-------
IB W- DJ
6(1-6)
- S)exp(2u + o2)(exp(o2) -
• - exp(u + O.So2)
(2.13)
where the D; equals individual detection limits for the non-detects, the 6; are the corresponding proportions of
not detected values with detection limit Di; and 6 = S8;.
2.4.2.1 Estimation of Long-Term Averages
Long-term averages were calculated for each sampling episode sample location separately. For the purposes
of estimating these long-term averages (equal to the expected value in the equation (2.14)),it was necessary to
divide the IWC data sets into two groups based on their size (number of samples) and the type of samples in
the subset Thus, the computations differed for each group:
Group 1: Less than 2 detected samples (NC) or less than 4 total samples.
Group 2: Two or more non-censored samples (NC) and 4 or more total samples.
For Group 1, the long-term averages were calculated as the arithmetic average of the samples, since the sample
sizes for either the discrete portion or the continuous lognormal portion of the data were too small to allow
distributional assumptions to be made. Specifically, Group 1 contained all data subsets with all non-detects
or only one detect. Detection limits were substituted as the values associated with non-detectable samples.
For Group 2, the long-term averages were calculated using the formula for E(U) in equation (2.14). p. and a
parameters were estimated as the mean and variance of the logged NC values.
Appendix!? presents summary statistics by analyte for each option and sampling episode combination.
2.4.2.2 Estimation of Variability Factors, Percentiles, and Limitations
After determining estimated long-term average values for each pollutant for each sample point location, EPA
developed 1-day variability factors (VF1) for each pollutant and either 4-day or 20-day monthly average
variability factors (VF4 and VF20) dependent on the assumed frequency of monitoring as outlined in Table
2-3. Appendix C presents estimated daily maximum limitations, monthly average limitations, and the
associated variability factors for each option that are calculated using pollutant-specific variability factors. The
estimation methodology is presented below.
2-9
-------
Table 2-3.
Assumed Monitoring Frequencies
Pollutant Category
Metals
COD
TDS
TSS
Frequency of Monitoring
Monthly (VF1.VF4)
Monthly (VF1, VF4)
Weekly (VF1, VF4)
Daily (VF1, W20)
Similar to the calculations for the long-term averages, the data were divided into the same two computation
groups based on the number and type of samples in each data subset:
Group 1:
Group 2:
Less than 2 detected (NC) samples or less than 4 total samples. Upper percentiles
and variability factors could not be computed using the modified delta-lognormal
methodology.
Two or more non-censored samples (NC) and 4 or more total samples. The estimates
of the parameters for the modified delta-lognormal distribution of the data were
calculated empirically in the log-domain. Upper percentiles and variability factors
were calculated using these estimated parameters.
2.4.2.2.1 Estimation of Facility-Specific 1-Day Variability Factors and 99th Percentiles
The 1-day variability factors are a function of the long-term average, E(U), and the 99th percentile. An iterative
approach was used in finding the 99th percentile of each data subset using the modified delta-lognormal
methodology by first defining D0=0, 80=0, and D^ = » as boundary conditions where Df equals half of the i"1
smallest detection limit, and 6; is the associated proportion of non-detects at the Ith detection limit. A cumulative
distribution function, p, for each data subset was computed as a step function ranging from 0 to 1. The general
form, for a given value c, is:
P -
(1 - 8)
D
c <
, m=0,l,...k
(2.14)
where $ is the standard normal cumulative distribution function. The following steps were completed to
compute the estimated 99th percentile of each data subset:
1. k values of p at c=Dm, m=l,...k were computed and labeled pm.
2. The smallest value of m, such mat pm :> 0.99, was determined and labeled as p,-. If no such m existed,
steps 3 and 4 were skipped and step 5 was computed instead.
3. Computed p* = p,- - 8j.
2-10
-------
4. If p* < 0.99, then P99 = DJ5
else if p* ;> 0.99, then
P99 = exp
ji + <£~1
'( J-1 }]'
0.99 -£oJ
1 1-0 ) &
(i - o) f.
(2.15)
5. If no such m exists, such that pm ;> 0.99 (m=l,...k), then
= exp
F.-1
0.99 - 8
(1-5)
The daily variability factor, VF1, was then calculated as
(2.16)
P99
E(U)
(2.17)
Appendix C displays long-term averages and 1-day variability factors by analyte for each option and sampling
episode combination.
2.4.2.2.2 Estimation of Facility-Specific 4-Day Variability Factors and 95th Percentiles of 4-Day Means
For all but TSS, it was necessary to calculate a variability factor for monthly averages based on the distribution
of 4-day averages, because EPA is considering proposing that these pollutants be monitored weekly
(approximately four times a month). In order to calculate the 4-day variability factor (VF4), the assumption
was made that the approximating distribution of U4, the sample mean for a random sample of four independent
concentrations, is also derived from this modified delta-Iognormal distribution, with the same mean as the
distribution of the concentrations. The mean of this distribution of 4-day averages is:
(2.18)
where (X4)D denotes the mean of the discrete portion of the distribution of the average of four independent
concentrations, (i.e., when all observations are not detected) and (X4)c denotes the mean of the continuous
lognormal portion of the distribution.
First, it is assumed that the probability of detection (6) on each of the 4 days is independent of that on the other
days, since the samples to be used for compliance monitoring are not taken on consecutive days, and no
correlation is expected to exist such that 84 = S4.
2-11
-------
Also, since
en
/= 1 o
and since E(tJ4) = E(U), then
= lo§
d-s4)
- 0.502,
The expression for o24 was derived from the following relationship:
(2.19)
(2.20)
(2.21)
Since
, and
then,
4
This further simplifies to:
k k
VarCC/4) =
4S2
W
(1 - 84)exp(2n4
and furthermore
k k
- 62(1 - 64)
/"= 1
(l-84)exp(2n4+o24)
- 1]
O.So2,)
(2.22)
(2.23)
(2.24)
(2.25)
2-12
-------
Then from (2.21) above,
exp(u4+ O.
(1-64)
d-S4)
— , since E(UJ = E(U)
(2.26)
and letting
il = E(U) - 63 ££,-£,, then, exp(fi4+ O.
Furthermore,
1
k k
(1-8)
(1-64)2
(2.27)
(2.28)
Since Var(U4) = Var(U)/4 and by rearranging terms,
0*4 = log
- 64) Var(U)
(1-
4tf
o2
I- 8rj
(2.29)
Thus, estimates of u4 and a4 were derived by using estimates of 81;...8k (sample proportion of non-detects at
observed detection limits D^—D^, u (mean of logged values), and a2 (MLE log variance) in the equations
above.
In finding the estimated 95* percentile of the average of four observations, four non-detects (not all at the same
detection limit) can generate an average that is not necessarily equal to DI, D2,...s or Dk. Consequently more
than k discrete points exist in the distribution of the 4-day averages. For example, the average of four non-
detects at k=2 detection limits, are at the following discrete points with the associated probabilities:
i Df, 8%
1
2
3
4
5
48,%
(2D.,+2D2y4
D
2-13
-------
In general, when all four observations are not detected, and when k detection limits exist, the multinomial
distribution can be used to determine associated probabilities, that is,
Pr
4!
(2.30)
The number of possible discrete points, k*, for k=l,2,3,4, and 5 are given below:
k
1
2
3
4
5
JL
1
5
15
35
70
To find the estimated 95th percentile of the distribution of the average of four observations, the same basic steps
(described in Section 2.4.2.2.1) as used for the 99th percentile of the distribution of daily observations were
followed with the following changes:
1. Change P^ to P9S, and 0.99 to 0.95.
2. Change Dm to Dm*, the weighted averages of the detection limits.
3. Change 6; to 8;*.
4. Change k to k*, the number of possible discrete points based on k detection limits.
5. Change the estimates of 6, n, and a to estimates of 84, jj.4, and a4, respectively.
Then, the estimate of the 95th percentile 4-day mean variability factor is:
F95
VF4 =
E(U)'
since
= E(U).
(2.31)
Appendix C displays long-term averages and 4-day variability factors by analyte for each option and sampling
episode combination.
2.4.2.2.3 Estimation of Facility-Specific 20-Day Variability Factors and 95th Percentiles of 20-Day Means
Since TSS is proposed to be monitored daily, the monthly average limitation was based on 20 days of sampling.
However, the data used to calculate the 20-day,variability factors for TSS cover only 5 days of sampling of
daily measurements. Therefore, at this time EPA does not have sufficient data to examine in detail and
incorporate any autocorrelation between concentrations for TSS measured on adjacent days. The
autocorrelation of TSS is further discussed in the preamble to the proposed regulation.
It is assumed that the concentrations for TSS are independent of one another, and
E(1720) = E(0) and
(2.32)
2-14
-------
where E(U) and V(U) are calculated as in equations (2.3) and (2.4). Finally, since U20 is approximately
normally distributed by the Central Limit Theorem, the estimate of the 95th percentile of a 20-day mean and
the corresponding 20-day average variability factor (VF20) are approximately
P9520 = E(U20)
(2.33)
and
VF20 =
20
P95.
20
E(U20)
(2.34)
where $"'(0.95) is the 95th quantile of the standard normal distribution.
As noted in Table 2-3, EPA assumed 20-day variability factors for TSS. See Appendix C for the TSS 20-day
facility-specific variability factors.
2-15
-------
-------
3. Estimation of Pollutant-Specific and Group-Level Variability Factors Resulting in Proposed Daily
Maximum and Monthly Average Numerical Limitations
This chapter describes the estimation of variability factors by pollutant ("pollutant-specific") and by group
("group-level"). Each group contained pollutants that were chemically similar. The pollutant-specific and
group-level variability factors were then used to develop limitations.
3.1 Estimation of Pollutant-Specific Variability Factors
After the facility-specific variability factors were estimated for a pollutant, the pollutant-specific variability
factor was calculated. The pollutant-specific daily variability factor was the mean of the facility-specific daily
variability factors for that pollutant in the option. Likewise, the pollutant-specific monthly variability factor
was the mean of the facility-specific monthly variability factors for that pollutant in the option. Appendix D
displays the pollutant-specific long-term averages and variability factors calculated as described above.
3.2 Estimation of Group-Level Variability Factors
After the pollutant-specific variability factors were estimated as described in section 2.1, group-level variability
factors were calculated for metals. These metal pollutants were considered to be chemically similar.
The group-level daily variability factor was the median of the pollutant-specific daily variability factors for
the pollutants within the group. Similarly, for the monthly variability factors, the group-level monthly
variability factor was the median of the pollutant-specific monthly variability factors for the pollutants within
the group. Appendix E displays the group-level long-term averages and variability factors calculated as
described above.
3.3 Estimation of Potential Daily Maximum and Monthly Average Limitations
For metals, potential daily maximum and monthly average limitations for each pollutant within each option
were set equal to the product of the pollutant-specific long-term average and the option group-level variability
factor. Appendix F presents potential daily maximum and monthly average limitations for each option that are
calculated using group-level variability factors.
3-1
-------
-------
APPENDIX A
RAW DATA LISTINGS
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APPENDIX C
FACILITY-SPECIFIC LONG-TERM AVERAGES,
VARIABILITY FACTORS, AND POTENTIAL LIMITATIONS
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-------
-------
APPENDIXD
POLLUTANT-SPECIFIC LONG-TERM AVERAGES
AND VARIABILITY FACTORS
-------
-------
Appendix D. Pollutant-Specific Long-term Averages
and Variability Factors
Estimates calculated as Mean of Facility-specific Results. No Imputation Performed
SCC Data Only
Option
A
A
Category
Classicals
Metals
B
Classicals
B
Metals
Analyte
COD
TDS
TSS
ALUMINUM
ANTIMONY
ARSENIC
BORON
CADMIUM
COPPER
IRON
MANGANESE
MOLYBDENUM
SELENIUM
TIN
TITANIUM
ZINC
COD
TDS
TSS
ALUMINUM
ANTIMONY
ARSENIC
BORON
CADMIUM
COPPER
IRON
MANGANESE
MOLYBDENUM
SELENIUM
TITANIUM
ZINC
Cas_NO
C-004
C-010
C-009
7429905
7440360
7440382
7440428
7440439
7440508
7439896
7439965
7439987
7782492
7440315
7440326
7440666
C-004
C-010
C-009
7429905
7440360
7440382
7440428
7440439
7440508
7439896
7439965
7439987
7782492
7440326
7440666
Long Term
Average (ug/i)
306,000.0
37,000,000.0
14,300.0
198.0
382.0
9.52
1,710.0
62.3
19.6
2,030.0
518.0
579.0
53.5
33.2
4.03
122.0
351,000.0
38,200,000.0
5,840.0
161.0
347.0
8.27
1,730.0
22.0
10.3
130.0
545.0
581.0
26.7
7.38
24.3
Daily
VF
11.9
1.48
4.10
1.70
1.34
3.39
1.23
7.76
3.49
3.40
1.17
1.31
4.95
1.73
3.25
2.03
11.4
1.26
4.16
1.62
1.48
2.01
1.13
6.20
2.18
2.08
1.16
1.36
2.93
5.99
2.19
Monthly
VF
3.55
1.15
1.31
1.21
1.11
1.81
1.08
2.57
1.64
1.62
1.06
1.10
1.95
1.19
1.53
1.30
3.44
1.08
.1.28
1.19
1.15
1.96
1.04
2.24
1.28
1.31
1.05
1.12
1.46
2.16
1.45
For TSS, the monthly variability factors are estimated assuming 20 days of sampling.
For all other pollutants, monthly variability factors are estimated assuming 4 days of sampling.
D-l
-------
-------
APPENDIX E
GROUP-LEVEL VARIABILITY FACTORS
-------
-------
Appendix E. Group-level Variability Factors
SCC Data Only
Option
A
B
Category
Metals
Metals
Number of
Pollutants in
Each Group
13
12
Daily
VF
2.03
2.05
Monthly
VF
1.30
1.30
For Metals, monthly variability factors are estimated assuming 4 days of sampling.
Option A Metal Group defined as Al, Sb, As, B, Cd, Cr*, Cu, Fe, Pb*, Mn, Hg*, Mo, Se, Ag*, Sn*, Ti, Zn
Option B Metal Group defined as Al, Sb, As, B, Cd, Cr*, Cu, Fe, Pb*, Mn, Hg*, Mo, Se, Ag*, Sn*, Ti, Zn
*Implies Delta-Lognormal Estimation Criteria Not Met
E-l
-------
-------
APPENDIXF
POLLUTANT-SPECIFIC LONG-TERM AVERAGES
AND POTENTIAL LIMITATIONS
-------
-------
Appendix F. Pollutant-Specific Long-Term Averages and Limitations
Potential Limitations for Metals are Imputed Based on Group-level Variability Factors within Option
SCC Data Only
Option Category
A Metals
B Metals
Analyte
ALUMINUM
ANTIMONY
ARSENIC
BORON
CADMIUM
CHROMIUM
COPPER
IRON
LEAD
MANGANESE
MERCURY
MOLYBDENUM
SELENIUM
SILVER
TIN
TITANIUM
ZINC
ALUMINUM
ANTIMONY
ARSENIC
BORON
CADMIUM
CHROMIUM
COPPER
IRON
LEAD
MANGANESE
MERCURY
MOLYBDENUM
SELENIUM
SILVER
TIN
TITANIUM
ZINC
Cas_NO
7429905
7440360
7440382
7440428
7440439
7440473
7440508
7439896
7439921
7439965
7439976
7439987
7782492
7440224
7440315
7440326
7440666
7429905
7440360
7440382
7440428
7440439
7440473
7440508
7439896
7439921
7439965
7439976
7439987
7782492
7440224
7440315
7440326
7440666
Estimated
Long Term
Average (ug/i)
198.0
382.0
9.52
1,710.0
62.3
10.0
19.6
2,030.0
47.7
518.0
2.64
579.0
53.5
9.49
33.2
4.03
122.0
161.0
347.0
8.27
1,730.0
22.0
10.0
10.3
130.0
46.8
545.0
2.00
581.0
26.7
5.00
31.5
7.38
24.3
Potential
Daily
Maximum Limit
(ug/l)
401.0
775.0
19.3
3,460.0
127.0
20.3
39.8
4,120.0
96.8
1,050.0
5.36
1,180.0
109.0
19.3
67.3
8.18
248.0
330.0
709.0
16.9
3,540.0
45.1
20.5
21.0
267.0
95.7
1,120.0
4.09
1,190.0
54.6
10.2
64.4
15.1
49.8
Potential
Monthly
Average Limit (ug/i)
257.0
496.0
12.4
2,220.0
81.0
13.0
25.5
2,640.0
62.0
673.0
3.43
753.0
69.5
12.3
43.1
5.24
159.0
209.0
449.0
10.7
2,240.0
28.5
13.0
13.3
169.0
60.6
706.0
2.59
753.0
34.6
6.48
40.8
9.56
31.5
For Metals, monthly variability factors are estimated assuming 4 days of sampling.
Option A Metal Group defined as Al, Sb, As, B, Cd, Cr*, Cu, Fe, Pb*, Mn, Hg*, Mo, Se, Ag*, Sn, Ti, Zn
Option B Metal Group defined as Al, Sb, As, B, Cd, Cr*, Cu, Fe, Pb*, Mn, Hg*, Mo, Se, Ag*, Sn*, Ti, Zn
*Implies Delta-Lognormal Estimation Criteria Not Met
F-l
-------
-------