C omplementing near-real time satellite rainfall products with satellite soil moisture-derived rainfall through a Bayesian Inversion approach

Journal of Hydrology 573 (2019) 341–351

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Research papers Complementing

near-real time satellite rainfall products with satellite soil moisture-derived rainfall through a Bayesian Inversion approach

T

Christian Massaria, , Viviana Maggionib, Silvia Barbettaa, Luca Broccaa, Luca Ciabattaa, Stefania Camicia, Tommaso Moramarcoa, Gabriele Cocciac, Ezio Todinid ⁎

a

Research Institute for Geo-Hydrological Protection, National Research Council, Perugia, Italy George Mason University, Fairfax, VA, United States c RED – Risk Engineering + Development, Via Giuseppe Frank 38, 27100 Pavia, Italy d Italian Hydrological Society, Piazza di Porta San Donato, 40126 Bologna, Italy b

ARTICLE INFO

ABSTRACT

This manuscript was handled by Emmanouil Anagnostou, Editor-in-Chief, with the assistance of Yang Hong, Associate Editor

This work investigates the potential of using the Bayesian-based Model Conditional Processor (MCP) for complementing satellite precipitation products with a rainfall dataset derived from satellite soil moisture observations. MCP – which is a Bayesian Inversion approach – was originally developed for predictive uncertainty estimates of water level and discharge to support real-time flood forecasting. It is applied here for the first time to precipitation to provide its probability distribution conditional on multiple satellite precipitation estimates derived from TRMM Multi-Satellite Precipitation Analysis real-time product v.7.0 (3B42RT) and the soil moisture-based rainfall product SM2RAIN-CCI. In MCP, 3B42RT and SM2RAIN-CCI represent a priori information (predictors) about the “true” precipitation (predictand) and are used to provide its real-time a posteriori probabilistic estimate by means of the Bayes theorem. MCP is tested across Italy during a 6-year period (2010–2015) at daily/0.25 deg temporal/spatial scale. Results demonstrate that the proposed methodology provides rainfall estimates that are superior to both 3B42RT (as well as its successor IMERG-early run) and SM2RAIN-CCI in terms of both median bias, random errors and categorical scores. The study confirms that satellite soil moisture-derived rainfall can provide valuable information for improving state-of-the-art satellite precipitation products, thus making them more attractive for water resource management and large scale flood forecasting applications.

Keywords: Rainfall Soil moisture Predictive uncertainty Water resource management

1. Introduction Accurate quantitative precipitation estimation is of great importance for water resources management, agricultural planning, and monitoring/ forecasting of natural hazards, such as floods, drought and landslides. In situ observations are limited around the Earth, especially in remote areas (Kidd et al., 2017) thus Satellite Precipitation Products (SPPs) are often the only alternative for applications. However, their accuracy depends upon many factors (Ebert et al. 2007, i.e., type of storms, temporal sampling, season, etc…) thus limiting their real utility. Traditional SPPs sense cloud properties to retrieve instantaneous rainfall estimates by combining a range of observations from Geostationary (GEO) and Low Earth Orbiting (LEO) satellites. Rainfall can be inferred from different sensors: i) visible images since thick clouds – that are more likely to be associated with rainfall – tend to be brighter than the surface; ii) infrared (IR) radiances since heavier rain

⁎

tends to be associated with larger and taller clouds with colder tops; iii) passive microwave (PMW), as radiation and scattering caused by precipitating ice particles lead to a decrease in the signal. During the Tropical Rainfall Measuring Mission (TRMM) and the Global Precipitation Mission (GPM) eras (e.g., Hou et al., 2014; Huffman et al., 2007) several techniques have been developed to take advantage of the synergy between PMW observations derived from polar-orbiting satellites (infrequent, more direct) and IR radiances derived from GEO observations (more frequent but less direct). In particular, GPM has brought a significant advancement in the measurement of precipitation from space thanks to a new Dual-frequency Precipitation Radar (DPR) and a more accurate radiometer which are used to intercalibrate observations coming from a constellation of sensors (Chiaravalloti et al., 2018; Gaona et al., 2016; He et al., 2017; Liu, 2016; Prakash et al., 2016; Tan et al., 2017; Tang et al., 2016; Xu et al., 2017).

Corresponding author. E-mail address: [email protected] (C. Massari).

https://doi.org/10.1016/j.jhydrol.2019.03.038 Received 4 August 2018; Received in revised form 22 February 2019; Accepted 3 March 2019 Available online 15 March 2019 0022-1694/ © 2019 Elsevier B.V. All rights reserved.

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State-of-the-art SPPs rely upon combinations of instantaneous observations of precipitation and thus are inherently affected by sampling errors (Behrangi and Yixin, 2017). Due to the intermittent nature of rainfall and its high temporal and spatial variability, these errors can translate into significant biases and missed precipitation events (Ciabatta et al., 2016), which may impact their skills in operational applications like flood forecasting (Habib et al., 2009; Maggioni and Massari, 2018). Common bias correction methods only adjust the total rainfall volume and may result in underestimation or overestimation of instantaneous precipitation (AghaKouchak et al., 2012), as SPPs show strong dependence of the error on rain rate (Maggioni et al., 2016). Recent studies have proposed to use satellite soil moisture observations for correcting SPPs (Camici et al., 2017; Ciabatta et al., 2015; Crow et al., 2009, 2011; Pellarin et al., 2008, 2013; Wanders et al., 2015; Zhan et al., 2015). The underlying idea is that soil moisture can be used as a trace of precipitation, as after a rain event it can persist from a few hours to several days. In other words, soil moisture is informative of the amount of water stored in the soil after a rainfall event, which can be exploited to retrieve information on the fallen precipitation. On this basis, the SM2RAIN method developed by Brocca et al. (2014) inverts the soil water budget equation and uses two consecutive soil moisture measurements for estimating the precipitation fallen within the interval between two satellite overpasses. SM2RAIN rainfall estimate, derived from active and passive soil moisture products, was shown to perform fairly well over semi-arid areas of Africa and Australia and South America while less performing results were obtained over cold areas, mountainous regions and dense forests (Ciabatta et al., 2018; Massari et al., 2017). The conceptually different measurement method of SM2RAIN is complementary to (and independent from) the one of traditional SPPs. That is, while SPPs estimates are based on averaging instantaneous observations of rainfall, Soil Moisture-based Satellite Precipitation Products (SM-SPPs) provide estimates of the accumulated precipitation within a time interval. Therefore, the combination of these two different measurements can potentially reduce the problems related to temporal sampling. However, the merging (or integration) between SPPs and SM-SPPs is not a simple task and should account for the dependence of errors on rainfall intensity and seasonality. Although many interesting studies have been published dealing with the integration of satellite rainfall estimates, gauges, radars and model reanalysis (Li et al., 2015; Mazzetti and Todini, 2009; Xie and Xiong, 2011) – a notable example is the one of Beck et al. (2017) – only few of them (Ma et al., 2018) focused only on SPPs. SPPs have the main advantage of being available everywhere and in near-real time. Nevertheless, SPPs typically show non-stationary seasonal and rain rate-dependent biases (AghaKouchak et al., 2012; Ebert et al., 2007; Maggioni et al., 2016), which are difficult to predict. In this respect, blending SPPs with SM-SPPs adds an additional challenge, as the latter tend to underestimate high precipitation rates and are less reliable in humid climates. This is due to the fact that, when soil moisture reaches saturation, no change in soil water storage – and consequently in precipitation – can be recorded. SM-SPPs are also affected by noise in the satellite soil moisture signal, which can provide false precipitation rate especially during dry seasons (Zhan et al., 2015). Past attempts to merge SPPs and SM-SPPs (or correct SPPs with satellite soil moisture observations) have used either simple nudging schemes with constant weighing factors (Ciabatta et al., 2015, 2017) or data assimilation approaches based on sequential filtering techniques, e.g. Kalman-based methods (SMART, Crow et al., 2011) and particle filters (Pellarin et al., 2013; Román-Cascón et al., 2017; Zhan et al., 2015). As nudging schemes are not being able to account for the dependency of errors on rainfall intensity, they might provide suboptimal merged products. Sequential filtering techniques provide a dynamical correction of SPPs, but this correction is either assumed only on the random component of the SPP error (i.e., the systematic component is deliberately ignored, Crow et al., 2011) or, only applied to non-zero

satellite precipitation (zero precipitation values are left unchanged, Román-Cascón et al., 2017). It must be noted that neglecting the systematic component of the error means overlooking the problem of biases in SPPs. However, total and hit biases are extremely important for many applications, as they dictate the temporal trend of soil moisture and runoff errors within land surface and hydrological models (Gebregiorgis and Hossain, 2013). In this work, we investigate the use of a conceptually different technique – which is based on the predictive uncertainty (PU) concept – to combine SPPs and SM-SPPs. That is, SPPs and SM-SPPs are seen as the a priori information (predictors) about the precipitation (predictand) and are used to provide its real-time a posteriori probabilistic estimate by means of the Bayes theorem. We chose the Model Conditional Processor (MCP, Todini, 2008), which was originally developed to support real-time flood forecasting and thus to provide the a posteriori probabilistic estimates of water level/discharge based on the knowledge of multiple forecasting model outputs. MCP is based on the definition of the multi-variate conditional distribution, i.e., the density of the predictand (precipitation in this case) conditional on multiple deterministic model predictions or satellite observations (i.e., SPPs and SM-SPPs in this case). It is worth mentioning that the notion “Bayesian” must be intended here in terms of “prediction” and not as “Bayesian inference” (Draper and Krnjajic, 2013). Specifically, we used a special case of Bayesian Inversion, in which the predictand and predictor are available at the same time (i.e., filtering). Despite its conceptual difference with respect to previously developed techniques, this approach has the same final objective, that is, seeking an optimal way to include soil moisture information into SPPs with the aim of reducing their error and improving their quality. MCP’s advantages over other techniques (e.g., Beck et al. 2017; Ciabatta et al. 2015; Tarpanelli et al. 2017) is its ability to correct for the error as a function of rainfall intensity. Moreover, it not only corrects for random errors, which is the main target of sequential assimilation techniques (Crow et al., 2011; Pellarin et al., 2013; Román-Cascón et al., 2017; Zhan et al., 2015), but also provides a bias reduction in the final precipitation product. In this article, we use the two terms integration and merging interchangeably when referring to the MCP application. The main objective of this work is to assess the efficiency of MCP in integrating SPPs and SM-SPPs information, so that the resulting product is superior with respect to its parent products (i.e., it performs better in terms of error, correlation, and categorical scores). MCP is thus applied to merge the TRMM Multi-Satellite Precipitation Analysis real-time product v.7.0 (3B42RT, Huffman et al., 2007) and the SM2RAIN-CCI SM-SPP (Ciabatta et al., 2018), obtained from the inversion of the satellite soil moisture observations derived from the ESA Climate Change Initiative (CCI, Dorigo et al., 2017) via SM2RAIN. The analysis was carried out in Italy during a 6-year period (2010–2015) at daily/0.25° temporal/spatial scale. The time series was split into two sub-periods: 2010–2012 was used for calibration and 2013–2015 for validation. Both calibration and validation were performed against a dense rain gauge network covering the entire study area (see Section 3.2). The proposed merging technique and the statistical metrics adopted to evaluate the different precipitation products are presented in the next section. The study area and datasets are described in Section 3. Results are discussed in Section 4 and discussion and conclusions are drawn in Section 5. 2. Methodology The methodology used in this work provides the probabilistic estimate of rainfall through the application of MCP considering two satellite rainfall products, one of which derived from satellite soil moisture measurements. MCP was applied to mutual non-zero precipitation estimates whereas in the other cases a heuristic type of approach was used (see Section 2.3). 342

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2.1. The multi-model conditional processor (MCP)

and yk , whose empirical cumulative distribution functions are computed using the Weibull plotting position (see Eqs. (1) and (2)), are converted into their transformed values η and k , respectively, which are normally distributed with zero mean and unit variance. The probability of each element of η and k is the same as their original corresponding values in y and yk . Thus, the relation between the original variables and their transformed values is:

The Model Conditional Processor (MCP) is a Bayesian approach developed for estimating the flood predictive uncertainty (PU) defined as the probability density of a future outcome conditional on all the available information, usually provided by model flood forecasts. Therefore, MCP was originally proposed for PU estimate to support the flood forecasting activities in real-time and it is essentially based on the definition of the multi-variate conditional distribution, i.e. the density of the predictand (water level, discharge, etc.) conditional on multiple deterministic model forecasts (predictors). The analysis through MCP can be done by considering a unique model forecast, single-model approach, or multiple model forecasts, multi-model approach (Coccia and Todini, 2011). Moreover, MCP can be applied either on a unique forecast horizon, i.e., the single-temporal approach (Coccia and Todini, 2011), or on multiple lead-times, i.e., the multi-temporal approach (Barbetta et al., 2016, 2017). MCP identifies the multi-variate conditional distribution, i.e. the probability density of the predictant conditional on the model predictions. This distribution is obtained by dividing the joint predictandprediction distribution by the joint marginal distribution of the predictor(s) and is estimated by considering an identified calibration data period. Specifically, the calibration identifies the joint and marginal probability distributions required for Bayes theorem application. The main characteristics of MCP are presented in what follows, while for additional details the reader is referred to Coccia and Todini (2011) and Barbetta et al. (2016, 2017). The MCP application is based on four main steps (Fig. 1):

P(y < yi )=

i = P( < n+1

P(y k < yki )=

i = P( n+1

i)

k<

(1) ki

(2)

where i = 1, ….., n, n = number of observed available data, i = plotting position order. In the normal space, the joint probability distribution of observed and product variables, f ( , ) , with ^ = [ ^ , ^ ,…, ^ ], is assumed to be a Normal Multivariate Distribution 1 2 M (Fig. 1, red box on the left side). 2. The conditional probability distribution, i.e., the conditional predictive density, is obtained applying the following equation that is known in the literature as the Bayesian Inverse problem, which starts from the Bayes formula (Fig. 1, green box on the right side):

f ( , ^) f ( ^)= f ( ^)

(3)

where f ( ) = f ( 1, 2 , ., M ) is the conditional probability distribution, f ( , ) = f ( , 1, 2 , ., M ) is the joint probability density of predictand and predictors and f ( ) = f ( 1 , 2 , ., M ) the joint probability density of predictors. The conditional predictive density f ( ) is a Gaussian distribution with mean, µ ( ) , and variance, 2 ( ) , defined as:

1. The observed precipitation, y, and the precipitation products (i.e., the prior knowledge that we have on precipitation), yk (k = 1…M, with M being the number of precipitation products that are combined together), are converted into the normal space using the Normal Quantile Transformation, NQT (Fig. 1, blue box on the left side). Note that the use of NQT is not in contrast with the use of other approaches based on the fitting of the distributions like Gamma (Sloughter et al. 2007) but it is justified in this case by the relatively large sample size (three years of data). In practice, y

1

µ(

)=

1

2

. .

M

(4)

step 3) Predictive Uncertainty, PU, is obtained by the Bayes Theorem

step 1) Conversion from the Real Space to the Normal Space using the NQT

Historical data

Inverse NQT

NQT

Joint Pdf

step 2) Joint distribution is assumed to be a Normal Bivariate Distribution

step 4) PU in the Real Space is estimated by sampling the probability density function in the Normal Space and reconverting the obtained quantiles by the Inverse NQT.

Fig. 1. Diagram summarizing the four main steps of the MCP for Predictive Uncertainty (PU) estimation, as applied to probabilistic estimate of rainfall in this work (NQT = Normal Quantile Transformation; y = observed precipitation; y = precipitation product; η = transformed of y in the Normal Space (NS); = transformed of y in the NS; a = precipitation threshold; ηa = transformed of a in the NS; f ( , ) = joint probability distribution; f ( ) and f (y y ) = conditional probability distributions). 343

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C. Massari, et al. 2(

)=1

1

T

applied here only to hit cases (when both predictand and predictors are non-zero during calibration and when both predictors are non-zero during validation). The zero/non-zero cases can be either left unchanged and equal to the SPP or treated differently. Generally, the first option is favorable if the SPP detection quality (i.e., correctly identifying zero and non-zero precipitation values) is relatively good and the inclusion of soil moisture information has a limited (or even detrimental effect) on the final result. Otherwise (like in our case), it may be worth to investigate whether the soil moisture-based product can add any value to the detection capability of the final product. Let us assume that we have to merge two precipitation products, A and B. If both products record 0 mm/day, we trust they are correct and assign 0 mm/d to the merged dataset. However, if only one of the two is 0 mm/d (let us say product A) and the other one records rain (let us say product B), then we may have one of the two following cases: i) product A is correct and product B is giving a false alarm; ii) product A is missing rain and product B is correct. In order to properly merge the two products in these cases, we assumed the probability density function of false alarms (pf) following an exponential law as function of rain rate Rsat:

(5)

3. The PU in the normal space is finally reconverted to the real space through the inverse NQT (Fig. 1, grey box on the right side). MCP requires as input rainfall observations and the corresponding estimates derived from the satellite products. For each time step, MCP provides as output the expected value of the identified multi-variate conditional distribution, i.e., the mean value and the cumulative 0.05 quantiles (5%, 10%, ….., 95%). In this study, MCP was applied to two rainfall products, a SPP and a SM-SPP as predictors, yk while a ground-based rainfall product was used as predictand y. As resulting merged rainfall product, P-MCP, the expected value of the multi variate conditional distribution was selected. 2.2. Choice of the predictand for MCP For building the density probability function f described above, MCP requires the knowledge of “true” precipitation field (i.e., the predictand). However, in practice, this information is never known, thus substituting the ground-based value for the actual value is a common practice in flood forecasting applications (Coccia and Todini, 2011). This concept can be also applied to precipitation, as rain gaugebased precipitation products are often assumed as truth. For example, many SPPs are named “corrected” when rain gauge observations are used for bias adjustment (inherently assuming that gauges provide a better estimate of the true rainfall volumes). Further proofs of this assumption are the existence of a number of studies using rain gauge observations for validating SPPs and assuming that the biases of rain gauge observations are small when compared to the ones in satellite estimates (Xie and Arkin, 1995). It has been also demonstrated that hydrologic simulations carried out in poorly gauged regions using SPPs provide performance equal or inferior to simulations that employ even just a few rain gauges (Xie and Arkin, 1995). In this study, we used a gauge-based rainfall dataset as the predictand (see Section 3.2 for further details) to extract the a priori information provided by SPPs and SM-SPPs during a “calibration” period (i.e., 2010–2012). The knowledge of the predictand is no longer necessary after calibration and MCP can provide a posteriori probabilistic rainfall estimates in near real time. The MCP output is the multi-variate conditional distribution and the cumulated 0.05 probability quantiles (i.e., 5%, 10%, …, 95%) which can be easily used to extract confidence bounds and the expected value of the estimates. As highlighted before, the expected value was considered in this work as the final merged rainfall product (i.e., P-MCP). Other applications and users may be interested in adopting a particular percentile of the distribution.

pf (Rsat ) = exp (

(6)

Rsat )

The underlying assumption is that false alarm rates are highly dependent on the satellite precipitation algorithm and the recorded rain rate, that is, the higher the rain rate, the lower the probability of that being a false alarm (Hossain and Anagnostou, 2004; Hossain and Anagnostou 2006). The parameter was calibrated by fitting Eq. (6) to data (i.e., zero-non zero cases) collected during the calibration period (see Fig. 2 for a curve fitting example in a pixel in Central Italy). is spatially variable (i.e., can vary from pixel to pixel) and was calibrated for both product A ( A ) and product B ( B ). Then, based on pf(Rsat), at every time t0 and location x0 at which product A was 0 mm/d and product B was RsatB greater than 0 mm/day, the probability (PF) of false alarm of B, PFB, was determined by

PFB (x 0) =

RsatB 0

B (x 0) exp (

B (x 0) Rsat )) dRsat

(7)

If PFB was larger than 90%, then the algorithm trusted product A and assigned 0 mm/day to the merged product. On the other hand, if PFB was lower than 90%, then the algorithm averaged products A and B (since chances were that B was overestimating and A underestimating rainfall). The same procedure was applied to times/locations at which product B was 0 mm/d and product A larger than 0 mm/day, by using

2.3. A heuristic merging approach for categorical cases One of the underlying assumptions of MCP is the possibility to transform the joint probability distribution associated to different products and observations to the normal space. With water levels and discharges, this is usually possible (see Coccia and Todini, 2011 for further details). However, when dealing with precipitation, its intermittent nature and the large presence of zero values – especially at short time scales – determine a high probability of zero-non zero cases when comparing two different products. This translates into a joint distribution that is highly skewed and very difficult to transform into the normal space. This issue is marginal for precipitation accumulations across 5–10 days, but it can be significant at the daily scale, when cases with one product being zero and the other being non-zero are very common. Although more robust approaches are possible (e.g., Sloughter et al., 2007; Wang and Robertson, 2011), to overcome this issue, MCP was

Fig. 2. Probability density function of false alarms for a pixel located in central Italy. Dots represent the observations, while the black dashed line refers to the fitted exponential function. 344

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PFA. Although simple, this method accounts for false alarms and missed rainfall cases, which were shown in previous studies to significantly contribute to total precipitation (Maggioni et al., 2016). Different false probability thresholds were tested and the 90% provided the best categorical score and bias reduction. As for MCP, the reference dataset used for calibration was a gauge-based rainfall dataset, described in Section 3.2. The application of this categorical case integration method in addition to MCP inherently poses a problem concerning the evaluation of the benefit of MCP itself. Therefore, to demonstrate its effectiveness (of MCP alone), we also considered scores calculated only on hits (see Section 2.4). In addition, we verified that if the method described in this section is not implemented (not shown here), the overall scores do not change significantly (although the missed volume of precipitation remains relatively high). It has also to be noted that treating differently zero and non-zero precipitation values at a daily time scale has a limited but still recognizable benefit, however, this benefit would be much higher if shorter temporal scales were considered.

has the units of precipitation). R, RMSE, and BIAS were also calculated on hit cases only (Rhits, RMSEhits, BIAShit) to investigate the SPP performance only when rainfall is larger than zero (i.e., by considering only the effect of MCP). In addition to these scores, two volumetric bias scores were considered (as defined in Tian et al., 2009): missed volume of precipitation (MS-vol) and falsely detected precipitation volume (FAvol). 3. Study area and datasets The proposed merging technique is tested across the Italian peninsula, which offers large variability in terms of terrain complexity and precipitation systems. Italy is characterized by a complex topography that spans from the Mediterranean Sea to the Alps and Apennines. The climate includes hot-dry summers and humid-cold winters, with high rainfall variability. For instance, a clear contrast exists between the more-rainy northern part (Liguria and Friuli regions) and the drier southern area (Sicily, Calabria, and Sardinia). The median annual precipitation averaged over the region during the study period is 1149 mm/year, with a minimum of 450 mm/year in southern Italy and a maximum of 3200 mm/year in northern Italy (using the rainfall dataset described in Section 3.2). A significant seasonal variability also exists, with late autumn and early winter months (September–November) being the wettest (Crespi et al. 2018).

2.4. Performance scores The performance of P-MCP was assessed against a gauge-based rainfall dataset (see Section 3.2 for further details on this product). Continuous and categorical error metrics were adopted in this study based on a pixel-to-pixel (0.25°) comparison for daily rainfall accumulations. Specifically, three categorical scores were considered: Probability of Detection (POD), False Alarm Ratio (FAR), and Threat Score (TS). POD refers to the measures the likelihood of a SPP to detect an event when it in fact occurs (with 1 being the perfect POD), FAR measures the likelihood that a precipitation event does not occur when SPP estimates rain (with 0 being the perfect FAR), while TS provides an integrated measure of the two scores and measures the fraction of observed events that were correctly predicted by the SPP (the perfect TS is 1). They are defined as follows:

POD =

H H+M

(8)

FAR =

F F+H

(9)

H H+F+M

(10)

TS =

3.1. Satellite-based rainfall datasets TRMM Multi-Satellite Precipitation Analysis real-time product v.7.0 (3B42RT), developed by Huffman et al. (2007), is the SPP adopted in this work. The TMPA algorithm merges information from PMW sensors and geostationary IR observations. The PMW data are intercalibrated to the TRMM Combined Instrument (TCI), which includes information from the Precipitation Radar (PR), an electronically scanning radar operating at 13.8 GHz. TRMM started collecting data in 1998, but the mission came to an end in April 2015. The 3B42-RT product has a 9 hr latency, is available at 0.25°/3 hr spatial/temporal resolution, and has quasi-global coverage (50°N–50°S). SM2RAIN-CCI (Ciabatta et al., 2018) applies the SM2RAIN algorithm to the ESA Climate Change Initiative (CCI) soil moisture dataset (Dorigo et al., 2017). The algorithm has been calibrated during three different periods (1998–2001, 2002–2006 and 2007–2013) against the Global Precipitation Climatology Centre Full-Data daily dataset (GPCCFDD, Schamm et al., 2015). The quality flag provided within the raw soil moisture observations was used to mask out low quality data, as well as the areas characterized by high topographic complexity, high frozen soil and snow probability, and presence of tropical forests therefore this product does not cover all the Italian territory. The integration exercise was then carried out only the remaining pixels. In addition to 3B42RT and SM2RAIN-CCI, the GPM IMERG “early run” and “final run” precipitation products v 4.0 (Huffman et al. 2015) were used to intercompare the merged product across the study area. No merging was carried out with these products, given the relatively short record of rainfall observations currently available. The IMERG algorithm, firstly released in early 2015 (Huffman et al., 2015), is run at 0.1°×0.1° spatial and half-hourly temporal resolutions in three modes, based on latency and accuracy: “early” (latency of 4–6 h after observation), “late” (12–18 h), and “final” (∼3 months). The main difference between the early and final runs is – beyond the way the different sensor measurements are propagated in time – the fact that the early run has a climatological rain gauge adjustment, while the final run uses a month-to-month adjustment based on GPCC gauge data. In this study, the two daily IMERG products were upscaled to 0.25° by using a box-shaped kernel with antialiasing, which approach was found to outperform simple spatial averaging.

where H represents the number of rainfall events successfully detected (hits), M is the number of the missed events, and F is the number of norain events erroneously predicted as events (false alarms). For each given threshold, a rainfall event is scored as hit, miss, or false depending on how the observed and estimated rainfall behave with respect to that threshold: hit if they both reach it, miss if only the observed rainfall reaches it, false if only the estimated rainfall reaches it. Three rainfall thresholds were selected in order to guarantee both the evaluation of the products for low (< 2mm/day), medium (2–10 mm/ day) and high (> 10 mm/day) rainfall accumulation amounts and the representativeness of each class. Three continuous scores were also taken into account: Pearson correlation coefficient (R), adjusted R-squared (R2), Root Mean Squared Error (RMSE), and total bias (BIAS) which was evaluated both in terms of difference in millimetres between ground precipitation and estimated precipitation fallen during the study period and in relative terms (i.e., percentage). R provides a measure of the linear association between SPP and the observations (the perfect score is 1), whereas R2 indicates the percentage of variance explained by SPP with respect to the total observed precipitation variance. RMSE is the square root of the average of the squared differences between SPP and the observations and gives more weight to large errors than smaller errors, which may be good if large errors are especially undesirable, but may also encourage conservative precipitation estimation (the perfect score is zero and it 345

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Table 1 Median Pearson correlation, R, root mean squared error, RMSE (Rhits and RMSEhits refer to the same scores but calculated only for hit cases, i.e., when both the product and the reference are larger than zero), and percent BIAS of 3B42RT, SM2RAIN-CCI, and P-MCP during the calibration (2010–2012) and validation (2013–2015) periods. Values in brackets refer to the scores calculated from 12 March 2014 (the beginning of the availability of the IMERG products). CALIBRATION

R Rhits RMSE (mm/day) RMSEhits (mm/day) BIAS (%)

VALIDATION

Mar. 2014- Dec. 2015

3B42RT

SM2RAIN-CCI

P-MCP

3B42RT

SM2RAIN-CCI

P-MCP

IMERG-EA

IMERG-FR

0.553 0.406 6.146 12.524 −40.337

0.546 0.487 5.946 8.665 −27.539

0.678 0.608 5.172 7.590 −10.827

0.541 (0.589) 0.405 (0.475) 6.694 (6.337) 12.972 (12.433) −34.019 (−29.334)

0.572 (0.582) 0.532 (0.555) 5.852 (5.778) 7.835 (7.672) −19.681 (−20.209)

0.645 0.581 5.381 7.379 3.539

0.649 0.605 6.562 9.811 −15.316

0.767 0.722 5.453 8.506 −7.143

3.2. Ground-based dataset

(0.667) (0.628) (5.227) (7.051) (2.892)

good performance of the processor. During the calibration period, this happens for almost all the pixels (red line in Fig. 3a). During the validation period, the correlation scores (and the errors) are significantly increased (reduced). In particular, P-MCP correlation is close to 0.65 and median RMSE is reduced to a value close to 5 mm/day (about 1.3 mm/day and 0.5 mm/day lower than 3B42RT and SM2RAIN-CCI) while the BIAS is close to zero (about 4%). If compared against the IMERG early run product (IMERG-EA) during March 2014 – December 2015, P-MCP provides better scores in terms of R (and Rhits), RMSE (RMSEhits), and BIAS. With respect to the IMERG final product (IMERG-FR), however, correlations are worse, but errors and BIAS are lower, which is remarking as the final run version of IMERG is corrected with gauge observations and released only after a couple of months after sensing. Unlike calibration, the diagram in Fig. 3b obtained during the validation period indicates a deterioration of the performance with a slight tendency to underestimate the probability quantiles. This is likely due to the non-stationary nature of precipitation products, which might be affected by the climatic regime and type of precipitation and can vary from year to year (Ciabatta et al., 2016; Stampoulis and Anagnostou, 2012). Fig. 4a and b display daily and monthly rainfall time series – averaged across the study area – obtained by 3B42RT, SM2RAIN-CCI, and P-MCP during the validation period. P-MPC estimates show a better agreement with the benchmark dataset with respect to the parent products with a tendency of overestimating summer precipitation, particularly in the monthly time series accumulations (Fig. 4b). A bias improvement is also observed for most months, with better identified rainfall peaks. The problem of overestimation during summer could be simply avoided by a seasonal calibration of MCP. However, given the relatively short observational period available here, this is not a recommended solution as it reduces too much the number of days when both predictand and predictors are available

A ground-based rainfall dataset obtained from more than 3000 rain gauges over the Italian territory and covering the period 2010–2015 was used for the calibration (as predictand for MCP) and the validation of the procedure. The dataset is composed by hourly rain gauge measurements recorded by the national monitoring network of the Italian Civil Protection Department. The hourly rain gauge observations are spatially interpolated over the analysis grid (i.e., 0.25°) using the Random Generator of Space Interpolations from Uncertain Observations (GRISO; Ciabatta et al. 2015; Pignone et al., 2010) algorithm, and aggregated at daily time step. 4. Results Table 1 summarizes the performance of SM2RAIN-CCI, 3B42RT, and PMCP during the calibration period (2010–2012) in terms of median R, Rhits, RMSE, RMSEhits, and BIAS. A clear increase in correlation (∼0.15 in median) and a reduction in error (of 1 mm/day in median) and BIAS (20/ 30% in median) is observed when P-MCP estimates are considered with respect the two parent products. These increments and reductions are evident both when considering all the precipitation and when considering only hits (Rhits, RMSEhits), suggesting that the main driver of the improvements is the application of MCP to non-zero precipitation values. The P-MCP performance was also investigated through the comparison between the estimated non-zero P-MPC precipitation percentiles and the corresponding observed occurrences during the calibration and the validation period (Fig. 3a and b). Specifically, the performance of the processor was evaluated by plotting the cumulated 0.05 probability quantiles assessed by the P-MCP against the corresponding percentages of observed data that fall below each percentile. A line close to the one-to-one line (black diagonal) suggests a

Fig. 3. Comparison between the MCP estimated percentiles and the corresponding observed occurrences during the calibration (a) and validation (b) periods. 346

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Fig. 4. Mean daily (a) and monthly (b) time series of rainfall averaged across the study area during the validation period (2013–2015).

within the specific calibration season. Fig. 5 presents the yearly accumulated rainfall from a) gauges, b) 3B42RT, c) SM2RAIN-CCI, and d) P-MCP during 2013–2015 and the

related frequency of rainy days (where a rainy day is defined as a rainfall accumulation of more than 2 mm). Both SM2RAIN-CCI and 3B42RT provide a significant underestimation of the ground-based

Fig. 5. Yearly accumulated rainfall from ground observations (a), 3B42RT (b), SM2RAIN-CCI (c), and P-MCP (d) during 2013–2015 and frequency of rainy days for the same products (e-h). 347

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Fig. 6. Differences in R2 (a) and RMSE (b) between P-MCP and 3B42RT and corresponding scatterplots (c, d). In (a) and (b) blue denotes an improvement, while red denotes a deterioration in the statistical metrics. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6c and 6d show a general unconditional increase in R2 and a reduction in RMSE. The same plots, but relative to SM2RAIN-CCI, are reported in the Supplementary material (Fig. S1). They show 1) a slight tendency to worsen correlation in southern Italy by P-MCP due to the relatively low performance of 3B42RT here (correlation below 0.3) and 2) a clear pattern of improvement in RMSE with larger corrections in areas characterized by a relatively higher accumulation amounts (wetter areas). The sub-optimal P-MCP performance in terms of correlation obtained with respect to SM2RAIN-CCI in southern Italy, can be also due to the relatively short calibration period associated with 1) the larger rainfall temporal variability of the Mediterranean climate with respect to monsoon-type of climates and 2) the smaller number of rainy days caused by the more arid climate which causes a less robust MCP calibration. Bias scores are reported in Fig. 7. P-MCP shows a median bias close

rainfall field and a different spatial pattern. On the contrary, P-MCP provides a spatial rainfall accumulation distribution closer to the one of the rain gauges. The number of rainy days is significantly underestimated by 3B42RT and overestimated by SM2RAIN-CCI due to the noise contained in satellite soil moisture observations. In this respect, PMCP is mainly impacted by the overestimation of rainy days provided by SM2RAIN-CCI and shows a general overestimation across the study area. Fig. 6a (6b) shows the difference between R2 (RMSE) of P-MCP and R2 (RMSE) of 3B42RT. Correlation is generally improved all over the study area, except few pixels (< 25 out of 393) located in central and southern Italy. Similarly, RMSE is consistently improved with a few pixels deteriorated (4 out of 393). For these pixels, the correlation of 3B42RT is below 0.2 (not shown) and it is likely that MCP struggles to provide a reliable precipitation estimates in these cases. Scatterplots in 348

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Fig. 7. Total bias (a), false volume of precipitation (FA-vol)(b), missed volume of precipitation (MS-vol) (c), and hit-bias volume (BIAS-hit) (d) for 3B42RT, SM2RAIN-CCI and P-MCP during the validation period (2013–2015).

Fig. 8. Differences in FAR, POD, and TS of P-MCP with respect to 3B42RT as a function of rain rate classes.

to zero. The variability is also reduced with smaller 25th an 75th percentiles. On one hand, false precipitation volumes (FA-vol) are reduced with respect to SM2RAIN-CCI, but not with respect to 3B42RT. On the other hand, missed precipitation volumes are significantly reduced with respect to both products suggesting the effectiveness of the procedure described in Section 2.3. The sub-optimal behaviour in terms of false volume is also due to the low performance of SM2RAIN-CCI in this respect. This is likely due to noise present in satellite soil moisture observations that is mistakenly identified as rainfall (Zhan et al., 2015). Nevertheless, false precipitation volumes are much lower than missed and hit precipitation volumes and have therefore a relatively low impact on the total bias. In terms of hit-bias, the merged product performs well confirming the robustness of the MCP procedure in merging nonzero precipitation values. When compared with IMERG products during March 2014-December 2015 (Fig. S2 in the Supplementary material), PMCP shows better performance in terms of hit bias and again a slightly worse performance in terms false volume of precipitation. Fig. 8 displays the improvements obtained by P-MCP with respect to 3B42RT in terms of categorical scores (FAR, POD, and TS) for three representative rainfall classes, defined as light (< 2mm), medium (2–10 mm), and strong (> 10 mm). There is a deterioration of false

alarms in medium and strong rainfall regimes, but a significant improvement in terms of POD. On the contrary, for light rainfall, POD is slightly deteriorated and FAR is slightly improved. The resulting differences in TS are generally positive showing that a good compromise is reached by the merging procedure in integrating the parent products. The improvement with respect to SM2RAIN-CCI categorical scores is reported in the Supplementary material (Fig. S3). Here P-MCP consistently outperforms the parent products both in terms of FAR and POD, which results in an overall enhanced TS with respect to the parent product. Fig. 9 shows one of the most interesting results of this study: the 30day moving mean spatial correlation of 3B42RT, SM2RAIN-CCI and PMCP calculated for each day during the validation period. It can be seen that P-MCP consistently provides better correlations with respect to the parent products and performs either better or at least similar to the best parent product, even when 3B42RT’s spatial correlation becomes negative or close to zero. This confirms its ability to provide estimates that are closer to the benchmark rainfall fields.

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For instance, some deterioration was observed in pixels located in drier precipitation regimes where the relatively large number of zero precipitation values associated with the relatively short calibration period for that climatology (i.e., too few precipitation events) impacted the robustness of the MCP calibration process. This can be overcome with alternative calibration strategies, which involve the inclusion of spatial information on precipitation (i.e., more pixels at the same time) to increase the size of the calibration dataset. Moreover, a reliable calibration dataset (i.e., the predictand), which is not always available, is necessary. To overcome this difficulty alternative calibration strategies can be considered like the use of available observations in areas characterized by similar climatic and topographic conditions to the study area, or, calibrating MCP with rainfall observations available at much larger spatial scales (i.e., the Global Precipitation Climatology Centre, GPCC, provides long time series of daily rainfall observations at 1-degree spatial resolution). These two solutions are currently under investigation. The reader is also advised that the results obtained here refer to daily accumulation rainfall with a 0.25-degree spatial resolution, therefore their general validity cannot be guaranteed at other spatiotemporal scales. This limits in part the range of applications suitable for using the integrated product to (for instance) drought monitoring, large scale flood forecasting and crop modelling. Applications requiring subdaily precipitation values like flash flood forecasting are not covered due to the revisit time of current satellite soil moisture products (typically one day or more). Lastly, it must be stressed that only two products were integrated. Nevertheless, MCP could be easily applied to multiple products, thus opening new opportunities for the development of blended-global rainfall data in near-real time, which can be extremely useful for many operational applications.

Fig. 9. 30-day moving mean of the spatial correlation between 3B42RT, SM2RAIN-CCI, P-MCP and rain gauges.

5.

Discussion

and conclusions

This work investigates the potential of a novel and alternative approach for complementing SPPs (derived from 3B42RT) with SM-SPPs (derived from SM2RAIN-CCI), which is based on the use of the Model Conditional Processor. As the inclusion of zero-precipitation values in the procedure might significantly skew the probability distributions associated with the products, we applied MCP only to mutual non-zero precipitation values (i.e., hits) and implemented a heuristic-based approach to consider categorical cases (i.e., when one of the product is zero while the other is larger than zero and vice versa) because of the large precipitation volumes missed by the SPP. Results demonstrate that the proposed method overall improves both SPP and SM-SPP providing a valid alternative to complement SPPs with satellite soil moisture observations. Significant benefits are observed in terms of correlation, RMSE, categorical scores and biases particularly in terms of hits suggesting that the main driver of improvements is the MCP. The P-MCP product outperforms the parent products also in reproducing the spatial distribution of precipitation across Italy. This is due to the MCP main concept, which does not provide a simple merged product, but rather the probability (and therefore an estimate) of the “true” rainfall conditioned on the available information. This ability of correctly estimating the precipitation spatial distribution can be extremely important in many operational applications including flood forecasting, when the contribution of different catchments to the total flow at the outlet has to be considered. Relative to the other methods, like the ones proposed by Crow et al. (2011) and Pellarin et al. (2013), this method provides similar results in terms of correlation, RMSE, and categorical scores enhancement, but being conceptually different, it shows some additional potential to improve the systematic error component of SPPs. This is confirmed by 1) the better RMSE and BIAS scores obtained here with respect to past integration exercises carried out in the same study region by Ciabatta et al. (2015, 2017) and Tarpanelli et al. (2017) and 2) with respect to the performance of GPM products. Some limitations are found in terms of estimation of frequency of rainy days like in Zhan et al. (2015) which we found highly impacted by high-frequency noise in satellite soil moisture observations. This causes larger uncertainty in the false alarm rainfall which however has a marginal effect on the final total bias. We expect that new satellite soil moisture observations like the ones derived from the Soil Moisture Active and Passive mission (SMAP, Entekhabi et al., 2010) can significantly reduce this problem thanks to their recognized better signal to noise ratio. A strategy to solve this issue can be also the limitation of the integration to only non-zero precipitation values (e.g., Zhan et al., 2015 corrected only 3B42RT precipitation values larger than 2 mm) which has not been followed within this study because of the poor performance of 3B42RT in this respect. In this context, the consideration of the new IMERG products, characterized by significant better detection skills with respect to 3B42RT, can make the strategy of correcting only non-zero precipitation values more successful. The method potentially suffers from a few additional limitations.

Declaration of interests None. Acknowledgements This work is supported by the European Space Agency ESA (contract 4000114738/15/I-SBo) project SMOS+Rainfall Land II and by the ESA Climate Change Initiative (CCI, http://www.esa-soilmoisture-cci.org/, contract no. 400011226/14/INB). The Italian Civil Protection Department is gratefully acknowledged for providing the observed data from the Italian monitoring network. The authors also want to gratefully acknowledge the anonymous reviewers and the editor for their help in improving the manuscript. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.jhydrol.2019.03.038. References AghaKouchak, A., Mehran, A., Norouzi, H., Behrangi, A., 2012. Systematic and random error components in satellite precipitation data sets. Geophys. Res. Lett. 39 (9). Barbetta, S., Coccia, G., Moramarco, T., Todini, E., 2016. Case study: a real-time flood forecasting system with predictive uncertainty estimation for the Godavari River India. Water 8 (10), 463. Barbetta, S., Coccia, G., Moramarco, T., Brocca, L., Todini, E., 2017. The multi temporal/ multi-model approach to predictive uncertainty assessment in real-time flood forecasting. J. Hydrol. 551, 555–576. Behrangi, A., Yixin, W., 2017. On the spatial and temporal sampling errors of remotely sensed precipitation products. Remote Sens. 9 (11), 1127. Beck, H.E., van Dijk, A.I.J.M., Levizzani, V., Schellekens, J., Miralles, D.G., Martens, B., de Roo, A., 2017. MSWEP: 3-hourly 0.25° global gridded precipitation (1979–2015) by merging gauge, satellite, and reanalysis data. Hydrol. Earth Syst. Sci. 21 (1), 589–615. Brocca, L., Ciabatta, L., Massari, C., Moramarco, T., Hahn, S., Hasenauer, S., Kidd, R., Dorigo, W., Wagner, W., Levizzani, V., 2014. Soil as a natural rain gauge: Estimating

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