in the same way that industrial quality control schemes are compared. Further, it is suggested that since we are dealing with quality control schemes, we should at least consider schemes that have proved successful in industrial situations. Industrial Environmental Monitoring and Assessment 16: 19-37, 1991. 9 1991 Kluwer Academic Publishers. Printed in the Netherlands.

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T H O M A S H. STARKS A N D GEORGE T. F L A T M A N

quality control schemes are discussed in terms o f their possible application to the ground-water monitoring decision problem. The results of Monte Carlo simulations of a Shewhart-CUSUM quality control scheme operating under various conditions are given to show the possible strengths and weaknesses of this scheme if applied to RCRA ground-water monitoring.

2. Current EPA Considerations of the Statistical Decision Problem The latest rule (53 FR 39720, October 11, 1988, Statistical Methods for Evaluating Ground-water Monitoring Data from Hazardous Waste Facilities) for 40 CFR Part 264, Standards for Owners and Operators of Hazardous Waste Treatment, Storage, and Disposal Facilities, suggests four general statistical methods (parametric analysis o f variance, analysii o f variance based on ranks, tolerance or prediction interval procedures, and control charts) for decision rule procedures. In addition, another statistical test method may be submitted by the owner or operator for approval by the E P A Regional Administrator. The nature of the sampling is specified in 53 FR 39720 as follows: The owner or operator willdeterminean appropriate sampling procedureand interval for each hazardous constituent listed in the facility permit which shall be specifiedin the unit permit upon approval by the Regional Administrator. This sampling procedure shall be: (1) A sequenceof at least four samples, taken at an interval that assures, to the greatest extent technically feasible, that an independent sample is obtained, by reference to the uppermost aquifer's effective porosity, hydraulic conductivity, and hydraulic gradient and the fate and transport characteristics of the potential contaminants, or (2) an alternative sampling procedure proposed by the owner or operator and approved by the Regional Administrator. The complexity o f the problem o f 'real world' monitoring and statistical decision making requires this flexibility. However, this opportunity for improved monitoring and testing (i.e., improved relative to the methods required under previous regulations) will require a knowledge of the different procedures. To have good decision characteristics, any statistical method must conform to a realistic model for sources o f variance for data from the facility. The latest rule accepts the use of control charts. In addition, it acknowledges (i) their use o f the same well for both background and compliance data; (ii) their lack of a and/3 error probability calculations; and (iii) their need to propose facility specific control parameters. The latest rule states that any proposed statistical method must account for data below the limit o f detection, and be appropriate for the distributions o f measurements o f chemical parameters or hazardous constituents. In addition to enumerating possible statistical methods to be used in decision rules, the latest rule also points out some of their potentional deficiencies. A statistician working for an owner or operator should carefully study these options and their deficiencies before prescribing a decision methodology.

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3. Ground-Water Characteristics To develop a reasonable model for the measurements of a ground water quality parameter, some discussion of ground-water characteristics is required. The horizontal velocity of water in an aquifer is slow (a few meters per day is considered a high velocity) and lateral dispersion o f a contaminant is considerably slower than fluid flow, which means that plumes do not widen to any great extent as they extend through the aquifer (Freeze and Cherry, 1979, p. 26-29, 104, and 394-395). At a particular sampling time, the water at different wells near a HWS will have entered the aquifer at different times and, therefore, carry different concentrations of various monitored constituents. Hence, the value of a monitored parameter may be increasing at one well while decreasing at another. Concentrations of contaminants in samples of ground water may also change due to changes in water table. A high water table resulting from recent rains of flooding may cause a reduction in measured concentrations of a pollutant because of dilution of the contaminant or because the contaminant is floating at the top of the aquifer which is now above the well screen. On the other hand, a high water table may cause an increase in the concentrations of some pollutants owing to leaching activity in the vadose zone. In any case, changes in water table may have similar and virtually simultaneous effects on all wells near a HWS. In addition, well construction and the topmost aquifer characteristics will typically differ from location to location. Because of this, one can expect well location to be a source o f variability in measurements taken at a site. Any statistical decision procedure for RCRA site ground-water monitoring should be based on a model that is not in disagreement with these ground-water and well-location characteristics.

4. Quality Control Schemes Since the slow flow of water and the narrowness of plumes in the aquifer may make tests of upgradient versus downgradient means, and tests of time-by-location interactions, inappropriate (Starks, 1988), and since outlier tests based on small data sets are lacking in power, what other approach to decision procedures might one apply? First, one should note that in a RCRA ground water monitoring situation a decision is made at the end of each sampling period as to whether additional regulatory action (e.g., increased monitoring, or some remedial action) is needed. Hence, one is dealing with a sequence o f decisions rather than just one decision. The one-decision case is a test of hypothesis situation in which it makes sense to consider significance level and power of the test. However, in a situation where decisions are made sequentially over time, one is dealing with a quality control scheme and should be interested in the distributions of run lengths. An-in-control length is the number of sampling period from start-up until a decision is made, on the basis of water sample measurements, that additional regulatory action is required when, in fact, there is no leakage

22

THOMAS H. STARKS AND GEORGE T. FLATMAN

from the HWS. An out-of-control run length is the number of sampling periods from the time that a pollutant plume originating from the HWS intercepts a well location until a decision is made that additional regulatory action is required. Naturally, one wants to use a quality control scheme that has, on average, long in-control run lengths and short out-of-control run lengths. In addition, one wants the probalility of short in-control run lengths (i.e., early false positives) to be quite small. From this perspective, we see that all decision procedures that have been suggested for monitoring ground water at RCRA sites are in truth quality control schemes and should be compared on the basis of their run-length distributions. It seems reasonable to think that when choosing a quality control scheme, schemes that have proved successful in other applications should be considered. That is, one should consider industrial quality control schemes. In fact, Vaughan and Russell (1983) have suggested the use of industrial quality control schemes for monitoring effluent from waste treatment plants, which is somewhat similar to, but not so difficult as, the problem that is encountered in RCRA ground-water monitoring. A prime consideration in using industrial quality control methods to monitor groundwater quality at a HWS is that they remove effects of location and well construction from the decision process. Instead of comparing the measurement of a ground water quality parameter (GQP) at a particular well with measurements at other wells, one compares the current measurement of the GQP with the past history of the GQP measurements of water from this well (or compares the current average GQP measurement over wells with the history of such averages from the same set of wells). Discussion in this section is restricted to one-sided industrial quality control schemes where the common concern in monitoring is to detect an increase in the GQP. The extension to two-sided schemes is straightforward if they are needed for indicators such as pH. SHEWHART CHARTS The Shewhart (1931) quality control chart is one of the oldest and simplest of the industrial quality control procedures. The chart is simply a graph of time of sampling, or sample number if samples are equally spaced in time, versus the sample mean value for the quality parameter being monitored. Time, or sample number, is the abscissa and sample mean value the ordinate of a point on the graph. Typically the horizontal axis is positioned so as to intersect the vertical axis at the steady-state mean value/~ for the quality parameter. A horizontal line is also drawn to intersect the vertical axis at/t + Ztr n where Z is the upper a quantile of the standard normal distribution and an is the long-run standard deviation of the sample means (i.e., the standard error of the mean, an=a/~/n, where tr is the standard deviation of an individual measurement and n is the number of independent measurements averaged). This line is called the upper control limit, and when a point falls above the line the process is declared out of control. The average in-control run length (i.e., the average number of samples between declarations that the system is out of control,

RCRA G R O U N D - W A T E R M O N I T O R I N G DECISION P R O C E D U R E S

23

when in fact it is in-control) is I / a if the sample means have a normal sampling distribution. The commonly used value of Z is 3, for which the corresponding value of a is 0.0013. In industrial quality control the sample sizes are usually somewhere between 5 and 10 depending on cost and internal variability between members of a sample. Lorenzen and Vance (1986) give a procedure for determining n, Z, and the time between samples, on an economic basis. It would appear that their approach could be generalized to other control schemes and to ground-water monitoring situations. A second control chart is often kept for the variability of the product. It is similar to the Shewhart chart for the sample means, only now the ordinate is the sample range, or sample standard deviation, and the horizontal lines representing upper and lower control limits are located on the basis of distribution of the statistic (sample range or sample standard deviation) under the assumptions of a normal distribution for the quality parameter measurements. In practice the lower limit is seldom used (Guttman et al., 1982, p. 275). This chart is not nearly as robust with respect to the assumption of normality as is the chart for the sample means, and so out-of-control situations for variability must be viewed with more skepticism than similar results on the means chart. If the variability of measurements of the quality parameter has changed, the height of the upper control line on the Shewhart chart for means is adjusted accordingly, or action is required to bring the variability back to its previous level. THE CUSUM QUALITY CONTROL SCHEME

The CUSUM (for cumulative summation) control scheme derives from a paper by Page (1954) and is somewhat more complicated than the Shewhart chart. (A review article by Lucas, 1985, gives the current state of development of this procedure). The CUSUM control scheme makes use of information in the present sample and in the previous samples in reaching decisions as to whether the process is in-control, whereas the Shewhart chart makes decisions on only the current observation (i.e., the Shewhart chart is a graphical representation of a sequence of individual tests of the mean; whereas, the CUSUM scheme is a sequential probability ratio test of the mean). The one-sided CUSUM scheme involves the computation of a cumulative sum S which for t h e / t h sample is given by the formula, S i = m a x I 0 , z i - k + S i_ 11

where zi is the standardized ith sample mean (i.e., zi = [X;-/l]/tr,,), k is a parameter of the control scheme, and typically So is taken to be zero. When Si exceeds a specified value h, the process is declared out of control (i.e., in ground-water monitoring, a decision is made to begin additional monitoring activity). The values of h and k are chosen to obtain desired average run lengths (ARL) under in-control and specified out-of-control situations. For a scheme designed to be sensitive to changes in mean quality of size Don, k is usually chosen to be D/2, and h is selected to give the largest in-control ARL consistent with an adequately small out-of-control ARL.

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THOMAS H. STAR.KS AND GEORGE T. FLATMAN

5-

h

S. 4" I 32I 1 I

2

3

4

5

6

7

8

9 I0

i

Fig. 1. The CUSUM control chart (k=0.5, h = 5).

181614~

1

12I0 8-

I

I

I

I

I

I

I

I

2

3

4

5

6

7

8

9 10

I

6Fig.

2. Shewhart quality control chart ~ = 10, on=2 ).

E X A M P L E : To illustrate the Shewhart and C U S U M schemes, r a n d o m normal ~ = 1 0 , tr2=4) deviates were drawn f r o m a table of such numbers. To obtain an out-of-control situation, 2 was added to each of the numbers drawn after the fourth (i = 4). These numbers are to represent the sample means o f a process that went out of control between the fourth and fifth samplings. The in-control situation has sample means distributed N(10,4), so z i = ( X i - ~ ) / t r n = ( X i - 1 0 ) / 2 . The C U S U M scheme (Table I) indicates that a decision to take action (process is out o f control) should be made at i = 10. The C U S U M chart (Figure 1) gives a visual impression of when the process went out of control. The corresponding Shewhart chart, with an upper control limit of 0t + 3trn) = 16, is given in Figure 2. The example illustrates the weakness of the standard Shewhart chart in detecting small changes in the value of the mean. One way to reduce this weakness in the Shewhart chart is to declare the process out o f control whenever there are r successive sample means with value above/~. The value o f r is usually chosen to be 7 or 8. However, this procedure also reduces the in-control ARL.

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RCRA GROUND-WATER MONITORING DECISION PROCEDURES TABLE I CUCUM quality control scheme example (k = 0.5, h = 5) In-control

Out-of-control at i = 5

i

X

z

S

X

z

S

0 1 2 3 4 5 6 7 8 9 10

14.504 11.108 7.594 7.580 11.588 12.002 10.434 9.378 10.708 11.278

2.252 0.554 - 1.203 - 1.210 0.794 1.001 0.217 -0.311 0.354 0.639

0.000 1.752 1.806 O. 103 0.000 0.294 0.795 0.512 0.000 0.000 0.139

14.504 11.108 7.594 7.580 13.588 14.002 12.434 11.378 12.708 13.278

2.252 0.554 - 1.203 - 1.210 1.794 2.001 1.217 0.689 1.354 !.639

0.000 1.752 1.806 O. 103 0.000 1.294 2.595 3.312 3.501 4.355 5.494

X

THE COMBINED SHEWARTCUSUM SCHEME The Shewhart scheme is better than the CUSUM scheme in quickly detecting large (>3On) shifts in the mean/~; whereas, the CUSUM scheme is usually faster in detecting a small change in # that persists. Bissell (1984) has also shown that the CUSUM scheme is to be preferred when the mean is increasing in a linear time trend. To take advantage of the good properties of both schemes, Lucas (1982) suggested combining the two procedures. This is accomplished by declaring the process out of control if any sample mean is above a specified upper Shewhart limit or if the CUSUM Si is above a specified limit h. To keep a reasonable in-control ARL, Lucas suggests using an upper Shewhart control level of ~ + 4On). Lucas calculated that if this upper Shewhart control limit is used with CUSUM parameter values k = 0.5 and h = 5 , then the in-control ARL is 459 while the out-of-control ARL is 10.4, if the true mean shifted upward by a,,, and only 1.6 if the mean shifted upward by 4tr n. If the nature of the data is such that there are occasional outliers (perhaps due to contamination of samples during handling and processing), Lucas and Crosier (1982) suggest the use of a two-in-a-row rule. That is, require values in two successive samplings above the upper Shewhart control limit before declaring an out-of-control situation based on the Shewhart control limit. MULTIVARIATE QUALITY CONTROL SCHEMES In monitoring ground water at a HWS, the concentrations of several GQP may be measured at each of several monitored wells. If a quality control chart is kept for each parameter, at each well, then one has the problem that while the chance of a false alarm is kept small for each chart, the overall probability of a false alarm becomes large (i.e., the in-control ARL for the whole set of charts may be quite small in spite of large in-control ARL's for each individual chart). This problem is

26

THOMAS H. STARKS AND GEORGE T. FLATMAN

addressed in survey papers on multivariate quality control schemes by Jackson (1985) and Alt (1985). There are two major problems in employing multivariate quality control schemes in ground water monitoring. The first problem is that measurements are expensive and tend to be accumulated slowly, which means that monitoring must proceed for a very long time before a reasonably good estimate of the covariance matrix needed in this procedure can be attained. The second is that out-of-control signals may occur when none of the measurements of GQP is larger than expected. For these reasons, we do not consider multivariate quality control schemes further.

5. Monte Carlo Investigation of Run-Length Distributions In industrial quality control, the period between samplings is typically short (minutes or hours), and the cost of sampling and measurement is inexpensive relative to the value of the product. For this reason, the learning period required to obtain excellent estimates of the process mean quality,/,, the process quality standard error of the mean, an, and the appropriate data transformation is short. After these three items are obtained, it is possible to mathematically determine the distribution of the average run lengths of the control process for in-control and for out-of-control (OOC) situations (Brook and Evans, 1972). In the RCRA hazardous waste site quality control situation, sampling and measurement is expensive, and the time between samplings is long (typically, three or six months). For this reason, quality control decision procedures must be in place before good estimates are available and adjustments must be made in the values used for/J and a n while the quality control process is in operation. This, in turn, implies that one cannot use the usual mathematical procedure to determine the properties of average run lengths. The alternative that is employed in this study is to estimate the distributions of average run lengths by using Monte Carlo procedures. With the Monte-Carlo procedure pseudo-random numbers are generated by a computer to represent measurements (sample means) of GQP at wells around a HWS. A specified quality control procedure is followed and one observes how many simulated sampling periods are passed through before an out-of-control signal is produced. This process is repeated 100 times, and information as to the distribution of the run lengths is observed. For determining the distribution of run lengths in an OOC state, the mean of the random deviates being generated is increased at some randomly chosen point in the process, and the number of sampling periods required to obtain an out-of-control signal after the change is recorded. This process is repeated 100 times to give an empirical distribution of OOC run lengths. The first step in the development of a combined Shewhart-CUSUM quality control scheme is to choose a learning procedure and to determine appropriate parameters U, h, and k, for the control scheme. For the Shewhart-CUSUM scheme, one declares an out of control situation at sampling period i if for the first time, Si=Max[0, zi-k+Si-iI > h , or zi > _ U, where zi is the standardized /th observation (i.e., Z~=(X~-/.t)/trn). In the learning procedure, it is necessary to decide on the number

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27

of sampling periods to use to obtain initial estimates for the target quality value and for the standard deviation of the quality measurements. Next, a decision is needed as to how often and when to update the estimates. Finally the parameters U, h, and k, have to be determined so that the quality control scheme will have good operating characteristics under ideal conditions (i.e., steady state, independence of all measurements, normal distribution or transformation to a normal distribution of GQP measurements from a well under control). Once a procedure that works well under ideal conditions is found the next step is to see how it will work when some of the ideal conditions are absent. As a starting point, it was decided to use U = 4 , h = 5, and k = 0 . 5 in the ShewhartCUSUM scheme. Pseudo-random N(0,1) deviates were generated using the subroutine G G N M L in the International Mathematical and Statistical Library (IMSL) of computer subroutines for F O R T R A N programs. In-control GQP mean values, /a i, were determined for the w wells b y c h o o s i n g w random deviates from a N(0,1) distribution, multiplying each by 0.5, and adding 6 to each such product to obtain w N(6,0.25) deviates. For each well i in each sampling period, j, a pseudo-random N(0,1) deviate was added to the mean value, ~ti, of the GQP for the well to obtain G Q P measurement, Xij, for the sampling period. (This simulated G Q P measurement may represent the sample mean of measurements made on n samples taken from the well at a particular sampling time). The control scheme started on the ninth sampling period and employed the sample means and standard deviations obtained over the first eight simulated sampling periods as surrogates for the true unknown means and standard deviations. If no OOC signal had been obtained, the estimates of the process quality mean and standard error were recalculated after every fourth sampling period up to the 32nd using all available data. Sample means and sample standard deviations were calculated for each well and for the sampling period averages over all wells. That is, for a given contaminant, control charts were kept for each well, and an additional chart was kept for the sampling period G Q P average over wells. If upper control limits were exceeded on any one of these (w + 1) charts, an OOC signal was given. To save computer time the simulation process stopped automatically if the simulation went through 1000 sampling periods (after the eight period learning stage) without an OOC signal. The occurrence of an OOC signal or the completion of sampling period 1000 marked the completion of a trial. With w = 4 wells, 100 trials gave a median run length to a false positive of 76.5 and a sample mean run length of 160.8. None of the 100 trials ran through the possible 1000 sampling periods. Since these average run lengths were considered too small, h was changed from 5 to 5.5, and the simulation tried again. The median in-control run length increased to 144.5, and the (Winsorized) sample mean increased to 242, with 5 trials running the full 1000 possible sampling periods. (Sample mean is 'Winsorized' in the sense that trials that would have reported more than 1000 sampling periods if the simulation process had been allowed to continue were reported as 1000 in calculating the mean). While this modification was an improvement, results were not as good as hoped for, and it also had a bad effect on average run length in OOC situations.

28

THOMAS H. STARKS AND GEORGE T. FLATMAN 1 Read m, w, Xinc, L, Dseed [

ABBREVIATIONS re=number of wells intercepted by plume w = number of wells at HWS X~nc = increase in mean value of GQP when plume intercepts a well L =number of samplings in learning period Dseea = seed number for random number generator

/

For each well i, randomly choose mean value: #i=6+0.5 * random N(0,1) deviate

I START LEARNING PERIOD For each of the w wells i, and for each of the L sampling periods j generate a GQP measurement: Xiy =/~ + random N(0,1) deviate, then calculate fi,= ~ . X J L , fi= F, ~ Xij/(L*w), and d = 4 [ ~ [ ~ . X 2 u - ( ~ X e ) 2 / L ] / [ ( L - l)*w]l END OF LEARNING PERIOD

I START OF MONITORING If Xi~r > 0, generate a random integer S between 1 and 48, inclusive, from a Uniform distribution to be the number of the first sampling period, after the learning period, in which a plume has intercepted m well(s).

I

If j = 5 , 9, 13, 21, or 33, recalculate ~,/2, and d using all data obtained prior to this sampling

I I

I Generate GQP measurements for each of the w wells: I Xi-'=~i+ random N(0,1) deviate if Xi~=0, o r j < S or i>m, ~=#~+Xinc+random N(0,1) deviate

if Xi~o>O,j~_S, and i ~ m

I For any well i, is [ Z, = (X, - 12i)lO>- U or Sj = Max[O, Sj_, - k + zi} > 5

Z"=(X- fi)(~/w)/d>_U or Tj= Max[O, Tj- - I

-- k + Z I i~" 5 where U=4.5 and k = l i f j < 1 2 , and U = 4 and k=0.75 if j > 127

NO -NO~

YES

I Is J = 1000? ~ ' - ~ YES

I~-~

Print 'run length =j' ]

I Print 'run length= 1000'

I - - N O 4

[ Isn=100?

/

.AYES

I Calculate sample statistics on 100 run lengths [

Fig. 3. Schematic of Monte Carlo simulation under ideal conditions.

RCRA GROUND-WATER MONITORING DECISION PROCEDURES

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For this reason, h was returned to the original value 5, and k was increased from 0.5 to 0.75. This had a more dramatic effect. The median in-control run length increased to 619.5 and the (Winsorized) sample mean increased to 577. There was no false-positive signal over the 1000 sampling periods on 35 of the 100 trials. While the last mentioned procedure has some good in-control characteristics, it still has some problems such as many short in-control run lengths, and many long OOC run lengths. Additional modifications were tried until a procedure was arrived at that seemed a good compromise. Under this compromise, U = 4.5, h = 5, and k = 1 for the first 12 sampling periods after the eight period learning stage, and then U is reduced to 4.0, k is reduced to 0.75, and these two parameters are held at these values for all subsequent periods. Adjustments in sample means and sample standard deviations are made after sampling periods 4, 8, 12, 20, and 32, following the learning stage. Figure 3 gives a schematic of this Monte Carlo simulation procedure. Now, for w = 4 wells, in two separate simulation runs, the Monte Carlo procedure gave median in-control run lengths of 502 and 637.5, and (Winsorized) sample-mean run lengths of 550.5 and 589.6. No OOC signals were given on 33 trials of the 100 trials on one simulation run and on 41 trials on the other. While these averages are no better than those obtained earlier, there were far fewer very short in-control run lengths, and better results were obtained in OOC situations. As the number of wells at a site is increased, the number of control charts that can give a signal increases, and thereby, the average of in-control run lengths decreases. For eight wells, the median in-control run length over 100 trials was 228 on one simulation run and 249.5 on the other, and the (Winsorized) sample means were 342.5 and 385.5. Information on in-control lengths is given in Table II. Under the ideal conditions and the procedure specified in the preceding paragraph, OOC situations were investigated and the OOC run length (number of simulated sampling periods until OOC signal after increase in one or more well G Q P means) empirical distributions were obtained. These OOC situations involved m wells (out of the w wells being monitored) whose mean contaminant concentrations were increased by r standard deviations at a randomly selected period (from 1 to 48, with equal probabilities) after the learning stage (see Figure 3). The results of these OOC

TABLE II In-control run-length statistics for w wells under ideal conditions (N -- 100 trials) w

Median

Mean a

# ~