#### 3.1. Estimation of approximate true surface areal rainfall

Based on the network spatial arrangement within the study area, two HRAP pixels of the MPE products are covered by multiple gauges (one pixel is covered by six gauges and another pixel is covered by four gauges; Figure 1). Within these two pixels, the inter-gauge distances are in the order of 1–2 km, which is smaller than the correlation distance of hourly rainfall in this area (Habib *et al.*, 2009). Therefore, by averaging observations from multiple gauges within each of the two pixels, a fairly reliable estimate of the unknown true pixel-average surface reference rainfall (which we refer to as *R*_{s}) that is not significantly contaminated by the point-to-area errors typically associated with single-gauge observations can be obtained. In the remainder of this study, the estimation error of a certain MPE product is defined as the deviation between MPE estimate and the corresponding *R*_{s} value. For consistency, the MPE-*R*_{s} samples are based upon paired datasets excluding hours when rainfall was not recorded by *R*_{s} or any of the MPE products.

#### 3.2. PED method for radar-rainfall uncertainty modelling

A complete description of the Product-Error-Driven (PED) method is given in Ciach *et al.* (2007) and Villarini and Krajewski (2010a): only a brief overview is provided herein. The PED approach starts with estimating and removing the product overall bias (*B*_{0}):

- (1)

- (2)

where *B*_{0} is the bias factor, *R*_{s, i} is the *i*^{th} hourly surface reference rainfall (obtained by averaging observations from multiple gauges within each pixel), and *R*_{r, i} are the corresponding hourly radar-product rainfall values (before and after correction for overall bias, respectively). The summation is taken over all hours available during the entire 2 year sample.

After removing the overall bias, the relation between surface rainfall *R*_{s} and the radar-product rainfall can be described as a combination of a systematic function (*h*) and a random component (ε) using two possible formulations, additive and multiplicative:

- (3)

- (4)

Both *h* and ε are functions of the radar-rainfall values. The systematic function characterizes the conditional bias in the radar-rainfall product and can be estimated as a conditional expectation function:

- (5)

where *r*_{r} and *r*_{s} represent specific values of the random variables *R*_{r} and *R*_{s}.

The systematic function (*h*) can now be removed to yield the random component in either the multiplicative or the additive error forms:

- (6)

- (7)

After removing the conditional and unconditional biases in *R*_{r}, it can be reasonably assumed that the conditional means of ε_{m}(*R*_{r}), *E*[ε_{m}|*R*_{r} = *r*_{r}], and ε_{a}(*R*_{r}), *E*[ε_{a}|*R*_{r} = *r*_{r}], are equal to 1 and 0, respectively.

The conditional variances of ε_{m}(*R*_{r}) and ε_{a}(*R*_{r}) can be expressed as:

- (8)

- (9)

Following Ciach (2003) and Ciach *et al.* (2007), a kernel regression approach is used to obtain a non-parametric estimate of the two conditional statistics, *h*(*r*_{r}) *and* σ_{ε}(*r*_{r}) using a moving-window averaging formula (see equations ((6)) and ((9)) in Ciach *et al.* (2007) and equation ((10)) in Villarini and Krajewski (2010a)). For a comparison between parametric and non-parametric approaches, the interested reader is pointed to Villarini *et al.* (2008).

To provide further characterization of the distribution of the conditional random error, the PED method also computes the conditional quantiles (*q*_{p}) for various levels of probabilities *P*:

- (10)

- (11)

Following the same non-parametric estimation method, weighted-point-counting procedure was used to estimate *q*_{p} (see equation (11) in Ciach *et al.* (2007)).

Besides marginal statistics, the PED method includes characterization of spatial and temporal autodependencies of the random error using the Pearson's product-moment correlation coefficient. The spatial auto-correlation of the random component is estimated by calculating the correlation between the MPE errors at the two neighbouring multiple-gauge HRAP pixels. The limited spatial coverage of the current gauge network allows for estimating the auto-spatial correlation at one spatial lag only (4 km). The temporal auto-correlations can be computed at various separation time lags (1–6 h) using the random error sample over a single pixel.