Sensitivity of evapotranspiration deficit index to its parameters and temporal scales 1 2

8 Sound estimates of drought characteristics are very important for planning intervention 9 measures in drought-prone areas. Among many drought indices used in estimation of drought 10 characteristics in many parts of the world, evapotranspiration deficit index (ETDI) is 11 increasingly used to estimate agricultural drought. However, in most studies ETDI has been 12 computed using the specific ETDI formula. Thus, there is no clear information about 13 sensitivity of ETDI to its parameter and temporal scales. In this study, the general ETDI 14 formula homologous to the specific ETDI formula was introduced and used to test sensitivity 15 of ETDI to its parameters and temporal scales using time series of remotely sensed 16 evapotranspiration data in the Ruvu River basin (Tanzania). The parameter sensitivity test 17 revealed that ETDI is sensitive to its parameters. Different parameter combinations resulted 18 into different drought characteristics. In order to reduce this uncertainty, the general ETDI 19 formula might require parameter calibration. On the other hand, the temporal scales 20 sensitivity test showed that drought characteristics such as number of drought events and the 21 total drought durations decreased as the size of temporal scales increased. Thus, inappropriate 22 temporal scales may lead to misrepresentation of drought characteristics. In order to increase 23 accuracy of drought characteristics derived from ETDI, small temporal scale data are highly 24 recommended. Therefore, this study has provided useful information for improving 25 application of ETDI in estimation of agricultural drought characteristics. 26


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Drought is an environmental disaster that brings severe social, economic, and environmental 33 impacts around the world. Thus, drought is usually categorized into four main operation-34 based types, namely, meteorological drought, hydrological drought, agricultural drought and  Basin, Ethiopia. In all those studies, ETDI was computed using the specific ETDI formula, 64 thus sensitivity of ETDI to its parameters and temporal scales is hardly known.

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Therefore, the objective of this study was to investigate sensitivity of ETDI (1) to its 66 parameters, and (2) to temporal scales. To address this objective, firstly the general ETDI 67 formula homologous to the specific ETDI formula was introduced. Then by using the general 68 ETDI formula, sensitivity of ETDI to its different parameter combinations was tested.

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The layout of the rest of the paper is as follows: Section 2 provide explanations about the 72 case study, main data used, evapotranspiration deficit index approach, parameter sensitivity 73 test and temporal scale sensitivity test. Section 3 presents results and discusses findings about  The case study used was the Ruvu River basin. The Ruvu River basin is located between 82 6°18'S-7°46'S and 37°15'E-38°58'E in east Tanzania (Fig. 1). Its headwaters originate on     Narasimhan and Srinivasan 2005). WS ranges from 0 (ET is the same as PET) to 1 (no ET).
Where, the subscript i represents a period (i.e., an 8-days, 16-days or 1-month) in year j. The   134 Narasimhan and Srinivasan (2005) invented the specific ETDI formula which states that, at a 135 particular point in time the current ETDI (ETDI t ) is the sum of half of the previous ETDI 136 (ETDI t−1 ) and the current WSA (WSA t ) (Eq. A1 in Appendix A). Although the specific 137 ETDI formula shows that ETDI t linearly depends on both ETDI t−1 and WSA t , the coefficient 138 of the latter was ignored or assumed unit. Moreover, the constant term (intercept plus error) 139 was also not addressed by Eq. (A1). In this study, the general ETDI formula was introduced 140 as a multivariate linear equation homologous to the specific ETDI formula. The general ETDI 141 formula has three variables and three unknown coefficients including the constant term (Eq.
Where, t represents continous timestep (it replaced period i in year j from Eq. 2). α modulates 146 the long-term memory of ETDI. β converts WSA value into ETDI and γ is the constant term.

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By considering that ETDI is scaled between -2 and +2 like the standard precipitation index      The ETDI (0.0,2.0) curve was highly correlated to the ETDI (0.1,1.8) curve (Table 2) Table 3). This means that small α-parameters of these two curves reduced the 233 influence of ETDI t−1 while large β-parameters allowed dominance of WSA t (Eq. 4). This is 234 inversely demonstrated by the ETDI (0.9,0.2) curve which had the lowest number of drought 235 event and the highest duration per event (10 months per event, Table 3). Here, large α-236 parameter allowed dominance of ETDI t−1 , but small β-parameter had already smoothened 237 peaks of WSA t (Eq. 4), thus causing wide but few peaks. In addition, the ETDI (0.9,0.2) and 238 ETDI (0.8,0.4) curves were highly correlated (Table 2), but they had substantially different 239 number of events and total drought durations (Table 3). High correlation between the two 240 curves was due to similarity of their patterns which were not affected by minor parameter 241 differences. However, the differences in drought characteristics were mainly due to the β- that the influence of their ETDI t−1 and WSA t were reduced to almost half by α-parameters 246 but after being almost fully allowed by β-parameters (Eq. 4), respectively.

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Therefore, an arbitrary choice of a parameter combination has drastic effects on drought   (Table 4). The difference in number of drought events between 286 consecutive time scales was mainly between 1 and 2 except at points P4 and P11 where the 287 differences between 16-days and 1-month time scales were relatively large (about 5 drought 288 events). The large differences in drought events could be attributed to local effects because 289 the two points are found in the northern part of the river basin (cf. Fig. 1). Although differences between numbers of drought events were not too large, their 295 corresponding total drought durations differed by very large number of months (Table 4). The 296 total drought durations of 8-days ETDI curves were almost two-times and three-times those 297 of 16-days ETDI curves and monthly ETDI curves, respectively. Thus, total drought 298 durations decreased as the size of time scales increased. Moreover, almost all points in the 299 river basin had duration per event ranging from 5 months for 8-days ETDI curves to 2 months 300 for monthly ETDI curves (Table 4).

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This study used MODIS ET time series from twelve points spatially distributed in the Ruvu 328 River basin to test sensitivity of ETDI to its parameters and temporal scales. Parameter 329 sensitivity test revealed that ETDI is less sensitive when the (α, β)-parameters ranges from 330 (0.1,1.8) to (0.5,1.0) inclusive, and more sensitive when they approach (0.9,0.2). Since 331 ETDI is sensitive to different parameter combinations, the selection of an appropriate 332 parameter combination might rely on information from specific locations. Moreover, an 333 appropriate parameter combination can also be obtained when ETDI is compared against 334 other drought indices. Therefore, in reducing uncertainty of selecting an appropriate 335 parameter combination, the general ETDI formula might require parameter calibration.

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Temporal scales sensitivity test at twelve points in the river basin showed that the number of

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If ETDI is scaled between -2 and +2, at a boundary condition (i.e., very dry condition), WSA t 349 equals to -1, both ETDI t and ETDI t−1 equal to -2. By substituting WSA and ETDI values in