3.2.2. Optimum Interpolation (OI)

OI analysis requires an initial estimation field, such as a set of gridded satellite precipitation data. By calculating the error weight function of the initial estimation field and the observation field point by point, the target grid points for analysis can be corrected. Thus, when developing OI-based data merging algorithms, the key is to quantify the error structure of the initial estimation field and the observation field. Unlike other data-merging methods, OI considers not only the autocorrelation of various errors but also the correlation between different observations. Moreover, OI involves solving for the optimal value within a certain range of each analysis point and thus is particularly suitable for the analysis of single variables with large spatiotemporal variability, such as precipitation [34]. In the OI analysis conducted in this study, the OIMERG precipitation data were used as the initial estimation field, and the station precipitation observations were used as the observation field. The final analysis result for the precipitation value (*Ak*) at each grid point is equivalent to the first guess (*Fk*) at this grid point plus the deviation between the observed value and the initial estimated value at the grid point. This deviation is obtained through weighted estimation based on the deviations between the known observed values (*Oi*) and initial estimated values (*Fi*) from *n* grid points within a certain range, and it represents the maximum distance correlated with the target grid point. The formula is as follows:

$$A\_k = F\_k + \sum\_{i=1}^{n} W\_i (O\_i - F\_i) \tag{1}$$

where *k* is the grid point to be analyzed, *i* is an index representing the "valid grid boxes" (boxes in the satellite precipitation grid containing at least one gauge station), and *Wi* is a weight coefficient assigned to the deviation between the observed value and the initial estimated value in the *i*th grid box during estimation. Note that in an area with a sparse station network, the analysis radius should be continuously adjusted to ensure that a sufficient number of valid grid boxes can be searched, from which several valid grid boxes nearest to the target grid point are then selected for inclusion during OI. In this study, the analysis radius was set to 100 km, and the 9 nearest valid grid boxes to the target grid point were selected for OI [35].

In Equation (1), the weight coefficients (*Wi*) are determined by the variance in the minimum error on the precipitation value (*Ak*) at the target point:

$$E^2 = \overline{(A\_K - T\_K)^2} \tag{2}$$

where *TK* represents the observed value at point *k*.

Based on the assumptions that the observation field and the initial estimation field are unbiased and that the observation error is not related to the error of the initial estimation field, the weight coefficients *Wi* in Equation (1) can be obtained by solving the following linear equation [34]:

$$\sum\_{j=1}^{n} \left( \mu\_{ij}^f + \mu\_{ij}^o \lambda\_i \lambda\_j \right) \mathsf{W}\_j = \mu\_{ki}^f \tag{3}$$

where μ*<sup>f</sup> ij* represents the co-correlation of the initial estimation field error, <sup>μ</sup><sup>0</sup> *ij* represents the co-correlation of the observation error, and λ*<sup>i</sup>* is the ratio between the standard deviation of the observation error (σ*<sup>o</sup> i* ) and that of the initial estimation error (μ*<sup>o</sup> ij*) at point *<sup>i</sup>*. In OI, the calculation of the *Wi* requires that <sup>μ</sup>*<sup>f</sup> ij*, μ*o ij*, <sup>σ</sup>*<sup>o</sup> i* , and σ*<sup>f</sup> <sup>i</sup>* are known values, which, in turn, requires the pre-estimation of the observation error and the satellite-retrieved precipitation error as well as the correlation between these errors. Here, the term "pre-estimation" means that these parameters need to be estimated in advance. In this study, this pre-estimation was performed based on a statistical analysis of the sample data within the study period [35].

The weight coefficients (*Wi*) were determined based on Equation (3), and then, the final precipitation values (*Ak*) were obtained based on Equation (1).
