1

NOAA Technical Memorandum GLERL-118

CLIMATE-CORRECTED STORM-FREQUENCY EXAMPLES

Thomas E. Croley II

NOAA, Great Lakes Environmental Research Laboratory, Ann Arbor, MI 48105

Great Lakes Environmental Research Laboratory

Ann Arbor, Michigan

August 2000

UNITED STATES

DEPARTMENT OF COMMERCE

Norman Y. Mineta

Secretary

NATIONAL OCEANIC AND

ATMOSPHERIC ADMINISTRATION

D. James Baker

Under Secretary for Oceans

and Atmosphere/Administrator

NAT OI NAL OCEANIC AND ATMOSPHERIC ADMINISTRAT OI N

U.S.DEPARTMENT OF COMMERCE
2

NOTICE

Mention of a commercial company or product does not constitute an endorsement

by the NOAA Environmental Research Laboratories. Use of information from

this publication concerning proprietary products or the tests of such products for

publicity or advertising purposes is not authorized. This is GLERL Contribution

No. 1174.

This publication is available as a PDF file and can be downloaded from GLERL’s

web site: www.glerl.noaa.gov. Hard copies can be requested from the GLERL

Publications Unit, 2205 Commonwealth Blvd., Ann Arbor, MI 48105.

pubs.glerl@noaa.gov.
3

Table of Contents

Abstract .............................................................................................................................................. 5

1. Introduction ................................................................................................................................... 5

2. Storm Frequency Estimation......................................................................................................... 5

3. Example Storm Frequencies ......................................................................................................... 6

4. Matching Probability Forecasts .................................................................................................... 9

5. Optimum Solution ....................................................................................................................... 11

6. NOAA and EC Forecast Example............................................................................................... 11

7. ENSO Conditional Forecast Example ........................................................................................ 20

8. Multiple Solutions ....................................................................................................................... 25

9. Summary and Observations ........................................................................................................ 25

10. References .................................................................................................................................. 26

Figures

Figure 1.--Annual Maximum (Calendar Year) Daily Maumee River Flow Exceedance Frequency,

made in September 1999 for September 1999—August 2000 (35.31 cfs = 1 m3

s-1) ....... 8

Figure 2.--Annual Maximum (Water Year) Daily Maumee River Flow Exceedance Frequency,

made in September 1999 for September 1999—August 2000 (35.31 cfs = 1 m3

s-1) ....... 8

Figure 3.--NOAA Outlook of Meteorology Event Probabilities for

October-November-December 1999, made 16 September 1999 ................................... 12

Figure 4.--EC Outlook of Most-Probable Meteorology Event for

September-October-November 1999, made 1 September 1999 .................................... 13

Figure 5.--Mixed NOAA and EC Meteorology Probability Forecasts made in

September 1999 over the Maumee River Basin ............................................................ 14

Figure 6.--Alternate Representation of (24), (11), and Equations in Figure 5 as (20) ................... 19

Figure 7.--ENSO Event Conditional Probability Forecast Equations for Maumee

River Basin..................................................................................................................... 22

Figure 8.--Alternate Equations Set for La Nina and El Nino Examples as in (20) ........................ 23

Figure 9.--ENSO Annual Maximum (Calendar Year) Daily Maumee River Flow Exceedance .... 24

Figure 10.--ENSO Annual Maximum Daily Maumee River Basin Precipitation Exceedance ...... 24
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Tables

Table 1.--Annual Daily Maxima for the Maumee River Basin (35.31 cfs = 1 m3

s-1) ........................ 7

Table 2.--Average Air Temperature over the Maumee River Basin (o

C) ......................................... 16

Table 3.--Average Daily Precipitation on the Maumee River Basin (mm) ...................................... 17

Table 4.--Historical Reference Quantiles of Average Air Temperature over the Maumee

River Basin (o

C)................................................................................................................ 18

Table 5.--Historical Reference Quantiles of Average Daily Precipitation on the

Maumee River Basin (mm) .............................................................................................. 18

Table 6.--El Nino and La Nina event onset years ............................................................................ 21
5 

CLIMATE-CORRECTED STORM-FREQUENCY EXAMPLES1

Thomas E. Croley II 

ABSTRACT. Storm frequency estimates (e.g., maximum precipitation or flow probabilities) allow engineers and hydrologists to assess risks associated with their decisions during 

the design, construction, and operation of water resource projects. Storm frequencies for 

the future are often estimated directly from past historical records of sufficient length. The 

estimation requires no detailed knowledge of the areaís meteorology, but presumes it remains unchanged in the future. However, the climate seldom remains static. Numerous 

climate forecasts of meteorology probabilities over extended periods are now available to 

the water resource engineer and hydrologist. It is possible to use these meteorology forecasts directly in the estimation of storm frequencies from the historical record. It is more 

desirable to do so now than at any time past, since meteorology forecasts have been improving and are now better than their predecessors. A heuristic approach is defined here to estimate storm frequencies that recognize forecasts of extended weather probabilities. Basically, those groups of historical meteorology record segments matching forecast meteorology probabilities are weighted more than others, during the estimation of storm frequencies. 

(Affiliated groups of hydrology record segments may be similarly weighted for hydrological 

estimation; e.g., flood frequency estimation.) Examples include frequency estimation of 

maximum daily precipitation and maximum flow, using currently available agency meteorological forecasts in the US and Canada as well as El NiÒo and La NiÒa conditional probabilities. 

1. INTRODUCTION 

The purpose of this report is to present application examples for modifying storm frequency estimates to reflect outside probability information that might be available, such as meteorology 

probability forecasts or El NiÒoóSouthern Oscillation (El NiÒo and La NiÒa) expectations. The 

report consists of a brief overview of storm frequency estimation and a brief discussion of the 

application of techniques to alter this estimation for meteorology probability forecasts. Details of 

the theory and discussion of the techniques presented here are available elsewhere (Croley 1996, 

1997a,b, 2000a,b). This report details the application examples omitted in the theoretical description presented by Croley (2000b). 

2. STORM FREQUENCY ESTIMATION 

Since storm frequencies are unknown, they are estimated from the historical record, which is assumed ergodic and treated as a ìrandom sample.î Successive observations are considered identically distributed and equally likely to occur (both in the past and future). Likewise, the observations must be defined so they can be considered as independent of each other. (Two successive 

 

1

 GLERL Contribution No. 1174 
6 

storms occurring very closely may result in a high degree of dependence of the second on the 

first.) Defining long event inter-arrival times or record pieces can minimize temporal dependence. For example, annual maximum floods or rainfalls (inter-arrival time on the order of a year) 

are often taken as time independent, as are 1-year record segments. 

Storm frequencies or ìexceedance probabilities,î P[ ] X x  can be estimated directly from the 

historical record. Suppose all values, i x , in a random sample of annual maximums ( i x , i  1, ..., 

n ) are ordered from largest to smallest to define the ordered variable values ( y ,   1 , ..., n ), 

where i( ) y  x   and i( )  is the number of the value in the unordered sample corresponding to the 

 th order. There are several methods to estimate exceedance probabilities from annual exceedance series (Chow 1964); without loss of generality, the popular ìWeibullî method is used 

here as an example: 

 

1

1 à 1 , 1, , 1 1 i

PX y n

n n 



  





  (1) 

The caret, ì^,î denotes an estimate of the characteristic named underneath. Other methods also 

could be used. 

This estimator is called ìnon-parametricî since knowledge of the underlying distribution and its 

parameters is not required. Other estimators (called ìparametricî) derive from knowledge (or 

supposition) of the type of underlying distribution. Functions of a random sample may be used 

as estimators of the parameters of the underlying distribution. Several of interest here are the 

ìsample mean,î à , ìsample variance,î 2 à , and ìsample skew coefficient,î à : 

1

1 à n

i

i

x

n





  (2) 

 2 2

1

1 à à 1

n

i

i

x

n





  (3) 

    

3 3 2

1

à à à 1 2

n

i

i

n

x

n n

 



  (4) 

They are estimators of distribution mean,  , variance, 2  , and skew coefficient, , respectively. 

Other estimators (Koutrouvelis and Canavos 1999) also could be used with no loss of generality. 

3. EXAMPLE STORM FREQUENCIES 

Daily precipitation data from 32 stations were assembled over 1948-1995 for the Maumee River 

basin, defined at its outflow point into Lake Erie with an area of 16,806 km2

. (Widespread meteorological observations began in 1948.) The annual maximum of the areal-average (Thiessenweighted) daily basin precipitation for this data set is given in columns 2 and 6 of Table 1. Additionally, daily flow records of the Maumee River at Waterville, Ohio (basin area = 16,394.7 km2

) 

were searched for this period, and the annual maximum daily flows are given also in Table 1. 
7 

Table 1. Annual Daily Maxima for the Maumee River Basin (35.31 cfs = 1 m3s

-1). 

Year Calendar 

Year 

Water 

Yearc

 Year Calendar 

Year 

Water 

Yearc

Precip.a

(mm) 

Flowb

(cfs) 

Flowb

(cfs) 

 Precip.a

(mm) 

Flowb

(cfs) 

Flowb

(cfs) 

(1) (2) (3) (4) (5) (6) (7) (8) 

1949 3.674 45100 45100 1973 2.490 40000 42800 

1950 4.984 92400 92400 1974 2.295 69600 69600 

1951 3.412 53100 53100 1975 2.690 49400 49400 

1952 3.931 53100 53100 1976 2.532 68500 68500 

1953 3.010 33200 33200 1977 4.151 64000 56200 

1954 3.753 23400 23400 1978 2.518 86400 86400 

1955 3.563 45900 45900 1979 3.185 53400 53400 

1956 4.232 42700 42700 1980 4.511 44400 44400 

1957 4.661 62400 62400 1981 4.930 85400 85400 

1958 3.276 29700 40300 1982 3.776 113000 113000 

1959 4.274 80000 80000 1983 3.510 54200 54200 

1960 2.594 44800 44800 1984 2.420 51300 51300 

1961 2.614 53500 53500 1985 2.545 91100 91100 

1962 2.335 45800 45800 1986 4.018 36200 36000 

1963 2.795 35200 35200 1987 3.206 23500 36200 

1964 2.533 46800 46800 1988 3.779 22900 23500 

1965 3.259 36200 36200 1989 3.370 42700 42700 

1966 3.561 79000 26200 1990 4.164 82000 62500 

1967 2.592 48900 79000 1991 4.187 86700 86700 

1968 3.537 56900 56900 1992 3.625 54000 36700 

1969 4.511 67500 67500 1993 2.359 65000 65000 

1970 3.580 33300 33300 1994 2.685 63900 63900 

1971 3.785 38900 38900 1995 2.918 51000 51000 

1972 5.575 46900 46900 

a

Areal-average daily precipitation for Maumee River basin at Lake Erie, Ohio. 

b

Maumee River at Waterville, Ohio, Lat. 41:30:00, Long. 83:42:46. 

c

ìWater Yearî is defined by its end (e.g., water year 1949 ends 30 September 1949). 

Both the calendar year of 1 Januaryó31 December (columns 3 and 7) and the water year of 1 

Octoberó30 September (columns 4 and 8) were used to define the annual maximum daily flow. 

The exceedance frequencies for the annual maximum flows were estimated with (1) and plotted 

in Figures 1 and 2 as ìnon-parametric sans forecast.î 

The log-Pearson Type III distribution also was fit to the data sets of Table 1. This distribution 

results from supposing the natural logarithms of the data in Table 1 [Z = ln(X)] are distributed 

as a three-parameter gamma distribution: 

  

1 1 ,00

z c

Z

z c f z e cz zc



  

              	 
   


 (5) 
8 

0

0.2

0.4

0.6

0.8

1

0 20000 40000 60000 80000 100000 120000

Annual Maximum Daily Flow (cfs)

Exceedance Probabilit

y

Non-parametric sans Forecast

Parametric sans Forecast

Parametric with Forecast

Figure 1. Annual Maximum (Calendar Year) Daily Maumee River Flow Exceedance Frequency, made in September 1999 for September 1999óAugust 2000 (35.31 cfs = 

1 m3

s

-1). 

0

0.2

0.4

0.6

0.8

1

0 20000 40000 60000 80000 100000 120000

Annual Maximum Daily Flow (cfs)

Exceedance Probability

Non-parametric sans Forecast

Parametric sans forecast

Parametric with forecast

Figure 2. Annual Maximum (Water Year) Daily Maumee River Flow Exceedance Frequency, made in September 1999 for September 1999óAugust 2000 (35.31 cfs = 

1 m3

s

-1). 
9 

where fZ ( )z PZ z  z





 , ( ) is the gamma function, and  , , and c are distribution parameters. Parameter estimates are given in terms of (2)ó(4) defined on the natural logarithms of the 

data (USWRC 1967) by replacing expected values from (5) with sample moments: 

 2

 à  2 à (6) 

à 2    à à 2 (7) 

2

càà à   2 à (8) 

The estimated log-Pearson Type III distributions are shown also in Figures 1 and 2 as ìparametric sans forecast.î See Koutrouvelis and Canavos (1999) for other parameter estimators. 

4. MATCHING PROBABILITY FORECASTS 

The probability of any event  , P[ ]  , can be inferred with the estimator, à

P[ ]  , defined as the 

number of observations in the random sample for which  occurs (i.e., for which the event  is 

true), n , divided by the total number of observations in the sample, n : 

 1 à 1

i

n

P

n n





   (9) 

In (9), the sum is taken over all i (members of the random sample) for which  occurs, denoted 

as i  . The estimate in (9) is seen as the ìrelative frequencyî of  in the random sample. 

Croley (1996, 1997a,b, 2000a,b) biased samples, by multiplying observations by non-negative 

weights, wi , to calculate probabilities matching othersí multiple probability forecasts: 

 1 à i

i

P w

n 

  (10) 

1

n

i

i

w n



  (11) 

Consider, for example, that othersí forecasts of event probability can be interpreted in m 1 probability equations (Croley 1996) and forecasts of most-probable events can be interpreted in u

probability inequalities (Croley 2000a). They are expressed in terms of relative frequencies over 

a random sample as follows: 

 

 

à , 2, ,

à , 1, ,

k k

k k

P ak m

P a k m mu



   



 (12) 

where k a are the forecast probabilities. Equation (10), when applied to match the forecasts of 

meteorology probabilities in (12) and added to (11), yield a system of equations to be solved for 

the weights: 
10 

1

, 2, ,

, 1, ,

n

i

i

i k

i k

i k

i k

w n

w na k m

w na k m m u









 

 











 (13) 

Equivalently, 

,

1

,

1

, 1, ,

, 1, ,

n

ki i k

i

n

ki i k

i

we k m

w e k m mu









 

 









 (14) 

where k i , = 1 for k = 1 or for k > 1 and k i  [inclusion of the i

th random sample value (ith event 

or segment of the historical record) in the event of the k

th probability statement]; otherwise it is 

zero. Also, ek = n for k = 1 and ek = k n a for k > 1. Any weights that satisfy (14) yield weightedsample relative frequencies of events that match forecasts of meteorology probabilities. These 

weights also yield other corresponding biased sample estimators; e.g., (1)ó(4) become: 

  

1

1 à , 1, , 1 i k

k

PX y w n

n 



 



  (15) 

1

1 à n

i i

i

w x

n





 (16) 

 2 2

1

1 à à 1

n

i i

i

w x n  



  (17) 

    

3 3 2

1

à à à 1 2

n

i i

i

n

w x

n n

 



  (18) 

Generally, some of the equations in (13) or (14) may be either redundant or infeasible (nonintersecting with the rest, resulting in no solutions) and must be eliminated. (If the number of 

equations is greater than the number of weights, then some of the equations must be either redundant or infeasible.) In practice, one could assign each equation in (13) or (14) a ìpriorityî reflecting its importance. [The highest priority is given to the first equation in (13) or (14) corresponding to (11), guaranteeing that all relative-frequencies sum to unity.] Each equation (starting with 

the second highest priority equation) is compared to the set of all higher-priority equations and 

eliminated if redundant or infeasible. Thus (14) can always be reduced so that the allowed number of forecasts of meteorology probabilities is less than or equal to the number of historical record pieces (sample size). If less, then there are multiple solutions to (14), and a choice must be 

made as to which solution to use. 
11 

5. OPTIMUM SOLUTION 

If there are multiple solutions to (14), the identification of the ìbestî requires a measure or 

objective function for comparing them. Solutions of (14) with larger values of this measure can 

be judged ìbetterî than those with smaller values. One such measure is the probability of a selected event. If the objective function is always a statement of maximizing or minimizing a 

probability, then it can be added to the problem statement of (14) to yield an optimization problem. Objective functions that use probability statements can be expressed in the general form: 

0,

1

max 

n

i i

i

 w



 (19) 

where 0, i are defined similarly to (14) in which the objective function is equation 0. The problem of solving (14) can now be formulated as an optimization, maximizing the objective function 

subject to a ìconstraint setî of equations: 

0,

1

,

1

,

1

max subject to

, 1, ,

, 1, ,

0, 1, ,

n

i i

i

n

ki i k

i

n

ki i k

i

i

w

we k m

w e k m mu

w in













 

 

 













 (20) 

Equations (20) are amenable to standard ìlinear programmingî optimization techniques. An algebraic procedure, termed the ìSimplexî method, has been developed (Hillier and Lieberman 

1969) which progressively approaches the optimum solution through a well-defined iterative 

process until optimality is finally reached. Croley (2000a,b) describes a procedure for applying 

the Simplex method in a two-stage optimization. The first stage finds a feasible solution to the 

constraint set in (20) and the second searches systematically from that feasible solution to the optimum solution. Multiple optima are possible, depending upon the objective function and the 

constraint set. 

6. NOAA AND EC FORECAST EXAMPLE 

The estimates of Figure 1 are modified by incorporating selected forecasts from the National 

Oceanic and Atmospheric Administration (NOAA event probability forecasts) and Environment 

Canada (EC most-probable event forecasts); see Croley (1996, 1997b, 2000a). An example for 

NOAA is given in Figure 3. The NOAA outlook estimates probabilities of average air temperature and total precipitation falling within pre-selected value ranges. The value ranges (low, normal, and high) are defined as the lower, middle, and upper thirds of observations over period 

1961ó90 for each variable. The climate outlooks presume that one of only four possibilities exists for the probability distribution type for each variable: 1) the probability of being in the high 

range exceeds one-third, and the probability of being in the low range is reduced accordingly (it 
12 

Figure 3. NOAA Outlook of Meteorology Event Probabilities for October-NovemberDecember 1999, made 16 September 1999. 

remains at one-third for the normal range)óreferred to as being ìabove normalî; 2) the probability of being in the normal range exceeds one-third, and the probabilities of being in the low and 

high ranges are reduced accordingly and are equalóreferred to as being ìnormalî; 3) the probability of being in the low range exceeds one-third, and the probability of being in the high range 

is reduced accordingly (it remains at one-third for the normal range)óreferred to as being ìbelow 

normalî; or 4) skill is insufficient to make a forecast, and so probabilities of one-third in each 

range are usedóreferred to ìclimatological.î From Figure 3 at the Maumee River basin (near 

the southwest edge of Lake Erie, marked by a red asterisk), the probability of OctoberNovember-December (OND) total precipitation in the upper third of historical observations is 

forecast to rise by about 0.05. According to the convention of NOAAís definitions then, the corresponding probability of OND precipitation in the lower third of historical observations is forecast to drop by about 0.05, and the probability of OND precipitation in the middle third of historical observations is forecast to remain unchanged at one-third: 

99 ,0.333

,0.333 99 ,0.667

99 ,0.667

à à 0.333 0.05 0.283

à à à 0.334

à à 0.333 0.05 0.383

OND OND

OND OND OND

OND OND

P Q

P Q

P Q



 



     

    

 	  
  

 (21) 
13 

Figure 4. EC Outlook of Most-Probable Meteorology Event for September-OctoberNovember 1999, made 1 September 1999. 

where Qg = total precipitation for period g and , àg  is the  -quantile for period g precipitation 

estimated from a reference historical period (1961ó90) such that: 

, à à P Qg g        (22) 

Likewise, air temperature, Tg , and its quantiles, , àg   , are defined similarly to (22). Noting that 

the second line of (21) is redundant with the rest of (21) and the fact that all probabilities must 

sum to unity, we need only keep the top and bottom lines in (21). 

An example for EC is given in Figure 4. The EC outlook indicates which of three pre-selected 

ranges of values (lower, middle, or upper thirds of observations from 1961ó90 for precipitation 

or from 1963ó93 for temperature) are most likely to occur for 1- and 3-month air temperature 

and 1- and 3-month total precipitation. Figure 4 at the Maumee River basin (marked with a red 
14 

'99 , 0.333 à à P Q OND OND    0.283   (1) '00 , 0.333 à à 0.333 P TFMA FMA       (21) 

'99 , 0.667 à à P Q OND OND    0.383   (2) '00 , 0.667 à à 0.333 P TFMA FMA       (22) 

'99 , 0.333 à à 0.273 P Q NDJ NDJ      (3) '00 , 0.333 à

P T MAM MAM   à 0.333   (23) 

'99 , 0.667 à à 0.393 P Q NDJ NDJ      (4) '00 , 0.667 à

P T MAM MAM   à 0.333   (24) 

'99 , 0.333 à à P Q DJF DJF    0.273   (5) '99 , 0.333 à à 0.333 P Q SON SON      (25) 

'99 , 0.667 à à P Q DJF DJF    0.393   (6) , 0.333 '99 , 0.667 à à à 0.334 P Q   SON SON SON      (26) 

'00 , 0.333 à à 0.133 P Q JFM JFM      (7) '99 , 0.667 à à 0.333 P Q SON SON      (27) 

'00 , 0.667 à à 0.533 P Q JFM JFM      (8) '00 , 0.333 à à 0.333 P Q JJA JJA      (28) 

'00 , 0.333 à à 0.273 P Q FMA FMA      (9) , 0.333 '00 , 0.667 à à à 0.334 P Q   JJA JJA JJA      (29) 

'00 , 0.667 à à 0.393 P Q FMA FMA      (10) '00 , 0.667 à à 0.333 P Q JJA JJA      (30) 

'00 , 0.333 à à P Q MAM MAM    0.273   (11) '99 , 0.333 à à 0.333 P TSON SON       (31) 

'00 , 0.667 à à P Q MAM MAM    0.393   (12) , 0.333 '99 , 0.667 à

P T    à à SON SON SON   0.334   (32) 

'99 , 0.333 à

P T OND OND   à 0.333   (13) '99 , 0.667 à à 0.333 P TSON SON       (33) 

'99 , 0.667 à

P T OND OND   à 0.333   (14) '99 , 0.333 àP T DJF DJF   à 0.333   (34) 

'99 , 0.333 à à 0.333 P TNDJ NDJ       (15) , 0.333 '99 , 0.667 à

P T    à à DJF DJF DJF   0.334   (35) 

'99 , 0.667 à à 0.333 P TNDJ NDJ       (16) '99 , 0.667 à

P T DJF DJF  	 à 0.333   (36) 

'99 , 0.333 à

P T DJF DJF   à 0.273   (17) '00 , 0.333 à à 0.333 P TJJA JJA   	    (37) 

'99 , 0.667 à

P T DJF DJF   à 0.393   (18) , 0.333 '00 , 0.667 à à à 0.334 P T JJA JJA JJA        (38) 

'00 , 0.333 à à 0.263 P T JFM JFM      (19) '00 , 0.667 à à 0.333 P TJJA JJA       (39) 

'00 , 0.667 à à 0.403 P T JFM JFM      (20) 

Figure 5. Mixed NOAA and EC Meteorology Probability Forecasts Made in September 

1999 over the Maumee River Basin. 

asterisk) shows that above-normal air temperatures are expected for the September-OctoberNovember period, which is interpreted as follows: 

99 ,0.333

,0.333 99 ,0.667

99 ,0.667

à à 0.333

à à à 0.333

à à 0.333

SON SON

SON SON SON

SON SON

P T

P T

P T







    

    

    

 (23) 

All 28 NOAA forecast maps, like Figure 3, which were published in September 1999, were selectively read and interpreted as in (21) and the results are summarized in Figure 5 as equations 1 

through 24. All eight EC forecast maps, like Figure 4, which were available in September 1999, 

were also selectively read and interpreted as in (23) and the results are summarized in Figure 5 as 

equations 25 through 39. The forecasts summarized in Figure 5 are in priority order with the ear-
15 

liest-made forecasts first (NOAA equalities precede EC inequalities), precipitation before temperature second, and chronologically third. 

Note that the precipitation forecasts in Figure 5 are for high precipitation with only one exception 

(the EC SON forecast). The objective in matching these forecasts is therefore (arbitrarily) taken 

as maximizing the probability that precipitation over the period November 1999óJuly 2000 will 

be in the upper third of its historical range (determined from 1961ó1990): 

'99 '00 , 0.667 max à à P Q Nov Jul Nov Jul       (24) 

Daily precipitation and air temperature data from 32 stations were assembled over 1948-1995 for 

the Maumee River basin. (Widespread meteorology observations began in 1948.) The data were 

used to determine the Thiessen-averaged air temperature and the total precipitation, for the periods shown in Figure 5 and (24), which are given in Tables 2 and 3. According to the agencies, 

the NOAA temperature and precipitation forecasts and the EC precipitation forecasts are defined 

relative to historical reference quantiles estimated over the 1961-1990 period. Likewise, the EC 

temperature forecasts are defined relative to historical reference quantiles estimated over the 

1963-1993 period. By ordering data from these periods, the reference quantiles are estimated in 

Tables 4 and 5. 

Consider the objective function of (24) and the forecasts of Figure 5 to apply prior to and through 

the beginning of each year in the sample, so that the time lag accounts for meteorology driving 

hydrology. (The Maumee River annual maximum flow typically occurs as spring snowmelt.) In 

other words, each year of record is to be weighted to reflect the objective of (24) and the beginning winter as forecast in Figure 2 (a total period from September of the year before through the 

following August). For example, the first value in Table 1 for calendar year 1949 corresponds to 

the objective and forecast values for September 1948óAugust 1949. The coefficients in (20) are 

derived from the data set, Figure 2, and (24); see Croley (2000a). In the ensuing optimization, 19 

weights are zeroes, indicating that some of the historical record is not used. However, all but the 

last three equations in Figure 2 are used (corresponding to all forecasts except the EC mostprobable JJA air temperature forecast). 

Suppose now that a hydrologist has acquired, in September 1999, the forecasts of Figure 5 and 

wishes to make an estimate of storm frequencies (annual maximum Maumee River flows) at that 

time for the coming winter and following spring and summer (September 1999óAugust 2000). 

He or she would consider each of the possibilities in Table 1 as a possibility for this period. (The 

Maumee River annual maximum flow typically occurs as spring snowmelt.) The objective function of (24) and the forecasts of Figure 5 would apply prior and through the beginning of each 

year in the sample, so that the time lag accounts for the meteorology driving the hydrology. In 

other words, each year of record is to be weighted to reflect the objective of (24) and the beginning winter as forecast in Figure 5 (a total period from September of the year before through the 

following August). For example, the first value in Table 1 for calendar year 1949 corresponds to 

the objective and forecast values for September 1948óAugust 1949. The coefficients in (20), 

k i , , have values of 1 or 0 corresponding to the inclusion or exclusion, respectively, of each 

variable in the sets indicated in the variable subscripts in (24) and Figure 5. For (24), the reader 
16 

Table 2. Average Air Temperature over the Maumee River Basin (oC). 

 Start in Year Start in Following Year 

Year SON OND NDJ Nov-Apr DJF JFM FMA MAM JJA 

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 

1948 12.28 5.93 2.76 3.69 0.40 1.66 4.64 9.95 23.21 

1949 11.35 6.63 2.30 1.99 0.09 0.10 1.67 7.82 20.70 

1950 11.04 3.55 -1.74 0.80 -3.06 -0.26 3.39 9.47 21.17 

1951 10.49 4.12 -0.54 1.82 -0.94 0.67 4.23 9.03 23.24 

1952 10.99 5.26 2.08 3.22 0.27 1.60 4.38 9.87 23.12 

1953 12.69 6.81 1.61 3.64 0.26 0.73 5.72 9.33 22.08 

1954 12.22 5.47 0.13 2.67 -1.85 -0.36 5.27 11.21 23.02 

1955 11.50 4.18 -1.38 0.75 -2.80 -1.16 2.93 8.38 21.88 

1956 12.08 7.17 0.20 2.41 -1.34 -0.88 4.67 9.61 21.89 

1957 10.82 5.42 1.00 1.78 -2.20 -1.94 2.58 9.28 20.57 

1958 12.01 4.10 -1.90 0.81 -4.52 -1.59 3.59 10.35 23.02 

1959 11.21 5.15 0.75 1.23 -0.81 -2.59 1.73 7.30 21.17 

1960 12.50 4.42 -1.35 1.40 -3.11 0.09 4.21 8.53 21.29 

1961 12.72 5.02 -1.11 0.80 -3.96 -2.41 2.76 10.30 21.49 

1962 11.20 4.18 -2.96 -0.22 -6.56 -3.49 2.58 9.55 21.07 

1963 13.40 5.50 -0.44 1.45 -3.69 -0.58 3.39 10.22 21.84 

1964 11.46 4.82 0.33 1.00 -2.77 -2.50 1.69 8.88 20.58 

1965 11.76 6.22 0.57 1.99 -1.89 -1.20 3.45 8.16 21.97 

1966 10.39 4.40 0.64 1.84 -2.63 -0.91 3.08 8.74 20.88 

1967 9.77 4.51 -1.22 1.14 -3.31 -1.89 3.55 9.38 21.94 

1968 11.74 4.97 -0.70 1.26 -3.11 -2.00 3.26 9.25 21.26 

1969 10.50 3.63 -2.98 -0.15 -4.94 -3.64 2.73 9.46 21.74 

1970 12.10 5.49 -0.97 0.72 -3.20 -2.60 2.45 7.97 21.65 

1971 13.25 7.41 0.69 1.42 -1.96 -2.09 2.17 8.81 20.58 

1972 10.09 3.75 0.09 2.46 -1.81 1.08 4.89 10.14 22.40 

1973 13.01 5.95 0.26 1.95 -2.85 -0.89 3.68 9.42 21.40 

1974 10.40 4.70 0.74 1.30 -1.67 -0.74 1.87 8.52 21.96 

1975 11.53 6.21 0.17 2.98 -1.98 0.29 5.85 10.03 21.08 

1976 8.22 0.94 -5.94 -0.52 -7.41 -3.35 5.02 12.58 21.39 

1977 11.19 3.77 -2.40 -1.65 -7.55 -6.56 -0.90 7.74 21.31 

1978 11.82 4.78 -1.26 0.02 -5.88 -4.05 1.32 9.24 20.96 

1979 11.22 5.48 0.47 0.76 -3.10 -3.04 1.05 8.27 22.06 

1980 10.47 3.50 -1.96 1.07 -3.65 -1.76 4.17 9.22 21.64 

1981 10.34 3.91 -2.20 -0.58 -5.68 -4.16 1.08 9.31 20.75 

1982 11.73 7.16 2.33 3.24 0.51 0.84 4.18 8.52 23.24 

1983 12.12 3.69 -2.96 -0.27 -4.64 -3.06 2.47 6.53 22.05 

1984 11.66 6.72 -0.14 2.22 -3.05 -2.04 4.63 11.79 20.71 

1985 12.50 4.45 -0.89 1.61 -4.19 -0.62 4.16 10.75 21.25 

1986 11.57 5.04 -0.28 2.33 -1.46 0.21 5.00 11.13 22.74 

1987 11.06 5.19 0.78 1.68 -2.88 -2.11 2.60 9.73 23.51 

1988 10.21 3.71 1.16 1.75 -1.94 -0.26 2.36 8.47 21.53 

1989 10.50 2.11 -1.35 1.82 -2.48 2.27 5.05 9.52 21.07 

1990 12.00 6.13 1.04 3.15 -1.33 0.08 5.30 11.90 22.73 

1991 10.96 5.35 0.30 2.09 -0.48 0.36 3.92 8.62 19.72 

1992 10.55 4.80 0.75 1.22 -2.29 -1.83 1.71 8.49 22.16 

1993 10.06 4.22 -2.00 0.37 -4.98 -3.49 2.80 9.14 21.40 

1994 12.35 6.84 1.67 2.30 -1.83 -0.77 2.95 9.06 23.31 
17 

Table 3. Average Daily Precipitation on the Maumee River Basin (mm). 

 Start in Year Start in Following Year 

Year SON OND NDJ Nov-July DJF JFM FMA MAM JJA 

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 

1948 2.56 2.52 3.19 2.88 2.74 2.53 2.04 2.52 3.02 

1949 2.03 2.03 3.27 3.24 4.42 4.47 3.65 2.59 3.80 

1950 3.66 2.87 2.40 2.91 2.04 2.31 2.89 3.00 2.95 

1951 2.33 2.86 3.03 2.82 2.79 2.71 2.68 3.46 1.98 

1952 1.71 1.49 1.91 2.15 1.62 2.13 2.13 2.61 2.17 

1953 1.02 0.96 1.40 2.29 1.97 2.42 2.72 2.67 3.37 

1954 2.81 2.90 1.58 2.15 1.72 2.20 2.47 2.31 2.76 

1955 3.02 2.61 1.64 2.49 1.23 2.03 2.64 3.33 2.81 

1956 0.81 1.31 1.71 2.60 1.70 1.39 2.66 3.32 2.76 

1957 2.79 2.84 2.11 2.53 1.71 0.78 1.15 1.68 4.62 

1958 2.38 1.50 2.30 2.67 2.12 2.86 3.06 3.05 2.48 

1959 2.99 2.74 2.40 2.34 2.24 1.77 1.51 1.85 2.64 

1960 1.24 1.10 0.75 2.39 1.17 2.19 3.73 3.44 2.91 

1961 2.27 1.58 1.97 1.98 1.84 1.95 1.45 1.79 2.27 

1962 1.71 1.25 0.89 1.88 0.72 1.62 2.11 2.53 2.45 

1963 0.87 0.86 1.27 2.24 0.98 2.03 3.15 3.46 2.27 

1964 1.01 1.06 1.98 2.30 2.43 2.58 2.62 2.63 2.60 

1965 2.67 2.42 1.57 2.00 1.49 1.13 1.72 2.08 2.86 

1966 2.49 3.14 3.21 2.62 2.26 1.69 2.19 2.68 1.93 

1967 2.52 3.38 2.91 2.81 2.48 1.49 1.67 2.97 3.05 

1968 2.17 2.33 3.06 2.77 2.12 1.57 1.62 2.62 2.75 

1969 2.99 2.01 1.58 2.49 0.91 1.25 2.40 3.23 2.59 

1970 2.54 1.85 1.44 2.11 1.59 1.57 1.58 1.99 2.41 

1971 2.16 2.13 1.91 2.48 1.71 1.36 2.49 3.30 2.80 

1972 3.95 2.67 2.46 2.94 1.52 2.09 2.40 3.18 3.61 

1973 1.94 2.58 2.74 2.43 2.37 2.41 2.35 2.96 1.94 

1974 1.85 1.96 2.47 2.63 2.34 2.11 2.16 2.50 4.03 

1975 2.08 2.10 2.18 2.51 2.32 2.53 2.55 2.48 2.47 

1976 1.61 1.02 0.61 1.98 1.04 2.03 3.03 2.82 3.35 

1977 2.49 2.28 2.67 2.37 2.21 1.83 2.13 2.72 2.41 

1978 1.75 1.98 2.16 2.45 1.86 1.70 2.03 2.58 3.69 

1979 2.06 2.46 2.16 2.71 1.38 1.80 2.41 2.83 4.05 

1980 1.43 1.34 1.02 2.56 1.43 1.03 2.30 2.71 4.01 

1981 2.74 2.04 2.11 2.46 2.32 2.73 2.29 2.89 2.28 

1982 2.57 2.84 2.84 2.65 1.36 1.10 2.26 3.05 2.24 

1983 3.41 3.81 2.77 2.62 1.65 1.45 2.60 3.45 1.90 

1984 2.62 2.31 2.12 2.38 2.13 2.62 2.57 2.45 2.95 

1985 3.34 3.17 2.44 3.00 1.53 1.64 2.38 2.35 4.35 

1986 2.96 2.09 1.51 1.92 1.10 1.02 1.00 1.91 3.39 

1987 1.76 2.27 1.98 1.71 1.81 1.39 1.68 1.43 2.21 

1988 2.84 2.64 2.30 2.65 1.43 1.42 1.88 3.14 3.15 

1989 2.14 1.50 1.48 2.81 2.36 2.64 2.76 2.76 3.91 

1990 2.62 3.53 2.88 2.41 2.56 1.32 1.98 2.93 2.04 

1991 2.69 2.55 1.52 2.51 1.19 1.63 2.24 2.68 3.80 

1992 3.71 2.71 3.12 2.88 2.15 2.41 2.37 2.45 3.06 

1993 2.75 1.85 2.01 2.10 1.35 1.32 2.04 2.02 2.78 

1994 1.51 1.89 2.36 2.39 1.69 1.51 2.03 2.80 2.77 
18 

Table 4. Historical Reference Quantiles of Average Air Temperature over the Maumee River 

Basin (oC). 

Quantile Period, i 

SONa

 SONb ONDa NDJa Nov-Apra DJFa DJFb JFMa FMAa MAMa JJAb

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) 

, 0.333 ài 10.50 10.50 4.18 -1.22 0.80 -3.69 -3.54 -2.5 2.47 8.45 21.15 

, 0.667 ài 11.74 11.71 5.19 0.26 1.75 -2.63 -2.35 -0.89 3.68 9.31 21.84 

a

Quantiles based on the period: 1961-1990. 

b

Quantiles based on the period: 1963-1993. 

Table 5. Historical Reference Quantilesa

 of Average Daily Precipitation 

on the Maumee River Basin (mm). 

Quantile Period, i

 SON OND NDJ DJF JFM FMA MAM JJA Nov-July 

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 

, 0.333

ài 2.06 1.98 1.91 1.49 1.49 2.11 2.49 2.45 2.37 

, 0.667

ài 2.57 2.42 2.44 2.13 2.03 2.40 2.83 3.01 2.62 

a

Quantiles based on the period: 1961-1990. 

can see from inspection of column 5 in Table 3 that the relation, , 0.667 à qNov Jul Nov Jul     (or 

qNov Jul   2.62 mm ; see column 10 in Table 5) is satisfied by the following indices: 1 (corresponding to 1948), 2, 3, 4, 11, 19, 20, 21, 25, 27, 32, 35, 38, 41, 42, and 45 (corresponding to 

1992). [Index 19 (corresponding to 1966) does not appear to satisfy this relation because of 

round-off error, but in fact does.] Thus, (24) would be written, similar to the first line in (20), as 

 max (w1+w2+w3+w4+w11+w19+w20+w21+w25+w27+w32+w35+w38+w41+w42+w45) (25) 

Expressing (25) in vector form, 

max 11110000001000000011100010100001001001001100100  w (26) 

where w = the column vector of the weights. Likewise, the reader can see from inspection of Tables 2ó5 that (24), (11), and Figure 5 become, similar to (20) [with (11) first in the constraint 

set], the vector equation set shown in Figure 6. In the ensuing optimization of Figure 6, 19 

weights are zeroes, indicating that some of the historical record is not used. However, all but the 

last three equations in Figures 5 and 6 are used (corresponding to all forecasts except the EC 

most-probable JJA air temperature forecast). 

Climate-biased storm frequencies for the annual (calendar year) maximum daily flow can now be 

estimated by applying these weights to the data in column 3 of Table 1 by using (15)ó(18). 

Only results for the fitted Log-Pearson Type III distribution (to simplify the presentation) are 

given in Figure 1. Compare the Log-Pearson Type III distribution derived from the parametric 
19 

max [11110000001000000011100010100001001001001100100] w subject to

[11111111111111111111111111111111111111111111111] w = 47 

[00001100101011111000001000101010100000000100011] w = 0.283  47 

[10110011010100000011000011000001001101001011100] w = 0.383  47 

[00000111100010110100011100001000100000100101000] w = 0.273  47 

[11010000000000000011100011100100001100000010100] w = 0.393  47 

[00000001000010110100010000001001101000101001010] w = 0.273  47 

[11010000000100001011000001110100010000000110100] w = 0.393  47 

[00000000110000000101010100000000101100111010010] w = 0.133  47 

[11111111001010001000000011111000010010000100100] w = 0.533  47 

[10000000010101100101101000000010000000111010011] w = 0.273  47 

[01110111101010011000000100011001000110000100000] w = 0.393  47 

[00000010010101000100001000010000000011110000110] w = 0.273  47 

[00110001101010010001010111000000011100001010000] w = 0.393  47 

[00110001001000100000010010001100110100001100000] w = 0.333  47 

[11001110110000010100001101010001001010000011001] w = 0.333  47 

[00100001001010100001010000001110110100000100010] w = 0.333  47 

[11001100010100001110000100100001001000011011101] w = 0.333  47 

[00000000001001110000010000001110010101000000010] w = 0.273  47 

[11011110110100000100000110110000001000101111101] w = 0.393  47 

[00000000000100101000011000001111010100000000010] w = 0.263  47 

[11111110100010010000000010110000001001101111001] w = 0.403  47 

[01000000000100001000001100100111010100001000100] w = 0.333  47 

[10011110100010000000000010011000101011100111000] w = 0.333  47 

[01000001000100000100001000000101000100000000000] w = 0.333  47 

[10101110101001110001010011011000010011110110000] w = 0.333  47 

[01001100100010111000000001101011100000010000001] w < 0.333  47 

[01101111110110111100010011101011110111111011111] w < 0.666  47 

[00100011010100000100010010000000010111101011110] w < 0.333  47 

[00011000000001110010001001000100011100010010000] w < 0.333  47 

[00100011101110001100110100010000000010000000011] w < 0.334  47 

[00111011101111111110111101010100011110010010011] w < 0.667  47 

[00010000000000000011000010101000110000001100010] w < 0.333  47 

[01101001010100101000010000010101000010110001100] w < 0.334  47 

[01111001010100101011010010111101110010111101110] w < 0.667  47 

[00000000001001110000010000001110110101000000010] w < 0.333  47 

[00100001000010001011101001000001000010010100000] w < 0.334  47 

[00100001001011111011111001001111110111010100010] w < 0.667  47 

[10111111101111010101111011101101101101111010111] w < 0.667  47 

[00100000000111000000111001001100100001001000010] w < 0.334  47 

[10011111101000010101000010100001001100110010101] w < 0.333  47 

Figure 6. Alternate Representation of (24), (11), and Equations in Figure 5 as (20). 

estimates without the forecasts to that made with the forecasts in Figure 1. There is a large shift, 

making all flows more likely to be exceeded. 

Next, consider the annual maximum daily flow defined over the water year. Now, for example, 

the first value in Table 1, column 4, for water year 1949 (1 October 1948ó30 September 1949) 

corresponds to the objective function of (24) and forecast values of Figure 5 for September 

1948óAugust 1949 (to make spring snowmelt reflect the effect of the forecast). This leads to 
20 

the same equation set as obtained before in Figure 6, yielding the same weights. These weights 

are now applied to the water year data in column 4 of Table 1 by using (15)ó(18) to make the 

distribution fit shown in Figure 2. The shift to increased exceedance frequency occurs for flows 

greater than 36000 cfs. Below this flow, there is actually a slight decrease in the exceedance frequency. 

7. ENSO CONDITIONAL FORECAST EXAMPLE 

As a second set of examples, the El NiÒo and La NiÒa conditional probabilities are used to bias 

storm frequency estimates. First, consider where these conditional probability estimates might 

come from. [ENSO-type conditioning has been applied elsewhere for statistical downscaling; 

see for example Katz and Parlange (1993, 1996).] The two phases of the El NiÒo ñ Southern Oscillation (ENSO) are the El NiÒo and La NiÒa events and refer to the oceanic and atmospheric 

circulation in and over the equatorial Pacific. It is recognized that weather in many parts of the 

world is related to the occurrence of El NiÒo and La NiÒa. The study of historical El NiÒo and 

La NiÒa events in Table 6 can yield event probabilities useful in hydrology or other derivative 

outlooks. A simple technique may be applied to derive probabilistic meteorology forecasts that 

consider the influence of El NiÒo, La NiÒa. That is, probabilities of various meteorological 

events can be estimated from the historical meteorological record conditioned on the occurrence 

of El NiÒo, La NiÒa, or the absence of both (Croley 2000a). Then, given that one of these three 

events is occurring at the time of a forecast, the appropriate set of conditional probabilities can be 

used as a probabilistic meteorological forecast. 

The definition of the occurrence of these events is taken from Shabbar and Khandekar (1996). 

Strong to moderate ENSO years are defined as those in which the 5-month running Southern Oscillation Index (mean difference in sea-level pressure between Tahiti and Darwin) remained in 

the lower 25% (El NiÒo) or upper 25% (La NiÒa) of the distribution for 5 months or longer. This 

definition is consistent with that used by Rasmusson (1984), Ropelewski and Jones (1987), and 

Halpert and Ropelewski (1992). Table 6 contains the years of onset of strong or moderate El 

NiÒo and La NiÒa events, as given originally by Shabbar and Khandekar (1996) and corrected 

and extended by Shabbar et al. (1997). 

By inspecting the historical meteorology record for those years of El NiÒo or La NiÒa in Table 6, 

one can estimate the probability of any event following an El NiÒo or La NiÒa with the eventís 

relative frequency. For example, in the Great Lakes, there is much interest in the effects of 

ENSO on winter precipitation and air temperatures. Figure 7 presents selected relative frequencies of precipitation and air temperature over the Lake Erie basin (to be used over the Maumee 

River basin). Given that an El NiÒo or La NiÒa is occurring, the numbers in Figure 7 can be interpreted as forecast probabilities conditioned on the El NiÒo or La NiÒa occurrence. 

The La NiÒa probabilities in Figure 7 are similar to the forecasts of Figure 5; those forecasts are 

predicated on a La NiÒa occurring. Therefore the right-most 12 equations in Figure 7 are used, in 

order, with the objective function of (24) to estimate a La NiÒa influence. Again, one can construct an equation set similar to (20), as at the top of Figure 8, by inspecting Tables 2ó5 for values in (24), (11), and the right-side 12 equations in Figure 7. Optimization of the top equation 
21 

Table 6. El NiÒo and La NiÒa event onset yearsa. 

Year Event Year Event Year Event Year Event 

(1) (2) (3) (4) (5) (6) (7) (8) 

1900 1924 La NiÒa 1948 1972 El NiÒo 

1901 1925 El NiÒo 1949 1973 La NiÒa 

1902 El NiÒo 1926 El NiÒo 1950 La NiÒa 1974 

1903 1927 1951 El NiÒo 1975 La NiÒa 

1904 La NiÒa 1928 La NiÒa 1952 1976 El NiÒo 

1905 El NiÒo 1929 El NiÒo 1953 El NiÒo 1977 

1906 1930 El NiÒo 1954 1978 

1907 1931 1955 La NiÒa 1979 

1908 1932 1956 La NiÒa 1980 

1909 La NiÒa 1933 1957 El NiÒo 1981 

1910 La NiÒa 1934 1958 El NiÒo 1982 El NiÒo 

1911 El NiÒo 1935 1959 1983 

1912 El NiÒo 1936 1960 1984 

1913 1937 1961 1985 

1914 El NiÒo 1938 La NiÒa 1962 1986 El NiÒo 

1915 1939 El NiÒo 1963 1987 

1916 La NiÒa 1940 1964 La NiÒa 1988 La NiÒa 

1917 La NiÒa 1941 El NiÒo 1965 El NiÒo 1989 

1918 El NiÒo 1942 1966 1990 

1919 El NiÒo 1943 1967 1991 El NiÒo 

1920 1944 1968 1992 

1921 1945 1969 El NiÒo 1993 

1922 1946 1970 La NiÒa 1994 

1923 1947 1971 La NiÒa a

After Shabbar and Khandekar (1996) and Shabbar et al. (1997). 

set in Figure 8 matches the first 9 La NiÒa equations in Figure 7 to yield weights for estimating 

the La NiÒa climate-biased storm frequencies by using (15)ó(18). These are for the annual 

maximum (calendar year) daily flow data in column 3 of Table 1 and for the annual maximum 

daily precipitation intensity data in column 2 of Table 1. These are shown in Figures 9 and 10, 

respectively, along with the unbiased storm frequency estimates. (Again, only parametric estimates are shown to simplify the presentation.) 

The El NiÒo probabilities in Figure 7 appear more significant for air temperature than they do for 

precipitation. The objective in matching these forecasts is taken (arbitrarily here) as maximizing 

the probability that air temperature over the period November 1999óApril 2000 will be in the 

upper third of its historical range (determined from 1961-1990). 

'99 '00 , 0.667 à max à P TNov Apr Nov Apr        (27) 

The left-most 12 equations in Figure 7 are used in order with the objective function of (27) to 

estimate an El NiÒo influence. Again, one can construct an equation set similar to (20), as at the 
22 

Maumee River Basin ENSO-Conditional Probability Estimates (as Forecasts) 

SON 1999, DJF 1999, and MAM 2000 Air Temperature (T) & Precipitation (Q) 

(Based on 1948-1995 data and 1961-1990 quantiles) 

 Conditional on El NiÒo Conditional on La NiÒa 

'99 , 0.333

'99 , 0.667

'99 , 0.333

'99 , 0.667

'99 , 0.333

'99 , 0.667

'99

à à 0.364

à à 0.273

à à 0.273

à à 0.273

à à 0.273

à à 0.636

à à

SON SON

SON SON

SON SON

SON SON

DJF DJF

DJF DJF

DJF DJ

P T

P T

P Q

P Q

P T

P T

P Q















    

    

    

    

    

    

 , 0.333

'99 , 0.667

'00 , 0.333

'00 , 0.667

'00 , 0.333

'00 , 0.667

0.636

à à 0.273

à à 0.091

à à 0.364

à à 0.273

à à 0.364

F

DJF DJF

MAM MAM

MAM MAM

MAM MAM

MAM MAM

P Q

P T

P T

P Q

P Q











   

    

    

    

    

    

'99 , 0.333

'99 , 0.667

'99 , 0.333

'99 , 0.667

'99 , 0.333

'99 , 0.667

'99

à à 0.111

à à 0.444

à à 0.444

à à 0.444

à à 0.000

à à 0.444

à à

SON SON

SON SON

SON SON

SON SON

DJF DJF

DJF DJF

DJF DJ

P T

P T

P Q

P Q

P T

P T

P Q















    

    

    

    

    

    

 , 0.333

'99 , 0.667

'00 , 0.333

'00 , 0.667

'00 , 0.333

'00 , 0.667

0.222

à à 0.444

à à 0.556

à à 0.111

à à 0.222

à à 0.778

F

DJF DJF

MAM MAM

MAM MAM

MAM MAM

MAM MAM

P Q

P T

P T

P Q

P Q











   

    

    

    

    

    

Figure 7. ENSO Event Conditional Probability Forecast Equations for Maumee River 

Basin. 

bottom of Figure 8, by inspecting Tables 2ó5 for values in (27), (11), and the left-side 12 equations in Figure 7. Optimization of the bottom of Figure 8 matches all 12 El NiÒo equations in 

Figure 7 to yield weights for estimating the El NiÒo storm frequencies in Figures 9 and 10. 

Figure 9 shows that the effect of considering La NiÒa conditions, with the objective of maximizing higher precipitation probability, increases the probability for all flows to be exceeded. The 

effect of El NiÒo conditions, with the objective of maximizing higher temperature probability, 

increases the exceedance probability for flows below about 77000 cfs, but decreases it for flows 

above. In Figure 10, the effects on maximum precipitation intensity are more mixed. La NiÒa 

and the maximization of higher precipitation probability increase the exceedance probability for 

precipitation rates above about 3.5 mm/day but decrease it below. El NiÒo and the maximization 

of higher temperature probability decrease the exceedance probability for precipitation rates 

above about 2.7 mm/day and only slightly increase it above. 

Even though one can exactly match meteorological forecasts, that does not mean that the resulting climate-biased storm frequencies contain no other errors. Sampling errors still exist and can 
23 

THE OBJECTIVE OF (24), (11), AND THE RIGHT-MOST 12 EQUATIONS (LA NI—A) IN FIGURE 7: 

max [11110000001000000011100010100001001001001100100] w subject to

[11111111111111111111111111111111111111111111111] w = 47 

[00010000000000000011010010101000110000001100010] w = 0.111  47 

[10000110101011010100001101000010000101000010001] w = 0.444  47 

[01001100100010111000000001101011100000010000001] w = 0.444  47 

[00100011010100000100010010000000010111101011110] w = 0.444  47 

[00000000001001110000010000001110010101000000010] w = 0.000  47 

[11011110110100000100000110110000001000101111101] w = 0.444  47 

[00000001000010110100010000001001101000101001010] w = 0.222  47 

[11010000000100001011000001110100010000000110100] w = 0.444  47 

[01000001000100000100001000000101000100000000000] w = 0.556  47 

[10101110101001110001010011011000010011110110000] w = 0.111  47 

[00000010010101000100001000010000000011110000110] w = 0.222  47 

[00110001101010010001010111000000011100001010000] w = 0.778  47 

THE OBJECTIVE OF (27), (11), AND THE LEFT-MOST 12 EQUATIONS (EL NI—O) IN FIGURE 7: 

max [11011110110000000110000011010000001010100111001] w subject to

[11111111111111111111111111111111111111111111111] w = 47 

[00010000000000000011010010101000110000001100010] w = 0.364  47 

[10000110101011010100001101000010000101000010001] w = 0.273  47 

[01001100100010111000000001101011100000010000001] w = 0.273  47 

[00100011010100000100010010000000010111101011110] w = 0.273  47 

[00000000001001110000010000001110010101000000010] w = 0.273  47 

[11011110110100000100000110110000001000101111101] w = 0.636  47 

[00000001000010110100010000001001101000101001010] w = 0.636  47 

[11010000000100001011000001110100010000000110100] w = 0.273  47 

[01000001000100000100001000000101000100000000000] w = 0.091  47 

[10101110101001110001010011011000010011110110000] w = 0.364  47 

[00000010010101000100001000010000000011110000110] w = 0.273  47 

[00110001101010010001010111000000011100001010000] w = 0.364  47 

Figure 8. Alternate Equations Sets for La NiÒa and El NiÒo Examples as in (20). 

be pronounced if many meteorological time series segments are weighted by zero and effectively 

eliminated from the sample. This happened in the ENSO conditioned examples. Only 11 La 

NiÒa example weights were non-zero (out of 47) and only 19 El NiÒo weights were non-zero. 
24 

0.00

0.20

0.40

0.60

0.80

1.00

0 20000 40000 60000 80000 100000 120000

Annual Maximum Daily Flow (cfs)

Exceedance Probability

Max. Temp. El Nino

Normal Empirical

Max. Prec. La Nina

Figure 9. ENSO Annual Maximum (Calendar Year) Daily Maumee River Flow Exceedance. 

0.00

0.20

0.40

0.60

0.80

1.00

0123456

Annual Maximum Daily Precipitation (mm)

Exceedance Probability

Max. Temp. El Nino

Normal Empirical

Max. Prec. La Nina

Figure 10 ENSO Annual Maximum Daily Maumee River Basin Precipitation Exceedance. 
25 

8. MULTIPLE SOLUTIONS 

In the optimization of (20), all expressions are linear, including the objective function [first line 

in (20)]. This allows a linear programming optimization technique to be used as compared to 

earlier formulations (Croley 1996, 1997b). Those used the minimization of the sum of squared 

differences of each weight with unity,  2

 wi  1 , and employed classical differential calculus solutions for zero slope of the Lagrangian. The linear formulation proves superior in its ability to include alternate objectives (expressed as maximization of selected event probabilities). 

Also, the formulation of (20) allows non-negativity constraints on the weights to be explicitly included. This means that all solutions can be searched. The earlier formulations of Croley (1996, 

1997b) lacked explicit inclusion of non-negativity constraints for the weights. There, optimum 

solutions were considered and discarded (along with lowest-priority constraints) if non-negativity 

constraints were unsatisfied. But that also discarded the many other possibilities that, while not 

optimum, might satisfy all constraints. 

There is a trade-off however. Multiple optima solutions are now a possibility that did not exist 

before. In the search algorithms employed in the linear programming solutions, these multiple 

optima can be detected (that is, the existence of more than a single optimum can be discerned) 

but the systematic exploration of them can be extensive. Croley (2000a,b) describes this further. 

Here, it is sufficient to note that, in the examples of Figures 1 and 2, the optimum was unique. In 

the examples of Figures 9 and 10, the optima were not unique. The optimization for the La NiÒa 

examples yielded 8 solution points in the systematic search with the Simplex method. Four of 

these were unique but discovered twice. The optimization for the El NiÒo examples yielded over 

5000 solution points (the limit on the search); 36 were unique. In both of these cases, the average of the found unique optima were used for weighting (biasing) the sample in estimating storm 

frequencies. 

9. SUMMARY AND OBSERVATIONS 

The methodology described herein allows one to recognize changing climate in the estimation of 

storm frequencies, removing one of the worst assumptions associated with this, which is that future probabilities are the same as the past. Existing forecasts of meteorology probabilities can be 

used to bias storm frequency estimates for a changing climate. The methodology is adapted from 

earlier work that uses forecasts of meteorology probabilities to derive forecasts of consequent 

hydrology probabilities in an operational hydrology approach. The linear objective function used 

here enables incorporation of an event probability into the objective, use of existing optimization 

techniques, and direct inclusion of non-negativity constraints. 

The examples presented here are provided to supplement and further illustrate the methodology 

described elsewhere (Croley 2000a,b). These examples may be more representative of storm frequency estimation in an operational setting rather than in a design setting. Climate-biased storm 

frequencies were estimated by preserving meteorology forecasts. These conditions are current 

and are not generally regarded as applying over a very long time into the future. The resulting 

biased storm frequencies can only be considered applicable over the same time period as the meteorology forecasts or other event probabilities used to condition them. The examples given here 
26 

applied over the next several months, appropriate for use in an operational setting. If probabilities can be defined (estimated) corresponding to climate shifts expected from the present forward, then the resulting biased storm frequencies could be used in a design setting. 

Complete software, in the form of an easy-to-use interactive Windows graphical user interface, 

and worked examples are available free of charge over the World Wide Web. The software, examples, and tutorial materials may be acquired in a self-installing file by visiting the web site entitled: http://www.glerl.noaa.gov/wr/OutlookWeights.html and downloading. 

10. REFERENCES 

Chow, V. T., 1964. Handbook of Applied Hydrology. McGraw-Hill, New York, 8-29. 

Croley, T.E., II, 1996. Using NOAA's New Climate Outlooks In Operational Hydrology. Journal of Hydrologic Engineering, ASCE, 1(3):93-102. 

Croley, T.E., II, 1997a. Water resource predictions from meteorological probability forecasts. 

Proceedings, Sustainability of water resources under increasing uncertainty (Proc. Rabat 

Symp.), D. Rosbjerg et al., ed., IAHS Publication No. 240, IAHS Press, Wallingford, U.K., 

301-309. 

Croley, T.E., II, 1997b. Mixing Probabilistic Meteorology Outlooks In Operational Hydrology. 

Journal of Hydrologic Engineering, ASCE, 2(4):161-168. 

Croley, T. E., II, 2000a. Using Meteorology Probability Forecasts In Operational Hydrology. 

ASCE Press, American Society of Civil Engineers, Reston, Virginia, 214 pp 

Croley, T. E., II, 2000b. Climate-Biased Storm-Frequency Estimation. Journal of Hydrologic 

Engineering, ASCE, (in review). 

Halpert, M. S., and C. F. Ropelewski, 1992. Surface temperature patterns associated with the 

Southern Oscillation. Journal of Climate 5:577ñ93. 

Hillier, F. S., and G. J. Lieberman, 1969. Introduction to Operations Research. Chapter 5: Linear Programming, Holden-Day, San Francisco, 127-171. 

Katz, R. W., and M. B. Parlange, 1993. Effects of an index of atmospheric circulation on stochastic properties of precipitation. Water Resources Research 29:2335-2344. 

Katz, R. W., and M. B. Parlange, 1996. Mixtures of stochastic processes: application to statistical downscaling. Climate Research 7:185-193. 

Koutrouvelis, I. A., and G. C. Canavos, 1999. Estimation In The Pearson Type 3 Distribution. 

Water Resource Research, AGU, 35(9):2693-2704. 

Rasmusson, E. M., 1984. El NiÒo: The ocean/atmosphere connection. Oceanus 27(2):5ñ12. 
27 

Ropelewski, C. F., and P. D. Jones, 1987. An extension of the Tahiti-Darwin Southern Oscillation index. Monthly Weather Review 115:2161ñ65. 

Shabbar, A., B. Bonsal, and M. Khandekar, 1997. Canadian precipitation patterns associated 

with the Southern Oscillation. Journal of Climate 10:3016ñ27. 

Shabbar, A., and M. Khandekar, 1996. The impact of El NiÒoñSouthern Oscillation on the temperature field over Canada. Atmosphere Ocean 34(2):401ñ16. 

U.S. Water Resources Council, 1967. A Uniform Technique For Determining Flood-Flow Frequency. Bulletin 15, Hydrology Committee, Washington, DC, 1967.