INTRODUCTION
In order to obtain drought indices for any region,
generally several stages should be followed, includes: determining the
parameters type and their scale appropriate to the desirable goal, calculating
the moisture departures based on an assumed comparison value, scaling
the departures and finally classifying them in a predefined categories.
Among drought indices, Palmer Drought Severity Index, PDSI (Palmer, 1965),
in a way, between the drought indices, has been a landmark in the development
of drought indices, even though it is not without limitation (Heim, 2002).
Therefore, a review of the PDSI will be valuable, in the beginning, which
can be divided in two parts:
Initially, the actual and potential values of four parameters
i.e., evapotranspiration (ET), moisture loss (L), soil water recharge
(RE) and runoff (RO) are estimated. Then, the average ratio of the actual
factors to the corresponding potential values, are calculated, named water
balance coefficients (α, β, γ, δ). At last, the climatically
appropriate precipitation for the existing conditions, Pr_{C},
is approximated (Guttman, 1998):
Pr_{C} = α
x ETp + β x REp + γ x RO_{p}  δ x Lp 
(1) 
Then, the moisture anomaly, Z, is calculated
as difference of the actual precipitation, Pr and Pr_{C},
multiplied by climatic characteristics coefficient, K (Wells, 2002): 
Z = (Pr Pr_{C})
x K = d x K 
(2) 
where, d is moisture departure (with length dimension)
and K is a weighting factor that allows comparisons of deviations
to be made between locations and months. 
The moisture anomalies at any time scale (i),
then, changes to a classified form, X, on the bases of its previous
time step, X_{i1} (Alley, 1984): 
X_{i} = 0.9
X_{i1} + 0.3 Z_{i} 
(3) 
The coefficients of this equation derived by Palmer (1965)
in his special study and should be adjusted for any case study. In the
classification process, three different X are applied and the appropriate
index is selected by picking one of them according to a set of rules (called
Backtracking). Using this method, the historical perspective of the index
or its Inherent Memory (that is: an inherent time window over which it
evaluates the climate trend), reaches only to the start of the current
spell (Wells, 2002). However, the backtracking is more or less complex
and needs separate studies.
Based on the subject of the study, the following text
turns to «the generalizing procedure of the Climatic Characteristic
Coefficient»:
The Kformula had been initially developed by Palmer
(1965) for a limited set of data from nine climate divisions, which do
not represent the average climate of entire world (Wells, 2002). Hence,
many researchers like Akinremi et al. (1996), Toraabi (2002) and
Quiring and Papakryiakou (2003) implied that the Kvalues should be calibrated
for any case study. So, they revised the original values of K, though
in some other study they are used by their initial equations. In order
to calibrate the Kvalues for any case study, several steps should be
tracked:
At the beginning, for the first approximation of coefficients, it is
assumed that the driest periods were of nearly equal significance locally
(Palmer, 1965). So for the average departures of the driest (or wettest)
period with a length of n
at different locations in the study region (R), the following relation
can be written:
where,
represents the average weighting factor for different locations and apply
to the driest (or wettest) periods as a whole, rather than to each month
individually (Palmer, 1965). Equation 4 can be rewritten
as a ratio mode for each pair regions:
From the Eq. 5, it is noticeable that since the average moisture demand
in the two places is roughly the same, the constants
may depend on the average moisture shortage (or excess) in two places.
In other words, the less the supply in relation to the demand, the greater
the significance of a given shortage (Palmer, 1965). Therefore, the average
weighting factor can be estimated from a demandsupply ratio: PE, RE and
RO as the representative of the moisture demand and Pr with the previously
stored moisture (or expected moisture loss, L) as the moisture supply
indicators (Palmer, 1965):
In this equation, k is the first approximation of and
the upperlines represent average values of the parameters, described
before. However, it is necessary to develop an appropriate relation between
k and the engaged parameters. Based on the abovementioned principles,
various equations were offered in Table 1. The first
two equations in the table were proposed by Palmer (1965), the third by
Toraabi (2002) and the others have been suggested in our study. Palmer
(1965) showed that his first combination (i) did not work well in some
climates and finally he proposed the second form (ii) and implied that
different equations had to be checked for any case study. So, the ranges
of each equation in Table 1 are calculated, for dry and wet situations
obtained by substituting the equivalent parameters. It is expected that
the values of k be larger at dry condition than wet situations. This is
because, in dry timesteps (such as summer), the absolute values of d
are usually less than in wet timesteps. The ranges show that some of
equations, like (v), do not obey the converse trend between k and
d (Eq. 5).
Table 1: 
Various equations for first approximated
coefficient of the climatic characteristic 

However, after selecting suitable equations, k is used
to produce first estimation of the moisture anomaly index, z (Palmer,
1965):
Now, the first guessed values of k can be revised. The
reevaluation of the weighting factor is done in two stages:
Stage 1:
Obtaining a new annually mean weighting factor, ,
inspiring from the reversed form of Eq. 7 (Palmer,
1965): 
where, w and d are related to the wet spell or drought events and
is sum of d for the annually driest (or wettest) period, which is assumed
that represents extreme drought (or wet spell) in any area of study (Palmer,
1965; Akinremi et al., 1996):
In addition, 6
is especial Σz (comes from Eq. 7) that can be assumed
as extreme drought (or wet spell) over a 12months period. Determining
what Σz should be the extreme values, needs special route: in which,
the varying periods represented the maximum rate of Σz in different
area of study were selected. Then, the picked values (individually for
dry and wet spells) are plotted verses their length and fitted by an envelope
line (Palmer, 1965; Akinremi et al., 1996):
where, m and b are coefficients of the fitted line and i is drought or
wet spell length.
now, is calculated from the regression lines:
Stage 2: Determining K as a function of its relative aspects:
Palmer (1965) supposed that the Kvalues depend on the water balance parameters
(Eq. 6), as well as vary inversely with the mean of
the absolute values of d for the total length of n year :
So, after some experimenting with various empirical relationships,
the semilogarithmic plot shown in Fig. 1 was developed by Palmer (1965).
The generalized form of Palmer equation can be expressed as:
where, is
annually mean weighting factor (Eq. 8 or 9),
λ, θ and μ are the calibrated coefficients (equal to 1.5,
2.8 and 0.5 in Palmer study which are derived only for dry conditions
and via inch unit of parameters).
in these equations are annually average of k in each study region, that
for a selected form of Table 1, such as Eq.
(ii), is written as:
In this equation, the doublebars represent annually
average values of the parameters, which were described before. For example,
for Pr it can be described as:
The next step is to apply the empirical coefficients (λ, θ
and μ) and Eq. 17 and 18 (derived
for all of regions

Fig. 1: 
The extracting of
Kvalue from driest 12 months in Palmer`s (1965) study 

Fig. 2: 
Flowchart of calibrating the
climatic characteristic coefficient in PDSI 
in a study), to each of the 12 calendar months and thereby
deriving the 12 K`values for each place:
(where, k is approximated from the selected equation in Table
1 and
is computed as implied at Eq. 15 and 16.
Stage 3:
As final adjustment of the monthly Kvalues, coming back
to Eq. 4, it is expected that the average annual sum
of the weighted average departures
should be about the same for all study area. So, if all weighting factors
are adjusted so that all of the annual sums of equals
to an average of all regions, i.e., ,
then drought (or wet spell) analysis results should be more comparable.
The adjustment factor, , can be obtained by:
And the final equation of K would be, as; 
Afterward, it is necessary to rebuild the envelop line
(Eq. 12) using Zvalues comes from the following equations (although Palmer
did not change its initial envelop line):
The aforementioned process is summarized in Fig. 2,
which can help the user to understand and apply it straightforward.
Based on these generalized procedure (Fig. 2), which
was the main part of this paper objective, a case study was selected to
show the procedure via actual data.
MATERIALS AND METHODS
For putting into practice the method of PDSI scaling (weighting
factor, K), the data set of Fars province (south part of Iran; Fig.
3) was applied and the results were illustrated step by step based
on the aforementioned details. In this way, the outputs led the research
to some modifications, which were explained in accompany with the results.
The Maharlue catchment was divided in three areas of study (three subbasins)
and for each one of them, the area equivalent data of monthly rainfall
and temperature from the existing stations were calculated for 31 year
(which are out of question). However, the obtained moisture departures
of these areas were used as the input parameters for this study. The dvalues
of the first area are presented in Table 2.

Fig. 3: 
Location of Maharlue Basin (4270
km^{2}) in Fars province and the Country of Iran (Software
bank of Fars Province Water Organization, 2003) 
Table 2: 
Values of moisture departures
in mm at a solar calendar; study location of Maharlue1 

(Note: the solar
year 1350, month 7 related to the 10th month of 1971) 

Fig. 4: 
The changes of selected kequation
via absolute values of d, in the study region 
RESULTS
At first, due to precalculated d for the studyregion, different
kequations in Table 1 were evaluated and then, the
(iv) equation was distinguished more appropriate, because of its converse
trends to the absolute values of the moisture departures (Fig.
4), as it is expected (Eq. 5).
Then the selected kcoefficients, were applied (using Eq.
7) to calculate initial moisture anomalies, z. Subsequently, in order
to obtain the enveloped line of extreme dry and wet spells, the varying
periods represented the maximum rate of Σz in different areas of
study, were scanned from the ztime series. Figure 5
shows the results, from which the following equations were obtained for
dry and wet spells:
Now, the annually mean weighting factor, for
each region (Eq. 8 and 9) can be obtained
form extreme drought (or wet spell), which were calculated from Eq.
28 and 29 (by i = 12) and Eq. 10
and 11. Table 3 shows the results.
However, determining the relationships between ,
and
k(Eq. 17 and 18) showed that following
the explained procedure will be encountered with unacceptable values (negative
coefficients for the slope of Eq. 17 and 18,
also negative ).
This can be interpreted as the effect of significant vacillations in the
monthly distribution of parameters (especially d), whose annual averages
do not reflect them.
Therefore, the aforesaid procedure was retracked in the month scale
(Table 4, Fig. 6). It should be noted
that in the new proposed method, Eq. 17 and 18
(with annual scale) are not derived and Eq. 21 and 22
are directly obtained using monthly parameters.
At the end Stage, the final values of K were obtained by the fitted equations
(Fig. 6) and adjusting coefficients (Table
5). Now, these weighting factors (K_{w} and K_{d}
for

Fig. 5: 
Driving the drought severity
equations using extreme events, (from the three study regions of
Maharlue) 
Table 3: 
The annually driest (or wettest)
moisture anomalies and departures as well as the annually mean weighting
factor (study regions of Maharlue) 

Table 4: 
The requirement parameters for
extracting the empirical relationships of K`, (for the three study
regions of Maharlue) 


Fig. 6: 
The empirical equations of K
(study region of Maharlue) 
Table 5: 
The coefficients of climate characteristics,
K (study regions of Maharlue) 

each of the 12 calendar months) are capable to be employed as the climatic
characteristics coefficients for three regions of the case study, in order
to calculate moisture anomaly or Zvalues (Eq. 26, 27).
DISCUSSION
At the operational part of this research, the average of
in all study regions (the numerator of Eq. 23) were
obtained 107 and 90 separately for wet and dry periods (Table
5); while Palmer (1965) was calculated it 17.7 without separating
the periods. Also, in some other study, different value has been found
for this coefficient (),
like 14.2 by Akinremi et al. (1996) in the Canadian prairies and
264.3 by Toraabi (2002) in ZayandeRood basin (IsfahanIR of Iran). Comparison
of these results shows that Kvalues should be calibrated for any case
study and moreover, it may be significant calculating Kcoefficients separately
for dry and wet periods.
However, the generalized procedure of the climatic coefficient,
represented in this study, can be considered as a reference text for any
case study. It also can be helpful for other researches related to the
Kdeveloping or modification, such as the monthly modified procedure,
proposed in this study. So, it can be take into account as a turning point
in the Palmer scaling method.
The «monthly modified procedure» in this study, also confirmed
the problem, implied by Alley (1984), who stated: deriving monthly Kvalues
(Eq. 21, 22) using data on the annual
level (Eq. 17, 18), may not yield
the desired result of comparability of the index values between months.
Consequently, it is recommended that the generalized
process, for any case of study that needs to quantify drought severity
by PDSI, was followed step by step in accompany with the monthly modification
method. Afterward their results should be compared to decide that which
one could be obtained better outputs.