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Validity, reliability and certainty of PERSIANN and TRMM satellite-derived daily precipitation data in arid and semiarid climates

Abstract

Having doubts about the adequacy of reliability level of satellite-derived precipitation products, along with their application in large number of hydrological models, has led to many studies on evaluating the efficiency of such data. In this study, two new procedures were proposed to compute reliability and certainty degrees of PERSIANN and TRMM 3B42RT data sets, and six traditional indicators were used to evaluate their validation. In addition, the cumulative density function (cdf) of the above-mentioned data sets was compared with the ground-based observations in 23 synoptic stations in Fars, Iran. The Kolmogorov–Smirnov test was performed using the data sets at 5% significance level which led to the result of null hypothesis that was not being rejected, suggesting that the satellite-derived daily precipitation data (SDDPD) and ground-based observations are drawn from the same distribution. Results indicated that TRMM and PERSIANN follow quite similar probability pattern of ground-based observations in arid and semiarid climate, respectively. However, data probability pattern of TRMM cannot be considered similar to ground-based observations in arid region, neither can PERSIANN in semiarid climate. Among common cross-validating attributes, the values of ME and BIAS, in addition to RMSE and MAE, led to the conclusion that in PERSIANN, the rainfall daily rates are almost underestimated while TRMM overestimates the values mainly in semiarid regions. Moreover, the PERSIANN was found to be significantly correlated with IDM (De Martonne aridity Index), and the values of underestimation increased with growth of the index. The reliability values of SDDPD over the study area, for both TRMM and PERSIANN, show the reverse trend with increasing IDM in almost all acceptable error intervals. Along with effects of climate conditions, the reliability degrees of PERSIANN seem quite more consistent at different acceptable error intervals in comparison with the corresponding values of TRMM. In addition to validity and reliability, the error entropy of SDDPD, as an index for uncertainty degree, increases as the IDM rises, which is theoretically corresponds with reliability concept. However, in comparison with PERSIANN, TRMM data set, overall, has higher degree of uncertainty. In addition, to evaluate effect of daily rainfall intensity on the uncertainty degree of SDDPD, the uncertainty degree slightly increases as daily rainfall intensifies to about 15 mm/day. But for higher daily rainfall intensities, on the other hand, the uncertainty degree seems to gradually decline as the daily rainfall increases.

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Notes

  1. Standardized Precipitation Index.

  2. Anomaly Standardized Precipitation.

  3. De Martonne aridity Index.

  4. Pinna combinative Index.

  5. \({\text{erf}}\left( x \right) = \frac{2}{\sqrt \pi }\mathop \int \limits_{0}^{x} {\text{e}}^{{ - t^{2} }} {\text{d}}t.\)

  6. \(T\left( {x,a} \right) = \frac{1}{2\pi }\mathop \int \limits_{0}^{a} \frac{{e^{{ - \frac{1}{2}x^{2} \left( {1 + t^{2} } \right)}} }}{{1 + t^{2} }}{\text{d}}t.\)

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Appendix: Entropy of skew-normal distribution

Appendix: Entropy of skew-normal distribution

The skew-normal distribution is an extension of the normal (Gaussian) probability distribution, allowing for the presence of skewness. This model and its variants have attracted the attention of an increasing number of research. The probability density function (pdf) of the skew-normal distribution with parameter \(\alpha\) is given by:

$$f\left(x\right)=2\phi \left(\frac{x-\xi }{w}\right)\Phi \left(\alpha ,\frac{x-\xi }{w}\right),$$

where

$$\left\{ {\begin{array}{*{20}l} {\phi \left( t \right) = \frac{1}{{w\sqrt {2\pi } }}e^{{ - t^{2} }} } \hfill \\ {\Phi \left( {t,\alpha } \right) = \frac{1}{2}\left[ {1 + {\text{erf}}\left( {\alpha t} \right)} \right]} \hfill \\ \end{array} } \right.\quad \quad t = \frac{x - \xi }{{w\sqrt 2 }},$$

in which \(\phi \left(t\right)\) represents the normally distributed part and \(\Phi \left(t\right)\) tilted it to characterize the skewness.

Establishing some algebraic procedure to determine the entropy of the skew-normal distribution, it can be calculated as:

$${H}_{{\rm SN}}={H}_{{\rm N}}-E\left({\text{ln}}\left(2\Phi \left(x,\alpha \right)\right)\right),$$

where \({H}_{\text{N}}\) is entropy of normally distributed function (\({H}_{{\rm N}}=\frac{1}{2}+\frac{1}{2}{\text{ln}}\left(2\pi {w}^{2}\right)\)). The following procedure shows that expectation of \({\text{ln}}\left(2\Phi \left(x,\alpha \right)\right)\) is just related to skewness factor \(\alpha\).

$$\begin{aligned} & \Psi \left( \alpha \right) = E\left( {\ln \left( {2\Phi \left( {x,\alpha } \right)} \right)} \right) = \mathop \int \limits_{ - \infty }^{ + \infty } 2\phi \left( x \right)\Phi \left( {\alpha ,x} \right)\ln \left( {2\Phi \left( {\alpha ,x} \right)} \right){\text{d}}x \Rightarrow \\ & \Psi \left( \alpha \right) = \frac{1}{\sqrt \pi }\mathop \int \limits_{ - \infty }^{ + \infty } e^{{ - t^{2} }} \left[ {1 + {\text{erf}}\left( {\alpha \,t} \right)} \right]\ln \left[ {1 + erf\left( {\alpha \,t} \right)} \right]{\text{d}}t\ . \\ \end{aligned}$$

The value is defined as “entropy reduction parameter” and can be identified from the following diagram:

figure a

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Khojand, K., Shaghaghian, M.R., Ghadampour, Z. et al. Validity, reliability and certainty of PERSIANN and TRMM satellite-derived daily precipitation data in arid and semiarid climates. Acta Geophys. 70, 1745–1767 (2022). https://doi.org/10.1007/s11600-022-00801-y

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Keywords

  • PERSIANN
  • TRMM 3B42RT
  • Arid and semiarid climates
  • Validity
  • Reliability
  • Certainty