20170804IRP Exhibit 2.pdf

Exhibit No. 2
Historical Tem perature Glimate Report
Historica! Temperature Climate Report
Prepared for
lntermountain Gas Company
May 9,2OL7
By RussellJ. Qualls, Ph.D., P.E.
Climate Consultant
Historica! Temperature Climate Report
Prepared for lntermountain Gas Company
By RussellJ. Qualls, Ph.D., P.E.
!NTRODUCTION
This report provides estimates of design air temperature values that are likely to be equaled or
exceeded (in a "colder than" sense) in a year, with specified probabilities of occurrence and
with specified average return periods. The estimates are made for monthly and annual daily-
average temperatures, and for annual minimum daily average temperatures, at seven locations
in Southern ldaho used by lntermountain Gas Company (lGC) in its gas supply and storage
calculations. The estimated values are intended to assist IGC in developing its lntegrated
Resource Plan (lRP).
This report arose out of discussions with representatives of IGC regarding what information was
most needed for development of the lRPs, and provides an update to similar earlier reports
(Qualls, 2OO7; Molnau, 1994). Each of these reports relied upon probability distributions
generated from historical temperature values measured at or near the seven Southern ldaho
IGC calculation locations, and included the Normal Distribution ("NORM"; symmetric bell-
shaped distribution) and the Pearson Type lll Distribution ("Plll"; a skewed distribution which
can represent asymmetric data and converges to the Normal distribution for symmetric data).
Selecting design temperatures from values generated by these probability distributions is
preferable over using individual observations, such as the coldest observed daily average
temperature, because exceedance probabilities corresponding to values obtained from the
probability distributions are known. This enables IGC to choose a design temperature, from
among a range of values, which corresponds to an exceedance probability that IGC considers
appropriate for the intended use.
Each successive report incorporates temperature data which occurred and was measured after
the completion of the earlier reports. ln addition, this report includes temperature data from
as much of the entire Period Of Record (POR) at each location as was deemed reliable. This
extends the data pool for each location backward in time, making each dataset much larger
(i.e., covering a longer time period)than in the preceding reports. This has some statistical
advantages. First, it allows one to assess with greater confidence how well a particular
distribution represents the observed data, and secondly, it generally narrows the range of
uncertainty associated with a given probabilistic temperature value. This report includes
additional analyses, not included in the earlier reports, which quantify the goodness-of-fit of
each probability distribution and the range of uncertainty of each estimated value. These
additional analyses include:
t
1) Running hypothesis tests on each probability distribution fitted, to assess whether it
should be accepted as a good descriptor of the data (Chi-Squared Test)
2l Calculation of the 90% confidence interval for each probabilistic temperature estimate.
There is a 90% probability that the endpoints of the confidence interval, known as upper
and lower confidence limits, encompass the true probabilistic temperature value.
Further discussion of these additional analyses is provided in Appendix B.
The contents of this report may be compared with lntermountain Gas Company's
lntegrated Resource Plan (lRP)to estimate probabilities associated with design values
presented there.
DATA
The data used in this report were either provided by Lori Blattner of lGC, or obtained directly
from the National Centers for Environmental lnformation (NCEl, formerly National Climate Data
Center, NCDC). The data consist of daily observed maximum and minimum temperatures,
and/or daily averages calculated as the mean of the daily maximum and minimum values.
Table 1 provides the IGC Zone lD, location name, and starting Water Year for the data. A Water
Year (WY) begins on October 1st, and ends on September 30th of the following year, and is
numbered by the year in which it ends. That is, the 1905 Water Year for Caldwell begins on
October L, L9O4 and ends on September 30, 1905. A Water Year groups all winter months of a
particular season together. The analysis for each station extends to the end of the 2015 Water
Year (September 30, 2015).
Table 1: Weather Station Zones, Locations, and Starting Water Year (WY)
Zone lD Location Starting WY
3s0 Caldwell 1905
450 Boise L94L
500 Hailey 1909
600 Twin Falls 1906
700 Rexburg 1908
750 ldaho Falls L949
800 Pocatello 1939
Most long-term weather stations include occasional changes such as instrument replacements
or changes, or station moves. The data used in this analysis span these changes. Some of these
changes have occurred in the past 30 years, so the Molnau (1994) and Qualls (2007) reports,
and the data currently used by IGC have some of these changes embedded in them, as would
the current analysis even if it was limited to the past 30 years.
2
RESUTTS
For each IGC location, results calculated over the POR at each station from data aggregated at
the annualtime scale are presented in this section, and results with greater detail including
monthly analyses and additional figures are presented in the appendices. "Annual" in this
report refers to a Water Year. Table 2 presents POR summaries and statistics of station data
and values of exceedance temperatures for annual mean daily average temperatures and
annual minimum daily average temperatures. Exceedance temperatures are presented for a
range of return periods lT=2,5, LO,20,50 and 100 years) and their corresponding exceedance
probabilities, calculated byfitting both Normal (NORM)and Pearson Type lll (Plll) distributions
to observed data from each IGC location.
POR summaries and statistics of the data are presented in the top third of Table 2. ln this
section, the statistics are calculated from the annual values at a given station over the number
of years available for that station. For the annual mean daily average temperature shown in
the left half of Table 2, the POR mean at each station ranges from a low value of 43 "F (Station
3
Table 2: Annual Station data and exceedance temperatures based on NORM and Plll Distributions ('F)
Annual Mean Daily Average Temperature
Station
Mean
Std Dev
Skew
Max
Min
No Years
3s0 450 s00 600 700 750 800 3s0 450 s00 600 700 750 800
51
1.6
0.2
55
48
TLL
52
1.6
0.0
56
48
75
43
1.5
0.3
49
39
LO7
49
L.7
0.4
54
45
110
43
1.8
0.1
48
38
108
44
1..7
-0.4
48
39
67
47
L.4
-0.1
50
43
77
10
9.7
-0.5
28
-18
LLL
10
8.6
-0.6
24
-16
75
-L
7.2
-0.4
13
-23
LO7
6
8.4
-0.3
24
-15
110
-6
8.1
-o.2
t6
-27
108
-5
7.8
-0.3
10
-23
67
0
8.0
-0.3
15
-18
77
T P Norm Distributed Exceedance Temperatures Norm Distributed Exceedance Temperatures
2 0.5 51
50
49
49
48
48
52
50
49
49
48
48
43
42
4L
4L
40
40
49
47
47
46
45
45
43
47
40
40
39
38
44
43
42
4L
4L
40
47
46
45
44
44
44
10
L
-3
-5
-10
-13
10
3
-L
-4
-8
-10
-1
-7
-10
-13
-1,6
-18
6
-1.
-4
-7
-tL
-13
-6
-13
-16
-19
-23
-25
-5
-77
-15
-18
-21,
-23
0
-7
-10
-13
-16
-19
5 0.2
10 0.1
20 0.0s
50 0.02
100 0.01
T P Plll Distributed Exceedance Temperatures Plll Distributed Exceedance Temperatures
2 0.5 51
50
49
49
48
48
52
50
49
49
48
48
43
42
4L
4L
40
40
49
47
47
46
46
45
43
4L
40
40
39
39
44
43
42
4L
40
40
47
46
45
44
44
43
10
2
-3
-8
-13
-16
11
3
-L
-5
-10
-L4
-1
-7
-tL
-L4
-L7
-20
7
-L
-5
-8
-12
-15
-6
-13
-L7
-20
-23
-26
-4
-LL
-15
-18
-22
-25
0
-7
-10
-14
-77
-20
5 0.2
10 0.1
20 0.05
50 0.02
100 0.01
Annual Minimum Daily Average Temperature
700) to high value of 52 "F (Station 450) across the different locations. This is shown in the first
row below the station number, labeled "Mean". ln the second row below the station number,
the relatively small standard deviation shows that the collection of annual mean temperatures
do not spread out very far around the POR mean at each station. This can also be seen in the
relatively small difference between the Max and Min values in the fourth and fifth rows below
the station numbers, which represent the largest and smallest annual mean daily average
temperature for each station. The Max and Min values differ by no more than L0 "F at any of
the stations.
Because the spread of the annual mean daily average temperatures is small at each station, the
exceedance temperatures for different return periods also fall within a fairly narrow range, as
shown for the normal distribution in the left half of the middle third of Table 2. For example, at
station 350, the two-year return period exceedance temperature is 51 "F and the 100-year
return period exceedance temperature is only 3 "F colder at 48 "F. Furthermore, the asymmetry
is small as shown by the small value of the skew coefficients for the annual mean daily average
temperatures, in the third row below the stations numbers in Table 2. As a result, the Plll
distribution nearly converges to the normal distribution and in most cases they give the same
result for each return period (e.g., the 100-year return period event is the same or only slightly
different between the NORM and Plll distributions for a given station; compare left half of
middle and lower thirds of Table 2).
ln addition, the small spread of the annual mean daily average temperatures indicates that the
calculated exceedance values can be accepted with high certainty. Ninety percent confidence
intervals have been calculated for the different return period exceedance temperatures at each
station and for both the Norm and Plll distributions. Upper and lower Confidence Limits (CL
values) are given in tables in Appendix A. For the annual mean daily average temperatures,
Figure 1 shows the values of the 100-year exceedance temperature and their upper and lower
CLs for the NORM and Plll distributions at each station.
The range of the CLs around the 100-year return period value at each station is small, less than
1 "F at most stations, and the results are similar between the Norm and Plll distributions for a
given station. The CLs are even smaller for exceedance temperatures with shorter return
periods than 100 years. Thus, the exceedance temperatures for the annual mean daily
temperatures given in Table 2 can be accepted with a high degree of confidence. The annual
minimum daily average temperatures, summarized in the right half of Table 2, have a much
higher degree of variability. This is expected because these data and statistics come from the
single lowest daily average temperature observation from each year, in contrast to the annual
mean daily average temperatures discussed above which are comprised of the mean of a full
year's worth of daily values. Averaging, as in the case of the annual mean temperatures,
4
centralizes the results and reduces variability. The difference between the behavior of the
annual minimum and annual meon daily average temperatures is clearly visible in the time
series plots of each of these variables for each station shown in Appendix A. The upper line is
the annual mean and the lower line is the annual minimum over the POR for a given station.
The greater variability of the annual minimums is readily apparent.
Summary statistics for the annual minimum daily average temperatures given in the left half of
Table 2 reflect its greater variability. The standard deviation of the annual minimums is much
larger than that for the annual means, and many of the stations exhibit large negative skew.
The large standard deviation indicates that the uncertainty of the exceedance values of annual
minimum temperatures is larger than it was for the annual mean temperatures, consequently
the range of the confidence limits for the minimum temperatures is much wider. This is shown
in Figure 2 for the 100-year return period minimum temperatures, where the CLs span a range
from 3.5 to nearly 7 "F. The large negative skew suggests that the Pllt distribution should
provide a more realistic estimate of the minimum annual daily average exceedance
temperatures than would the NORM distribution. Results of Chi-Square tests of the "goodness-
of-fit" of the NORM and Plll distributions for each station generally confirm this as discussed in
Appendix B.
For the T-year return period exceedance values of annual minimum daily average temperature,
the Plll distribution provides better estimates than the NORM distribution, and the true values
5
Figure 1. 100-Year Annual Mean Daily Average Temperature
lro
o
=o
o0,
Eo
50.0
48.0
46.0
44.0
42.0
40.0
38.0
36.0
f
Norm Pilt Norm Pilt Norm Pilt Norm Plll Norm Pilt Norm Plil Norm Pilt
- Upper CL
- Lower CL
. T-Yr Temp
350 450 500 600 700 750 800
Station and Probability Distribution
Figure 2. 100-Year Annual Minimum Daily Average Temperature
-5.0
II
o,
o
oe
Eo
-10.0
-15.0
-20.o
-2s.0
-30.0
- Upper CL
- Lower CL
. T-Yr Temp
zoa3
!zoI3
!
=
zo
3
!zo
3
T'=zo
3
T'=zoa3 =
zo
3
!
=
3s0 450 500 600 700 750 800
Station and Probability Distribution
of the exceedance temperatures could be several degrees larger or smaller than the values
provided in Table 2 owing to the inherent uncertainty of this variable.
Although estimated annual Tavg exceedance temperatures are very similar between the Norm
and Plll distribution, for monthly Tavg and annual minTavg exceedance temperatures there are
a number of cases where the Plll distribution represents the data better than does the Norm
distribution and where the exceedance temperatures and confidence limits differ between the
two distributions. For this reason, to simplify the selection process, it is recommended that the
Plll exceedance temperatures be used in general. Where both distributions provide similar
results it doesn't matter, and where they differ, Plll is usually more representative of the data.
Routinely using the Plll values will avoid accidentally using the Norm exceedance temperatures
when they are not representative of the observations. Further explanation is provided in
Appendix B.
Multi-Year Time Horizon Probabilities
Probabilities of equaling or exceeding an event at least once during a multi-year period can be
calculated based on the return period, T. Average Return Period T and annual exceedance
probability have a reciprocal relationship , P=LfT. The exceedance probabilities, P, correspond
to the likelihood of observing temperatures less than or equal to the indicated value in any
single yeor.ln order to apply these numbers over a multi-yeor time horizon, one should
6
calculate the probability P1 that the temperature will be less than the specified threshold at
least once during the J-year period. P1 ma! be calculated as P1 = (1-(1-P)r). Values of P1 for J
equalto 5, 10, and 15 years are given in Table 3.
The single-year exceedance probability of 0.033 which appears in the third row up from the
bottom is the approximate exceedance probability corresponding to using the coldest day
observed in a T=thirty year period as the peak design day. Thus, the likelihood that a
temperature colder than the 0.033 or 3.3% exceedance temperature will be observed at least
once in the next five years is 0.15 or L6%o, as shown in the S-year (J=5) column . Similarly, there
is only a 5% chance of having a temperature occur at least once that is colder than the P=0.01
exceedance temperature (i.e., the T=100 year event)within the a 5-year span.
SUMMARY
T-year exceedance temperature values for monthly and annual daily average Temperatures
(Tavg) and annual minimum daily average temperatures (minTavg) have been estimated for
seven stations across Southern ldaho by fitting Normal and Plll distributions to long-term
observations from the stations. Results for annual Tavg and minTavg are presented in Table 2
and Figures 1 and 2 above and in Appendix A, which also includes results for monthly Tavg.
7
Table 3. Multi-Year Exceedance Probabilities corresponding to different time horizons (J=5,
10, and 15 years) for different values of single-year exceedance probability.
Single-Year Multi-Year Exceedance Probabilities, PJ
T P J=5 J=10 J=L5
2
5
10
20
30
50
100
0.5
o.2
0.1
0.05
0.033
0.02
0.01
0.97
0.67
o.4L
0.23
0.16
0.L0
0.0s
0.999
0.89
0.6s
0.40
o.29
0.1.8
0.L0
0.99997
0.96
o.79
0.54
0.40
o.26
o.t4
For annual Tavg exceedance temperatures, results are similar between the Norm and Plll
distribution estimates. For monthly Tavg and annual minTavg, results are often substantially
different owing primarily to skew in the distribution of the data so that the Plll distribution
provides a superior estimate of exceedance temperatures. The 12 (Chi-Squared) test applied to
the results generally confirms that Plll is as good as or better than the Norm distribution.
Because Plll usually provides a better estimate when results differ between the Norm and Plll
distributions, it is recommended to use the Plll results from this report. When the Norm and
Plll estimates are similar, this produces no negative consequences. However, when they differ,
routinely using the Plll results avoids accidentally using the less accurate results from the Norm
distribution.
A measure of uncertainty of the exceedance temperatures is given by 90% confidence limits in
Table 2 and Figures L and 2 above, and in tables given for each station in Appendix A.
8
REFERENCES CITED
Benjamin, J.R. and C.A. Cornell, Probobility, Stotistics ond Decision, for Civil Engineers, McGraw-
Hill, New York, 1970.
Chow, V.T., D.R. Maidment, and L. W. Mays, Applied Hydrology, McGraw-Hill, 1988.
Devore, J.L., Probobility ond Stotistics for Engineering ond the Sciences,znd Ed., Brooks/Cole,
Monterey, CA, 1987.
Haan, C.T., Sfotisticol Methods in Hydrology, lowa State University Press, Ames, lowa, L977
lnteragency Advisory Committee on Water Data (now combined into Advisory Committee on
Water lnformationl, Guidelines for determining flood flow frequency, Bulletin 778,
http ://water. usss.eov/osw/bu I I eti n 17blbu I I etin 178. htm l. 198 1.
Lapin, 1.1., Probability and Statistics for Modern Engineering, Brooks/Cole, Monterey, CA, 1983.
Pearson, E.S. and H.O. Hartley (eds.), The Biometrica Tables for Statisticians, vol. 1, 3'd ed.,
Biometrico, 7966.
9
Appendix A
Detailed Station Results
Detailed data and statistics for each station are presented in both tabular and graphic form in
this appendix. lnformation for each station is grouped together on a series of four pages in
order of station number. Each page lists the station number and name at the top. The first
page for each station lists the starting and ending water year, followed by a table similar to the
top one-third of Table 2 in the main body of the report, except that it contains data for one
station only, and contains monthly results for the mean daily average temperature, in addition
to annual results. ln these pages T.ru refers to the monthly or annual mean daily average
temperature; minTr* refers to annual minimum daily average temperature. The results in the
last two columns are identical to those in Table 2 for the respective station. The left figure
shows time series over the period of record (POR) of annual values of Trrg and minT.,r. These
time series plots illustrate how the data vary from year to year throughout the period of record,
and the difference in variability between T"r, and minTr,r. The two figures on the right present
the annual data sorted by magnitude for T.ru (upper)and minT.rg (lower).
The second and third pages for each station give tables of exceedance temperatures and Upper
and Lower Confidence Limits (CLs) for the Norm and Plll distributions, respectively. As in the
tables at the top of the first page for each station, monthly and annual values are included for
T.ru, and annual values for minTrrr. The exceedance temperature values for annualTavg and
minT*u given in these tables are the same as those given in Table 2 in the main body of this
report.
The fourth page for each station shows the exceedance temperatures and CLs graphically for
annual values of T.,, and minTr,, for the Norm and Plll distributions. Each graph presents
results for the full range of return periods analyzed in this report, that is, 2,5, L0,20, 50 and
100 years. Each left/right pair of figures can be compared to see the influence of fitting a
Normal versus Plll distribution to the data. T.,, is presented in the top pair of figures, and
minT*u is presented in the bottom pair.
Weather Station Zones, Locations, and Starting Water Year (WY)
Zone lD Location Starting WY
3s0 Caldwell 1905
4s0 Boise L94L
500 Hailey 1909
600 Twin Falls 1906
700 Rexburg 1908
750 ldaho Falls t949
800 Pocatello 1939
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Appendix B
Analysis Details
This appendix presents information about the methods used to analyze the data in this report.
The T-year exceedance temperatures presented in this report are statistical estimates based on
NORM and Plll probability distributions fitted using sets of sample observations. Exceedance
temperatures were calculated using the Frequency Factor method applied to the Norm and Plll
distributions (Chow et o1.,1988).
At least two sources of uncertainty are involved in these T-year exceedance temperature
estimates. One source is the selection of an appropriate probability distribution to represent
the sample observations; a second source is associated with how well the random sample of
observations collected represents the underlying population. The 12 lCfri-Squared) test is used
to determine if either the NORM or Plll distribution is more appropriate than the other to
represent the observations, and confidence intervals are calculated to provide a measure of
uncertainty in the estimated exceedance temperatures relative to their unknown population
values based on the fact that they are derived from a random sample of observations.
Selection of Distribution-12 (Chi-Squared) test
Assuming the sample observations adequately represent the underlying population, selection
of an unsuitable probability distribution will yield erroneous exceedance temperatures
wherever the fitted distribution deviates from the trend of the sample observations. For
example, application of the Normal distribution to observations with significant real skew will
induce errors in the estimated exceedance temperatures. One way this may be assessed is by
testing the "goodness of fit" of the NORM and Plll distributions to each set of sample
observations using a 1' lchi-Squared) test (Chow et o1.,1988). The 12-test is a hypothesis test in
which the null hypothesis, Ho, is that a proposed distribution together with its parameters fit
the observations well. The alternative hypothesis, Hr, is that the distribution and/or the
particular parameters are inadequate.
In the 12 test, the range of n sample observations is divided into k intervals, and the number of
observations D; occuffing in each interval is compared with the theoretical number of
observations expected within each interval based on the fitted distribution, given bV np(xl.
Here, p(x) is the theoretical probability of the random variable with cumulative distribution
function F(x) fallingwithin the rth interval bounded between xi and xi-r. That is, p(x) =F(xl- F(xrt)
It essentially compares the number of observations occurring in each bin of a histogram of the
LL
data, with the number of occurrences expected within the range of each histogram bin based
on the fitted probability distribution. The squared differences of the observed minus expected
number of occurrences in each interval are normalized by the expected number of occurrences
and summed over all the intervals to give the test statistic 12.
k
lni - np(x)12@
The sum 1'. is the test statistic which is compared with a 12 distribution limiting value. The null
hypothesis is accepted if the test statistic, 1'. is lower than the 12 distribution limiting value.
A 12 distribution is the distribution of the sum of squares of v standard normal random
variables, z. The number of degrees of freedom, v, is given by v=k-m-1, where k is the number
of intervals, and m the number of parameters fitted for a particular distribution (m=2 for the
Normal distribution, and m=3 for Plll). The effect of m is that the limiting value for the Plll
distribution is smaller than that for the Normal distribution, so that the test is a little more
stringent for Plll to account for the fact that Plll has three parameters which allows greater
flexibility in the distribution to fit the observations. The X2 u,t-rlimiting value has cumulative
probability 1- cr, where a is the significance level. A typical value is a=0.05; it gives the likelihood
of rejecting the null hypothesis when it is true. Tables of the X2 ,,r- o distribution function are
available in many statistics texts (e.g., Benjamin and Cornell, t97O; Devore, L987; Haan, L977;
Lapin, 1983; Pearson and Hartley, 1966).
Histograms are usually set up using uniformly sized increments of the variable for each interval
so that the histogram shape is similar to the shape of the probability density function fitted to
the data. However, for the 12 test, it is desirable to select the range of values for each interval
such that each interval has the same number of expected occurrences of the random variable
within it based on the fitted distribution (e.g., 20 intervals might be selected each with L/2Oth or
5% probability of occurrence), and the commonly recommended smallest number of expected
occurrences in each interval is 5 (Benjamin and Cornell, 19701. For any fitted distribution other
than a uniform distribution, this requires that the span or range of the values defining each
interval will vary. ln the analysis performed here, each dataset was divided into a number of
intervals, k, sized so that the expected number of occurrences in each interval was at least 5,
that is, k<n/5. For example, with Caldwell there were n=111 years of data. The number of
intervals was limited by k3n/5=22.2. Thus,22 intervals were used, each having probability
p=1/k=0.0455 (4.55%1.
The 12 test was applied to all the observations from each station, both on an annual basis for
Tavg and minTavg, and on a monthly basis for Tavg. On a WY basis, both the NORM and Plll
distributions passed the 12 test applied to Tavg and minTavg at each station. Generally, the Plll
Ii=l
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distribution passed the 12 test as well as or by a greater margin than did the Normal
distribution, especially when the observations contained significant skew. lt is recommended
to use the Plll distribution values: When the skew is significant, as it often is for minTavg, Plll
likely provides more accurate results; when the skew is near zero, as commonly occurs with WY
Tavg, the Normal and Pltl return period values are not very different, so selection of Plllto be
consistent induces no penalty and simplifies the selection process.
For the T.ru monthly data, both the Normal and Plll distributions passed the 12 test for most
months at most stations. Across twelve months at the seven stations, there were 84 station-
months tested (ie.,7 stations x l2months) for each distribution. For the Normal distribution, 78
station-months passed the 12 test, and 5 failed. For the Plll distribution, 81 passed and 3 failed.
Station 350 (Caldwell) passed the 12 test for all station-months for both distributions.
Considering the general recommendation of this report to use the Plll distribution, I will discuss
only the three 12 test failures of the Pllt distribution. These occurred for December at Station
450 (Boise), December at Station 700 (Rexburg), and November at Station 750 (ldaho Falls). Of
these, only December-Station 450 failed the 12 test for both the Normal and Pllt distributions.
For November-Station 750, the data exhibits modest skew (-0.2), and the largest difference
between the return period estimates of monthly Tavg from the Normal and Plll distributions
occurs for the 100-year event and is only 0.7 "F . Thus, I recommend accepting the Plll return
period values, as they provide conservative (lower) temperature estimates for a specified
return period, and are not very different from the Normal distribution values.
For December-Station 700, I also recommend using the Pllt results. The skew for this month is
nearly zero, causing the Plll return period temperature estimates nearly to converge with those
from the Normal distribution; there is at most on 0.2 "F difference between them.
December at Boise was the only month among all the station-months tested for which the Plll
distribution failed the 12 test by a significant margin. Comparison of the Boise data with that at
Caldwell and Twin Falls alleviates this concern. Both the Caldwell and Twin Falls data are
stongly correlated with the Boise data, having R2 values of 0.92 and 0.82, respectively, and both
passed the f test for December by a wide margin. The Boise data fails the f test because of a
concentration of data in the 30.4-31.3 'F range, for which the corresponding data at Caldwell
and Twin Falls is slightly more spread out, so that at Boise all of these data fall into a single
interval in the 12 test, whereas they spread across a couple of intervals at Caldwell and Twin
Falls. This concentration of data in a particular range at Boise could be the result of an observer
tendency to round to a particular number rather than another. This one interval at Boise is
responsible for its f test failure, and it is in a range of the data that is not of concern in this
analysis. Here the concern is to estimate the magnitudes of T-year return period events on the
13
cold extreme side. Caldwell and Twin Falls include data for many more years than at Boise, but
even when the probability distributions are fitted to data from the same period as Boise (WYs
L}4L-2OL5) at these two stations, they still both pass the 12 test. The similarity among the
Boise, Caldwell and Twin Falls observations suggests that all three datasets could likely be
represented by the same type of probability distribution (Plll) with similar parameters. lt is
notable that when the skew is calculated for December at Caldwell and Twin Falls for the same
time period as available at Boise, namely L94L-20L5, it is larger at each of these two stations
than when their full period of record data sets are used to calculate skew, and similar to that
obtained at Boise (approximately -1.4), and that this skew is very strongly affected by the very
cold December 1986 Tavg which occurred at all sites.
Most importantly, the regression line between the monthly Tavg values at Caldwell and Boise
has nearly a one-to-one slope and zero intercept, so that across the range of
December monthly Tavg values observed at either site, the estimated regression value for Boise
is never more than 0.5 'F different from the value at Caldwell. Thus, one would expect their
corresponding T-year return period values to be similar to each other. This is in fact the case:
they differ by less than 0.8 "F for the 100-year value, and only by 0.2 "F for the SO-year value.
Consequently, even though the probability distribution fitted to Boise's December data fails the
12 test, T-year return period values estimated from it are close to what should be expected
based on the similar dataset collected at the Caldwell station.
Confidence Limits
The temperature data collected at each station constitute a random sample of the underlying
populations of temperatures and these samples have been used to estimate the true frequency
curves of the corresponding populations. lf a random sample consisting of the same number of
observations could be selected from a different period of time, they would probably produce a
different estimate of the population frequency curve. How well the observations represent the
underlying temperature population depends on the number of observations (sample size), its
accuracy, and whether or not the underlying distribution is known.
Confidence limits provide a measure of the uncertainty of the exceedance temperature at a
selected probability or return period. A range or confidence interval which brackets the true
exceedance temperature with a specified probability or confidence level, 9, can be calculated
That is, for a two-sided confidence interval with an upper and lower confidence limit, and
confidence level p=Q.g, there is a 90% probability that the limits span or encompass the true
exceedance temperature. The significance level, o,, corresponding to the confidence level is
given by a=(1- pl/Z, or 0.05 for the selected 9=0.9.
L4
Approximate confidence limits for the T-year exceedance temperatures for the NORM and Plll
distributions were calculated using the lnteragency Advisory Committee on Water Data (1981)
method, also described in Chow, et al. (1988). Confidence limit values are reported in Table 2
in the main body of this report, and in Appendix A.
15