Daily Quantification of Drought Severity and Duration

Hi-Ryong Byun*
Department of Atmospheric Sciences, Pukyong National University, Republic of Korea

and

Donald A. Wilhite
National Drought Mitigation Center, University of Nebraska, Lincoln, Nebraska, USA

(*Currently visiting National Drought Mitigation Center at University of Nebraska)


Abstract

Common weaknesses of current drought indices were analyzed. First, most of the current indices are not precise enough in detecting the onset, end, and accumulating stress of drought. Second, they do not effectively take into account the aggravating effects of runoff and evapotranspiration, which build up with time. Third, they have a limited usefulness in monitoring ongoing drought because they are based on a monthly time step. Fourth, most of them fail to differentiate the effects of drought on surface and sub-surface water supply.

A new series of indices are proposed to solve these weaknesses and to improve drought monitoring. In the new indices, daily, rather than monthly, time steps are used. A new concept, Effective Precipitation (EP), the summed value of daily precipitation with a time-dependent reduction function, is proposed as a basic tool.

Three additional indices complement EP. The first index is each day's Mean of EP (MEP). This index shows climatological characteristics of precipitation as a water resource for a station or area. The second index is the Deviation of EP (DEP) from the MEP. The third index is the Standardized value of DEP (SEP). By using these three indices, the onset and ending date of a water deficit period is categorized.

With the duration categorized, five additional indices that can show drought severity are calculated : (1) Consecutive Days of Negative SEP (CDNS), which can show how long the deficit of precipitation has lasted; (2) Accumulation of Consecutive Negative SEP (ACNS), which shows the duration and severity of precipitation deficit together; (3) Accumulated Precipitation Deficit (APD), which shows precipitation departure from the norm al during a defined period; (4) Precipitation for the Return to Normal (PRN) : and (5) Effective Drought Index (EDI), a standardized index which can be used to assess drought severity worldwide. The merits and weaknesses of each index are compared. New quantified defini tions on drought and its onset, end, and duration are proposed.

These indices were tested in the High Plains region of the United States from 1960 to 1996. The results were compared to historical reports of drought. From this analysis, it was concluded that the new indices not only advanced objectivity and rationality, but also offered a number of advantages in practical use. These are (1) a more precise determination of drought duration, (2) the usefulness in monitoring an ongoing drought and (3) the variety of ways a drought's characteristics can be described.

[Software is available via ftp with no charge. Contact the author at byun@enso.unl.edu or click here to download it. After you download it, please untar ``DDAM.tar'' and read ``README.DDA'' first in the directory of ``DDAM''.]


1. Introduction

Most current drought severity indices are reviewed in this paper. They are the Bhalme and Mooly Drought Index (BMDI; Bhalme and Mooley 1980), Crop Moisture Index (CMI; Palmer, 1968), Deciles (Gibbs and Maher 1967), National Rainfall Index (RI; Gommes and Petrassi 1994), Palmer Drought Severity Index (PDSI; Palmer 1965), Percent Normal (PN; Willeke et al. 1994), Rainfall Anomaly Index (RAI; Rooy 1965), Reclamation Drought Index (RDI; Weghorst 1996), Standardized Precipitation Index (SPI; Mckee et al. 1993, 1995), Surface Water Supply Index (SWSI; Shafer and Dezman 1982), etc. Recently, the Soil Moisture Drought Index (SMDI; Hollinger et al. 1993) and Crop-Specific Drought Index (CSDI; Meyer et al. 1993; Meyer and Hubbard 1995) appeared after the CMI. Futhermore, CSDI is divided into a Corn Drought Index (Meyer et al. 1993) and Soybean Drought Index (Meyer and Hubbard 1995). Except these, the indices made by Penman (1948), Thornthwaite (1948, 1963), Keetch and Byrum (1968) had been used (Steila 1983, 1986; Hayes 1996). The characteristics of each index are summarized in Table 1. It is recognized that the PDSI is still the most widely used index and more advanced and scientific one is needed. This study starts assessing the usefulness and pertinency of each index, explores the weak points of them, and finally, new indices are presented as a solution.

Table 1: Characteristics of current drought indices
Name Factors used Time scale Main concept
PDSI r, t, et, sm, rf m Based on moisture inflow, outflow and storage.
SWSI P, sn m Like the PDSI except considering sn.
PN r m Dividing actual r by the normal value
Deciles r m Dividing the distribution of the occurrences over a long term r record into sections each representing ten percent.
SPI r 3m, 6m, 12m, 24m, 48m Difference of r from the mean for a particular time and dividing it by the standard deviation.
CMI r, t w Like the PDSI except considering available moisture in top 5 feet of soil profile.
SMDI sm y Summation of daily sm for a year.
CSDI ev s Summation of the value of calculated ev divided into possible ev during the growth of specific crops.
RI r y, c Patterns and abnormalities of r on a continental scale.
RAI r m, y r compared to arbitrary value of +3 and -3, which is assigned to the mean of ten extreme + and - anomalies of r.
BMDI r m, y Percent departure of r from the long term mean.
Abbreviations :
P: factors used in PDSI, r: precipitation, et: evapotranspiration, ev: evaporation, t: temperature, sm: soil moisture, rf: runoff, sn: snow pack, w: week, m: month, s: season, y: year, c: century, 3m: 3 months, etc.


2. Problems of Current Indices

2.1 Defining the period of water deficit

Drought occurs with the deficiency of fresh water resources from the climatological mean. Important is that it is not only the deficiency at a specific time, but the consecutive occurrence of a deficiency. Therefore, when water deficit period began and how long it has lasted are very important. But most of current indices only assess the deficiency of water from the climatological mean on some predefined duration. Futhermore, no objective method defining the duration is known.

2.2 Time unit

Most current drought indices use a monthly or longer time period as a unit, as shown in Table 1. No index uses a daily unit. But the day unit should be used because an affected drought region can return to normal condition with only a day's rainfall. Futhermore, it is important that the drought intensity be reevaluated frequently and be presented at any time. This would allow the general public to prepare against the risks.

2.3 Storing term of water resources

Drought damage can be categorized into two kinds of causes. One is damage from a shortage of soil moisture, another is from shortage of reserved water. Soil dryness is influenced by a recent short term deficiency of precipitation, and water resources deficit in reservoirs or other sources are affected by much longer term precipitation totals. It is not easy to imagine other drought damages which are not associated with these two categories. So, categorizing these two separately is better method to assess drought. But it is very hard to find current drought indices which divide the two.

2.4 Considering the diminishing effect of water resources with time passage

After rainfall, soil moisture diminishes day by day as a function of a runoff and evapotranspiration ratio. One day's diminishing of water is not small enough to simply be ignored. Therefore, because simple summation of the precipitation cannot provide good results, a time dependent reduction function is needed to estimate the current water deficiency. However, almost all current drought indices use simple summation of precipitation.

2.5 Data used

Besides precipitation, current drought indices are calculated from the data of soil moisture, inflow and outflow by waterways, evaporation and evapotranspiration etc. But most of the parameters are not observered but have to be estimated from some meteorological data. During the estimation, unreasonable simplification is inevitable because these parameters are strongly dependent on the nature of the soil and topography, which vary widely. Also, the important fact that the origin of water included in these parameters is nothing but the rainfall itself, may be disregarded. Olapido (1985), after comparison of the PDSI (Palmer 1965), with the RAI (Rooy 1965) and BMDI (Bhalme and Maher 1967), said that using only the precipitation data is better for the purpose of the meteorological use. Alley (1984) voiced the same opinion.

2.6 Various informations

When drought occurs, important informations to the public are, except the problens connected to prediction, how long the drought has lasted, how much the deficit of water occured and how much rainfall is needed today for the return to normal conditions etc. But, a very few indices (PDSI, SPI) only account for the precipitation value needed for the return to normal conditions in some manner.


3. Calculation and Application of Effective Rainfall Amount (ERA)

3.1 Calculation of ERA

A new concept of Effective Rainfall Amount is proposed to solve these problems and shown by the next equation:
\begin{displaymath} ERA_i = a\sum_{N=1}^iF\end{displaymath} (1)
where a is total constant, i is Summation Duration (SD) and F = summation of precipitation with daily reduction function, which is $(\sum_{m=1}^NP_m)*f(N)$. Here, Pm is the daily precipitation of the day D-m and f(N) is daily reduction function.

SD is the number of the days whose precipitation is summed for calculation. In this study, two dummy values of SD are used to detect real SD. One is 365 and the other is 14. Also, `a' is considered to be 1 and f(N) to be 1/N. The explanations the numbers of 365 and 14 and 1/N being chosen, are found in chapter 3.2 and 3.3. Then finally,

\begin{displaymath} ERA_{365} = \sum_{N = 1}^{365}\frac{\sum_{m = 1}^NP_m}{N} = \sum_{N = 1}^{365}P_ m\sum_{m = N}^{365}\frac{1}{m},\end{displaymath} (2)
\begin{displaymath} ERA_{14} = \sum_{N = 1}^{14}\frac{\sum_{m = 1}^NP_m}{N} = \sum_{N = 1}^{14}P_m\s um_{m = N}^{14}\frac{1}{m}.\end{displaymath} (3)

ERA365 denotes the water resources stored for 365 days and is used for the detection of a water resources drought. Also, ERA14 shows the water resources stored for 14 days and is used for the detection of soil moisture deficiencies.

For an easier understanding of this equation, let's assume a case of ERA2. When m is equal to 1, F equals P1/1, and when m is equal to 2, F becomes (P1+P2)/2. Because m varies from 1 to 2, ERA2 becomes (P1+(P1+P2)/2).

3.2 Discussion on the ERA

a. On daily reduction function

ERA365 of D-day is calculated by a summation of the precipitation from day D-1 till day D-365 along the equation (2). A difference of ERA365 from the simple summation of the 365 days daily precipitation data is that the precipitation of the D-N day is included in ERA365 as a form of mean precipitation averaged for N days from D-1 through the D-N day. This summation method works to produce a heavy storing effect for recent precipitation and a light storing effect for the precipitation many days before.

Several trials to find a better function than (1/N) failed to improve the results. The `a' of Equation (1) also can work as a function for reduction effect. It is used as 1 in this study and will be elaborated on more in the future. It was nominated as the Effective Rainfall Amount (ERA) for reduction function is used to represent the total stored water resources available at the pointed day.

b. On dummy SDs

First of all, the most inportant problem in detecting drought severity is to determine that how long the precipitated water have been in deficit. Dummy value of 365 was chosen temporary, because one year is the most dominant precipitation cycle over the whole world.

Otherwise, the factors which affect on soil moisture, such as topography, vegetation, soil characteristics, seasonal or other meteorological conditions etc., are so complex that ERA14 can not simply be a representative value of total water resources stored for a short period in soil. But, 14 days was chosen because it is apparent that at a plane bare top soil, soil moisture deficiency by 14 consecutive days of little precipitation during summer can cause damage. Also, no other exact solution exists. It may be linked to the same reasoning that a `Drought Spell' in Europe has been categorized as 14 days' consecutive little precipitation, a concept that has been used for a long time (Byun and Han 1994; Doornkamp et al. 1980).

c. Application using the ERA

Once the ERA (of 365 or 14 days) is computed, a series of calculations can be made to highlight different characteristics of a location's water resources. The first step beyond ERA is shown as the Mean of ERA (MERA) per day-number. Because a strong daily variation of MERA is not helpful for practical use, a 5-day running mean is applied. The second step is DERA, which is the deviation of ERA from the MERA as seen in equation (4):

DERAi = ERAi - MERAi.

(4)

The DERA shows the deficit or sufficiency of water resources for a particular date and place. The next step is SERA, which is the standardized (divided by standard deviation of each day-number's ERA) value of DERA. In this case also, standard deviation is used incorporating a 5-day running mean.

d. Quantification of dry duration

Because the negative values of the DERA or SERA denote a period of water deficit, a dry duration can be defined as the period of consecutive negative values of SERA (or DERA) and SD in equation (1) can be categorized by a similar method. For example, if 35 days of consecutive negative SERA365 occurred and ended at D-day, SD of D-day is 399 (365+35-1) and that of D-1 day is 398. The dry duration of this period are 35.

Drought duration should be different from dry duration because drought means not only a `long lasting' but also `severe' water deficiency. Table 2 and the following chapter will address this problem again.


Table 2: New definitions
Name Definition
Dry duration Period of consecutive negative values of SERA365 (or SERA14).
Summation duration (SD) Dry duration added to 365 (or 14) (SD) on D-day. Number of days whose precipitation is summed into the calculation. `i' in Equation (1).
Drought duration Period that the EDI shows values less than -1.0. Dry duration between drought durations is included if no positive EDI is involved.
Water Resources Drought indices Indices calculated from SERA365.
Soil Drought indices Indices calculated from SERA14.


4. Quantification of Drought Severity

After the SDs are found, many kinds of drought severity indices can be defined. More indices and their meanings, merits and demrits are as follows (Table 3).

4.1 Accumulation of consecutive NEgative SERA (ANES)

All positive SERAs are translated to zeroes within the ANES. Only consecutive negative SERAs are accumulated and make ANES. One benefit of the ANES is that drought duration is easily imagined by the ANES because the absolute value of SERA is almost always less than 2.0.

4.2 Accumulated Precipitation Deficit (APD)

APD is calculated by a simple accumulation of precipitation deficit as seen in equation (5):
\begin{displaymath} APD_i = \sum_{N = 1}^iP_N - AVG_i.\end{displaymath} (5)

4.3 Precipitation for the Return to Normal (PRN)

Negative values of DERA can be calculated directly into the today's Precipitation needed for the Return to Normal condition (PRN) as follows:
\begin{displaymath} PRN_i = \frac{DERA_i}{\sum_{N = 1}^i\frac{1}{N}}.\end{displaymath} (6)

Then, as an example, PRN400 shows the needed precipitation for recovery from the deficit accumulated during the last 400 days, in which daily deminishing effect of water resources are taken into account. It should be noted that PRN400 should be calculated by the use of ERA400, MERA400, and DERA400.

4.4 Effective Drought Index (EDI) and other indices

Other indices, which can represent drought intensity are possible like those displayed in equations (7) - (10):

PNSK = APDi/AVGi (7)
\begin{displaymath} PNS_{i1} = PRN_{i1}/ST[AVG_{365}], or \ PNS_{i2} = PRN_{i2}/ST[AVG_{14}] \end{displaymath} (8)
\begin{displaymath} EDI_{i1} = PRN_{i1}/ST[{MERA_{365}/\sum_{N = 1}^{365}\frac{1}{N}}]\end{displayma th} (9)
\begin{displaymath} EDI_{i2} = PRN_{i2}/ST[{MERA_{14}/\sum_{N = 1}^{14}\frac{1}{N}}].\end{displaymat h} (10)

Here, ST[f(N)] denotes the standard deviation of f(N), and i1 is SD decided by the SERA365 and i2 by SERA14. Equation (6) is known as the Percent Normal as a Second kind (PNS) because it is similar to the Percent Normal (Willeke et al. 1994) or Deciles (Gibbs and Maher 1967) except that duration is defined more scientifically in comparison by use of the SERA. Equation (8) can make another kind of index and helpful for the easy understanding of equations (9) and (10).


Table 3: Characteristics of each indices. The `i' is 365 or 14. The `j's are determined by consecutive negative SERAi.
Name Calculation Simplified meaning merits demerits
ANESi Accumulation of consecutive negative SERA Index shows the accumulated stress during drought a, b, c, d, e 3, 4
APDj Eq. 5 Accumulated deficit of precipitation f, g 1, 2, 4, 5
PRNj Eq. 6 Precipitation needed for the return to normal conditions a, c, e, f, g 2, 4
PNSKj Eq. 7 Ratio of APD to the average precipitation for the same duration a, c, e 1, 2, 4, 5
PNSj Eq. 8 Ratio of PRN to the standard deviation of Average i a, c, d, f 2, 4
EDIj Eq. 9 or 10 Ratio of PRN to the standard deviation of MERAi a, c, d, e 2
Abbreviations :
a) Takes into account the daily deminishing effects
b) Dry duration can be imagined without another index
c) No possibility to fail to represent drought severity
d) Efficient in comparison to other places' index whose
climatic condition is different
e) Efficient to represent drought severity in a timely manner.
f) More efficient in comparison to other years' index in same place than to other place's index.
g) Easy to understand for the general public.
1) Does not take into accounts the effects of run-off and evapo-transpiration.
2) Needs another value to show dry duration.
3) May mistake a period of consecutive weak water deficit as severe drought.
4) Not efficient in comparison to other places' index whose climatic condition is different.
5) Weak to represent the drought severity in a timely manner.


5. Quantification of Drought Duration

Drought duration can be defined as the consecutive days of EDI which is less than (-1.0). Also, the duration of consecutive negative SERA values between drought durations, although their value are larger than -1.0, have to be included in the drought duration if no positive EDI is involved. Because all the indices can be divided into two groups, drought definitions can also be divided into two groups. The drought associated with 365 days of ERA is nominated as a water resources drought and 14 days of ERA represents soil drought (Table 2).


6. Application to Real Data and Discussion

6.1 Data control

Initially, 37 years (from 1960 till 1996) of daily precipitation data for 193 stations of High Plains Region of U.S.A. were chosen for this study. And all of daily indices explained before were calculated per station during 36 (from 1961 till 1996) years. Data control was required and carried out because so many missing data were detected. After all, 113 stations' data were used.

6.2 Annual variations

In Fig. 1, from August 20, 1995 (day number is 232) till May 8, 1996 (493), values of ERA365 are smaller than that of MERA365 which shows one simple annual cycle. SERA365 translated these deviations to long lasting consecutive negative values, which is 627 days of dry duration. The largest value of SD is detected as 991. An abrupt rising of ERA365 and SERA365 at the May 9, 1996 means a large rain event started on May 8.

Drought is considered to be started on September 16, 1995 and lasted until May 8, 1996 in which 40 days of consecutive negative SERA365 which are bigger than -1.0 included in. Drought severity is shown by EDIi, ANESi, APDi, PRNi. It needs notice that each day has its own `i' decided by help of SERA365. The ANESi shows minimum of -299.8 at May 8, 1996. The APDi shows a minimum of 214.4 mm on April 29, 1996 (its day number in this figure is 484). The PRNi shows a minimum of 70.5 mm on the same day. The minimum EDIi appears on March 16, 1996 (439) with a value of -2.5. These discordance of minimum value along the indices means that ANES, APD and PRN is also needed for describing the drought situation although EDI shows drought severity well.

After heavy rains on May 8, 9 and 10, no negative values of SERA365 or DERA365 are detected throughout the remainder of the year. Then SD, which are zeroes, simplified all indices as zeroes. Figures associated with ERA14 (not shown) are not so simple as Fig. 1 due to the large seasonal fluctuation of precipitation. Heavy rains on May 8, 9, 10 affects the index for only a brief while and many negative value of SERA14, and DERA14 are detected during the year.

Fig. 1. Daily variations of an ERA365 series and drought severity indices at Hickman, Nebraska, USA from January 1, 1995 till December 31, 1996. SERA, ANES and EDI have no dimension. Other indices' units are mm.


6.3 Yearly variations

Fig. 2 shows each of the four indices' averages per year for all the 113 stations' yearly minimums. The four indices are nearly in-phase. Riesman et al. (1991) roughly figured that only the droughts of 1910, 1917, 1936 and 1956 in United States of America were severer than that of 1989. All of them except 1989 are out of the range of this study. The EDI, PRN and ANES show a minimum value at 1989 through the record. This fact partly verifies that the indices computed by ERA function is superior to APD in showing drought intensity.

Fig. 2. Time series of the averaged yearly minimum values of each drought index through 113 stations in the High Plain region of the United States of America.


7. Summary and Conclusions

Through comparative analysis, seven weaknesses of current drought indices were discussed and a new series of indices were proposed to address these weaknesses. A new concept of ERA was used for the solution. The MERA, DERA and SERA after ERA were used for detection of water deficit period. After deciding of water deficit period, the ANES, PRN, PNS, PNSK, APD and EDI were calculated as indices of drought intensity in assessing drought severity. The benefits and weaknesses of each index were discussed. It is also proposed that two kinds of time scales on drought, which are long lasting drought and short term drought needed to detect drought severity effectively. A new quantified definitions of drought duration, dry duration and SD were also made.

Applications using real data were carried out over the High Plains Region in the United States of America. It was found that all of new indices were good enough to show the drought intensity, the EDI was the best index in assessing drought severity worldwide and the PRN was the best suited for limited areas in timely manner etc. In this study, new techniques were used only to assess the water deficit, but this technique can be applied for assessing water surplus also. It is not so hard to imagine that the disasters associated with the lack or surplus of water resources can be assessed, monitored and predicted more objectively and quantatively and can be mitigated effectively by using these new techniques.

Acknowledgments. This study greatly benefited from several hours of sincere discussion and review with Dr. Donald A. Wilhite, Dr. Michael Hayes and Mr. Mark Svoboda at National Drought Mitigation Center and Dr. Yong-Qyun Kang of Pukyong National University, Dr. Ying-Hwa Kuo at NCAR. The author would also like to thank, Sam-Yeon Jeong, Tae-Kook Kim, Mo-Rang Her, Byung-Hown Lim and Hae-Yeong Ko for their help on the processes of long lasted trials and errors. Also, thanks to Jun-Suk Jung, Dong-Yl Lee and Mi-Hyun Lim for their help on data collection. This research was supported by the LG Yonam foundation.

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Prof. Hi-Ryong Byun
Dept. of Atmospheric Sciences
Pukyong National University
Namku, Pusan, 608-737, Republic of Korea
E-mail: byun@dolphin.pknu.ac.kr

Currently visiting at:
National Drought Mitigation Center
University of Nebraska
Lincoln, Nebraska, USA
E-mail: byun@enso.unl.edu