Abstract: The scaling technique in calculating Palmer Drought Severity Index (PDSI) has specific properties, which is considered in details, in this study. The Palmer scaling method contains a physically-based weighting factor called climatic characteristic coefficient, K. This factor, is the ratio of average moisture demand to average supply in a regional water balance, whose related coefficients and involved equation parameters should be calibrated according to any special case study. Therefore, in this study, the generalizing procedure of obtaining K-value with no dependency on particular area, in addition to some modifications in the process was analytically developed. The generalized procedure has an appropriate potential for various purposes such as; correctly identifying analytical K-formula, simplifying its application, recognizing its limitations and moreover it can be used in other researches for developing its advantages. The proposed procedure and its modifications also applied for a case study (e.g., Maharlue catchment, Fars province, Iran). The outputs of the application validated the suggested modifications; therefore, the obtained K-values can be used for other studies in the region.
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, PrC, is approximated (Guttman, 1998):
PrC = α
x ETp + β x REp + γ x ROp - δ x Lp |
(1) |
Then, the moisture anomaly, Z, is calculated as difference of the actual precipitation, Pr and PrC, multiplied by climatic characteristics coefficient, K (Wells, 2002): |
Z = (Pr- PrC)
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, Xi-1 (Alley, 1984): |
Xi = 0.9
Xi-1 + 0.3 Zi |
(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 K-formula 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 K-values 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 K-values 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
(4) |
where,
(5) |
From the Eq. 5, it is noticeable that since the average moisture demand
in the two places is roughly the same, the constants
(6) |
In this equation, k is the first approximation of
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):
z = dxk |
(7) |
Now, the first guessed values of k can be revised. The re-evaluation of the weighting factor is done in two stages:
Stage 1:
Obtaining a new annually mean weighting factor, |
|
for wet spells |
(8) |
|
for drought events |
(9) |
where, w and d are related to the wet spell or drought events and
|
for wet spells |
(10) |
|
for drought events |
(11) |
In addition, 6
(12) |
where, m and b are coefficients of the fitted line and i is drought or
wet spell length.
|
(13) |
|
(14) |
Stage 2: Determining K as a function of its relative aspects:
Palmer (1965) supposed that the K-values 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
|
(15) |
|
(16) |
So, after some experimenting with various empirical relationships, the semi-logarithmic plot shown in Fig. 1 was developed by Palmer (1965). The generalized form of Palmer equation can be expressed as:
for wet spells |
(17) |
for drought events |
(18) |
where,
(19) |
In this equation, the double-bars represent annually average values of the parameters, which were described before. For example, for Pr it can be described as:
(20) |
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 K-value 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:
|
for wet spells |
(21) |
|
for drought events |
(22) |
(where, k is approximated from the selected equation in Table
1 and
Stage 3:
As final adjustment of the monthly K-values, coming back
to Eq. 4, it is expected that the average annual sum
of the weighted average departures
|
(23) |
And the final equation of K would be, as; |
|
for wet spells |
(24) |
|
for drought events |
(25) |
Afterward, it is necessary to rebuild the envelop line (Eq. 12) using Z-values comes from the following equations (although Palmer did not change its initial envelop line):
for wet spells |
(26) |
for drought events |
(27) |
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 sub-basins)
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 d-values
of the first area are presented in Table 2.
Fig. 3: |
Location of Maharlue Basin (4270 km2) 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 Maharlue-1 |
(Note: the solar year 1350, month 7 related to the 10th month of 1971) |
Fig. 4: |
The changes of selected k-equation via absolute values of d, in the study region |
RESULTS
At first, due to pre-calculated d for the study-region, different k-equations 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 k-coefficients, 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 z-time series. Figure 5 shows the results, from which the following equations were obtained for dry and wet spells:
|
for wet spells |
(28) |
|
for drought events |
(29) |
Now, the annually mean weighting factor,
Therefore, the aforesaid procedure was re-tracked 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 (Kw and Kd
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 Z-values (Eq. 26, 27).
DISCUSSION
At the operational part of this research, the average of
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 K-developing 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 K-values (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.