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OCR for page 211
CAPPENDIX
Probable Maximum
Precipitation (PMP) Estimates
A brief account of the historical development of methods of estimating
maximum flood potentials is given in Chapter 4. Chapter 5 outlines the steps
involved in current procedures for estimating probable maximum precipita-
tion (PMP). In this Appendix the rationale and methods used in that estimat-
ing are discussed.
The methods used have been described in many publications, including a
series of Hydrometeorological Reports by the U.S. National Weather Service
(1943-1984~. These cover a fairly wide range in estimates from those for
specific drainages to those covering a large portion of the United States. A
World Meteorological Organization (1973) manual summarizes the various
techniques for preparing PMP estimates used in the United States ant] in
some other countries.
BASIC DATA
Since the need is average PMP values for drainage basins, maximum
observed values of average rainfalls over standard-sized areas provide the
most suitable basic data for PMP estimates. These eliminate, for the most
part, working with station or point extremes that cannot be adjusted by a
common method to areal average values. A joint effort of the Corps of
Engineers and the Weather Service has provided and kept current a loose-
leaf publication that contains data on the most extreme observed areal storm
rainfalls (U.S. Army Corps of Engineers, 1945~. The publication now con-
tains information for about 550 storms. Additional, not as detailed, storm
211
OCR for page 212
212 Appendix C
data for approximately 300 storms have been assembled by the Hydrome-
teorological Branch of the Weather Service. For each storm the greatest areal
depths are listed for standard-sized areas, usually from 10 square miles up to
20,000 square miles (depending on the area covered by heavy rain) for
standard durations usually from 6 up to 72 hours or more. Methods for
computing these areal storm rainfalls from amounts measured at gages are
described in a manual (World Meteorological Organization, 1969~. An im-
portant part of these storm studies is to analyze the meteorological features
associated with the heavy rainfall to better understand what factors are
important. These studies are necessary for other aspects of PMP estimation
described further on.
STORM TRANSPOSITION
An obvious question to ask concerning any major storm is: Why could it
not have occurred at some other locality? One type of limitation on a particu-
lar storm location is differences in terrain. Significant mountain features
(i.e., height, orientation, and steepness of slopes) affect the amount of rain
and where it is deposited. Thus, transposition of a storm without adjust-
ment, from a mountainous location to a nonmountainous location or vice
versa, is not reasonable.
Meteorological factors also limit where storms can be transposed. If other
storms, with the same basic atmospheric factors (moisture source, location
of high- or low-pressure centers, frontal positions, for example) have been
observed in other locations, this is usually a good indicator that the most
severe storm of that type can be transposed to other such locations with
appropriate adjustments.
A more generalized transposition procedure has been used for tropical
storm rainfalls along the Gulf and east coasts of the United States (Schreiner
and Riedel, 1978~. This gives adjustments to rain depths based on the de-
crease with distance inland from the coast observed in the major tropical
storms.
ADJUSTMENT FOR MOISTURE
Detailed study of major storms has shown that, if all other factors are the
same, the magnitude of rain is related to the moisture content of air flowing
into the storm location. Briefly, the mass of water vapor that a column of air
can hold is related to the surface dew point assuming the column is saturated
and the vertical temperature variation is pseudo-adiabatic. These condi-
tions are generally closely fulfilled in major storms. Thus, in estimating
maximum rainfall potentials, we have basis to increase the rainfall in a storm
OCR for page 213
Appendix C
213
by multiplying by the ratio of the moisture associated with maximum ob-
served dew point in the vicinity of the storm to that observed in the moist air
flowing into the storm. Generalized maps of maximum dew points have
been published (U.S. National Weather Service, 1968~. The dew points
representative of the inflowing air for major storms are routinely deter-
mined for major storms by the Weather Service.
The maximum dew point charts are also used for adjusting storms when
they are transposed. That adjustment is the ratio of the maximum moisture
in the transposed location to that where the storm occurred, both based on
maximum observed dew points.
ENVELOPMENT
Just as some methods of determining a probable maximum flood (PMF)
directly from floods used envelopment, similar envelopments are used for
PMP. Such envelopment takes into account the strong possibility that we
have not experienced equally extreme rainfall-producing parameters in the
observed storms, etc., for all area sizes and durations. Another envelopment
step is introduced by utilizing smooth lines and gradients for PMP estimates
when making generalized PMP estimates for a region. Such smoothness, of
course, is not realistic nor attempted in regions where precipitation is
strongly controlled by orography.
APPLICATION TO ESTIMATE FOR SPECIFIC BASIN
A brief summary of the determinations of PMP estimates for a specific
drainage basin in nonorographic regions is as follows.
· From meteorological studies, decide which storms in storm rainfall
(U.S. Army Corps of Engineers, 1945) can be transposed to the drainage
basin of interest. (The U.S. Weather Service has made such determinations
for many major storms.)
· Transpose each of these storms by multiplying the storm rainfall depth-
area-duration data by the ratio of moisture for the maximum observed dew
point at the transposed position to that for the dew point of the inflowing air
for the actual storm. This single adjustment increases the rainfall for maxi-
mum moisture as well as adjusts it for transposition to the drainage.
· Draw smooth curves for rainfall depth versus area and rainfall depth
versus duration enveloping the values determined from the transposed
storms. The resulting values from the curves corresponding to the size of the
drainage basin provide an estimate of PMP for that basin.
This procedure for estimating PMP involves the assumptions (1) that there
OCR for page 214
214 Appendix C
are a "sufficient" number of extreme storms and (2) that adjustment for
moisture and not for any other meteorological factor is sufficient to give an
upper limit. (It is assumed the other meteorological factors combined can be
looker] upon as "efficiency" and that among the transposed observed storms
this efficiency factor reached the highest value that can be expected at the
site.)
PROCEDURES FOR MOUNTAINOUS REGIONS
We have stated why storm transposition is not reasonable in mountainous
(orographic) regions. Thus, it is necessary to include other methods in such
regions. Several different techniques have been used for PMP estimates in
orographic regions. One of these is based on adjusting the PMP for the
nearest nonorographic location for topographic effects. The adjustment
could be based on comparison of extreme rainfall of various categories (sta-
tion and areal for various clurations, rainfall frequencies, etc. ~ . Another is to
apply an orographic precipitation computation model, which briefly stated,
increases observed storm rainfall for winds as well as moisture and adjusts
for differences in topographic features (U.S. Weather Bureau, 1961a). Use of
this orographic mode} must be restricted to substantial slopes of reasonably
long lateral extent facing inflowing moisture, and to regions where precipi-
tation in the cool season is most important. For the coterminous 48 states
there now remain the eastern Appalachian region and its extension into New
England where the published generalized PMP charts are for the most part
not applicable (Shreiner and Riedel, 1978~. In that report this questionable
region is stippled; * where the orographic portion of the total rain is small,
the regional study will normally provide only estimates of nonorographic or
convergence PMP. The user must evaluate the effects of topography on the
precipitation process and incorporate these into the final estimate (Miller et
al., 1984; Fenn, in press).
GENERALIZED PMP CHARTS
It has been found that more reliable and consistent PMP estimates result
from studies giving mapped values for a region. This is particularly the case
for orographic regions where discontinuities from drainage to drainage are
· . .
m~n~m~zec .
Procedures for determining generalized PMP estimates are basically the
*Another stippled region is shown in that report in the foothill or steep slope region of the
Continental Divide. This region is covered by a recent generalized PMP study that does take into
account the mountains (Miller et al., 1984).
OCR for page 215
Appendix C
215
same as those for a specific drainage. One added feature naturally is that
regional smoothing results in consistency from drainage to drainage within a
particular region.
The National Weather Service has completed detailed meteorological
studies and analyses of data leading to generalized PMP estimates for the 50
states and Puerto Rico. Figure C-1 shows the regions covered by individual
studies. Table C-1 lists the studies and other information. These generalized
PMP estimates have been adopted by all concerned federal agencies.
APPLICATION OF PMP
The listed PMP studies (Table C-1) give average rainfall values for specific
sized areas and for standard durations usually in sufficient detail to deter-
mine from the greatest, second greatest, etc., to the least 6-hour incremental
rainfall that may be expected in a 72-hour PMP storm. Detailed instructions
and recommendations for application of PMP for the United States east of
the 105th meridian have been published (Hansen et al., 1982~. This report
covers the spatial distributions within any specified drainage and the se-
quence of incremental 6-hourly rain depths in the PMP storm. Details of the
study will not be repeated here—rather the considerations used will be
discussed in general terms that may be helpful for other regions.
SPATIAL DISTRIBUTION
Should a severe rainfall actually have centered on any specific drainage, it
can be used to provide a rainfall pattern for the PMP storm. In nonoro-
graphic regions the hypothetical elliptical pattern of Hydrometeorological
Report (HMR) 52 (Hansen et al., 1982) can be applied. Some constraints in
orientations of the pattern, based on meteorological parameters and ob-
served pattern orientations, may be employed. Generally, the same pattern
is recommended for the four greatest 6-hour rain depths, and no areal pat-
tern for the remaining 6-hour depths. This is a simplification of patterns
observed in major storms some of which even show holes or depressions in
some 6-hour periods.
In mountainous regions no detailed studies have been made on spatial
distributions for the PMP. Use of an actual storm pattern, if well centered on
the basin, is preferred. Otherwise, the pattern of mean seasonal precipita-
tion or that of the rainfall frequencies (Miller et al., 1973) such as the 100-
year rainfall may be useful. Should orographic PMP distribution be given in
a National Weather Service report covering the basin, it could also give
guidance to the distribution of PMP.
In order to avoid possible excessive maximization, generally the National
OCR for page 216
216
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Appendix C
Weather Service assumes PMP only for the total area of the basin being
studied. Thus, less than the PMP is used for smaller areas within the basin
and for larger areas including and surrounding the basin. HMR 52 explains
in some detail how such specifications are used to determine rainfall within
and around the basin.
Generally, it is assumed the PMP storm is centered over the basin of
concern—with no travel or movement with time during the PMP storm
period. This is a simplification--resulting in possibly a higher or lower PMP
and PMF depending on the direction of streamflow relative to direction of
possible storm movement. For certain basins it is quite possible that PMP for
a smaller area within the basin could be hydrologically more critical than
PMP over the entire basin. HMR 52 discusses the rainfall that should be
assumed to occur over the residual or remaining part of a basin if the PMP is
assumed to cover only part of the basin.
219
TEMPORAL DISTRIBUTION
Values of PMP are usually given for durations up to 72 hours, unless the
basin of concern is small (less than a few hundred square miles), in which
case 6- or 12-hour PMP may be adequate. For relatively flat regions, studies
have shown that assuming PMP occurs for all durations, out to 72 hours is not
an undue maximization.
Sequences of the 6-hour PMP increments (that is, in what order they occur
relative to their magnitude) vary widely in major storms. Rain bursts lasting
from 12 to 24 hours are most common, although cases can be found with a
gradual increase to most intense rain near the middle of the storm. Another
consideration is maintenance of PMP for all durations. This can only be
obtained if the first and second greatest 6-hour increments are adjacent to
each other, and the third adjacent to these two, and so on. With these
considerations the following guidance based on storms in mountainous re-
gions is recommended:
· Group the four heaviest 6-hour increments of the 72-hour PMP in a 24-
hour sequence, the middle four increments in a 24-hour sequence, and the
smallest four increments in a 24-hour sequence.
· Within each of these 24-hour sequences, arrange the four increments in
accordance with the sequential requirements. That is, the second highest
next to the highest, the third highest adjacent to these, and the fourth highest
at either end.
· Arrange the three 24-hour sequences in accordance with the sequential
requirements, that is, the second highest 24-hour period next to the highest,
with the third at either end. Any of the possible combinations to the three 24-
OCR for page 220
220
Appendix C
hour periods is acceptable with the exception of placing the lightest 24-hour
period in the middle.
These sequences are similar to those observed in major storms. They also,
to a large extent, preserve the basic estimated PMP depths. For example, if
the greatest 6-hour increment were followed and preceded by much lesser
(smaller) rain increments, the 12-hour depth would be greatly reduced from
the estimated PMP value for 12 hours.
SEASONAL VARIATION OF PMP
Certain hydrologic problems require consideration of PMP for seasons
other than when the greatest or "all-season" value can occur. For example,
PMP for the spring (April or May) in Montana combined with optimum
snowpack and melt factors may give a more critical hydrograph than the all-
season (most often summer) PMP.
Published studies that include seasonal variation for areas in the United
States are listed in Table C-1. The studies, in general, are based on seasonal
variations in meteorologic data, including 1,000-millibar dew points (these
are an index to atmospheric moisture) ant] extreme rainfalls of various cate-
gories such as maximum station rainfalls of record for several durations,
maximum areal rainfalls from storm rainfall (U.S. Army Corps of Engi-
neers, 1945), and climatic summaries. It is believed such precipitation data
can be used to extend existing PMP evaluations.
Often when PMP is required for combination with a spring snowmelt
situation, optimum sequences of snowmelting parameters (winds, tempera-
tures, dew points, and solar radiation) need to be assessed, as well as an
estimate of the optimum snowpack that is available for melt. It is believed
that a combination of an estimates] maximum probable snowpack, most
extreme possible snowmelting parameters, and springtime PMP will give an
improbable flood. judgment needs to be taken to avoid compounding im-
probabilities. Where spring PMP is the major contributor to the flood for a
relatively small basin, a snowpack on the order of a 100-year event would] be
sufficient. Where snowmelt is the major flood producer, especially for a
large basin, at least the greatest snowpack of record (where such records are
available) should be used with the PMP. An extreme snowpack is sometimes
computed by a hypothetical combination of the greatest recorded precipita-
tion for each of the winter (or snow accumulation season) months. It has
been found for several basins in the lower 48 states that observed snowpacks
are greater than can be melted by critical sequences of the melt parameters.
Thus, something less than this should be used for greatest runoff.
Sequences of snowmelting parameters should be derived from observa-
OCR for page 221
Appendix C
tions either in the basin or adopted from those nearest the basin. A study for a
large basin in Alaska (U.S. Weather Bureau, 1966b) used an envelope of
highest observed temperatures and winds on the order of 100-year return
period values.
221
EVALUATION OF PMP
It is assumed in a study of PMP, that there is no climatic trend in the
meteorologic parameters that would increase or decrease the estimates in the
foreseeable future. PMP for the United States is based largely on the greatest
recorded rainstorms that have occurred mainly since about 1860 (a few on
record prior to this year). As an example consider the 177 storms east of the
105th meridian that have observed rainfall depths that are 50 percent or
more of the PMP where the storm occurs (Riedel and Schreiner, 1980) for at
least one area size of 10, 200, 1,000, 5,000, 10,000, and 20,000 square miles)
and at least one duration (of 6,12, 24,48, and 72 hours). The earliest of these
storms occurred in 1819, and the latest in 1979. For the decade 1930-1939
there are 30 storms and for the decade 1940-1949, 32 storms. The record
shows 15 storms for 1950-1959 and 12 for 1960-1969. At first glance, one
could think there may be a climatic trend in the number of extreme rainfalls.
This is most likely not the case—the increased data base and interest begun in
the 1930s and 1940s are important factors. In addition, the number of storms
analyzed is partly dependent on the number of projects and where they are
located. Experience with extreme point or station rainfalls Jennings, 1952)
also has not shown a significant climatic trend in their number or magni-
tude. Users of generalized PMP estimates like to know how rare the values
are. This question is difficult to answer, since rainfall data cover at most 100
years. What extrapolations have been attempted indicate the PMP at certain
stations range around a return period of millions of years. Relatively little
confidence can be placed to such extrapolations.
A comparison has been published (Riedel and Schreiner, 1980) of point
(station) 100-year rainfall and 10-square-mile PMP for selected durations.
Figures C-2 and C-3 are examples of these comparisons for 24-hour dura-
tion. These comparisons show ratios of PMP to 100-year rainfalls ranging
from 6 to 2. The highest ratios tend to be in the drier desert regions, the
lowest being along mountain ridges. Such variations can be expected, since
the more extreme the rainfall, the less the regional range due to topography
and other factors. Thus, there is a greater range in the 100-year events than
in PMP, and necessarily, there is a regional range in the PMP to 100-year
ratio. This ratio, however, is not a simple measure of the rarity of the PMP.
Some data are given in Appendix A on storm rainfalls that exceeded 50
percent of PMP. Figure C-4 shows a map of the storms that had rainfall
OCR for page 222
222
Appendix C
ll
64
AN 1
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rate
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kin ~ ~ STATUTE MILES
by/ J 1 oo 0 1 oo 200 300
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,' 100 0 100 300
KILOMETE RS
FIGURE C-2 Ratios of estimated PMP for 10 square miles to estimated 100-year frequency
rainfalls (both for 24-hour durations), eastern United States. Source: Riedel and Schreiner
(1980~.
depths for 10 square miles and 24 hours that were 50 percent or more of PMP
(Ho and Riedel, 1980~. There are 84 storm depths exceeding 50 percent for
this area size and duration for the region east of the 105th meridan and that
west of the Continental Divide. One may be concerned that there are rela-
tively few cases in Nevada, Arizona, and western Oregon and Washington.
It is believed this is partly due to much fewer rainfall observations in these
regions than, say, in California. Another factor may help explain the distri-
bution shown on the map: the PMP is a rarer event in some regions compared
with others. It seems reasonable to expect a relatively rare storm more fre-
quently on the Sierra slopes than in the deserts of Nevada and Arizona.
In an earlier section, several assumptions were listed that are inherent in
the procedure for estimating PMP. One assumes a "sufficient" storm sample.
If this is not met (a subjective decision) and if no other compensating proce-
dure is applied, it would indicate the PMP estimate is on the low side.
OCR for page 223
Appendix C
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FIGURE C-3 Ratios of estimated PMP for 10 square miles to estimated 100-year frequency
rainfalls (both for 24-hour durations), western United States. Source: Riedel and Schreiner
(1980).
The other assumption, that adjustment only for moisture is sufficient to
obtain an upper limit for storm rainfall, also probably results in a low PMP
estimate. However, a possible compensating factor is that the highest "effi-
ciency" (as indicated by extreme observed rainfall) when combined with
maximum moisture may not yield the most extreme rainfall or the upper
limit. Not enough is known about the "best', values of the factors for maxi-
mum rainfall.
OCR for page 224
224
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OCR for page 225
Appendix C
225
One bit of evidence lends credence to the assumption that current general-
ized PMP estimates are on the low side. Generalized studies, covering the
same region, if of the same degree of detail and endeavor, over a period of
years show an increase with time. Generalized studies for the United States
east of the 105th meridian made over a period of 31 years most nearly meet
these conditions (U.S. Weather Bureau, 1947; Riedel et al., 1956; Schreiner
and Riedel, 1978~. Figure 5-1 (see Chapter 5) shows mapped isolines of the
ratios of the PMP values for the 1978 study to the PMP values of the 1947
study. The PMPis for 200-square-mile areas for a duration of 24 hours. The
map indicates a maximum ratio of 1.37 with considerable areas with ratios
above 1.20. These changes in PMP estimates over a 31-year period may be
considered moderate, in light of the nature of the estimating process, and on
that basis, the PMP estimates may be considered fairly stable. However, one
conclusion can be made: that the assumptions on "sufficiency" and "mois-
ture maximization" have not quite stood the test of time. Another way of
putting it is that in the future there will be greater rainfalls (for some area
sizes and durations) at least in some portions of this particular region. This
conclusion includes the fact that lesser storms of certain critical types with
time will occur over wider "bounds"; thus, the transposition limits of some
most extreme storms will be increased. These two factors are most responsi-
ble for the increase in PMP with time.
The following equation (Hershfield, 1961) is sometimes used as a check on
PMP estimates or as the basis of design rainfall:
Xm = Xn + Km Sn'
where Xm is maximum observed rainfall, Xn is mean of the series of n annual
maxima, Sn is standard deviation of the series, and Km is number of standard
deviations needed to obtain Xm.
Using many series of station maximum annual 24-hour rainfall events,
mostly in the United States but also some in other countries, the greatest
value found of Km to obtain Xm was 15. Use of this value was suggested to
compute statistical PMP. In a later paper (Hershfield, 1965), K was found to
be correlated with the raininess of the region, the cirier locations having a
higher K.
A map covering the eastern two-thirds of the United States (Hershfield,
1961) shows isolines of statistical PMP using a K of 15. It was shown (Riedel,
1977) that this "statistical PMP" has been exceeded by at least 10 observed
10-square-mile storm depths (U.S. Army Corps of Engineers, 1945~. While
the later adjustment (Hershfield, 1965) was not used in the comparison,
which could possibly raise the K value to a maximum of 20, casual inspection
indicates these observed 10-square-mile depths would still exceed the statis-
tical PMP. Numerous other comparisons have been made between the deter-
OCR for page 226
226 Appendix C
ministic and this statistical PMP. One was for the southwest states region
(Hansen et al., 1977~. Here the variable K factor was used to compute the
statistical PMP for 98 stations. Only 2 of the 98 statistical PMP values ex-
ceeded the PMP of the report. Overall, the statistical PMP averaged approxi-
mately two-thirds of the report PMP.
From the several methods of evaluation, it can be concluded that the PMP,
although extremely rare, is probably on the low sicle with respect to its
definition.
Representative terms from entire chapter:
generalized pmp
