Vegetation closely interacts with soil moisture and is recognized as a good proxy of precipitation. The remote sensing technique provides us with various high-resolution vegetation products such as NDVI, EVI (enhanced vegetation index), and LAI (leaf area index). Among the vegetation indices, NDVI, an indicator of plant density and growth, is chosen as the proxy of precipitation in this study due to its wide availability. Considering the crucial role of NDVI in deriving precipitation estimates under our framework, we conduct an inter-comparison in data accuracy between two NDVI datasets (termed datasets A and B hereinafter) to reduce the error. First, the systematic errors of both datasets are eliminated by multiplying the reduction factor or using the simple regression model. After the correction, the final dataset is then obtained by selecting a better element between A and B if the quality criteria are satisfied, otherwise filling an anomaly value.

It should be noted that since the vegetation growth is suppressed or promoted on some land covers (e.g., rivers, lakes, snow and ice, and urban areas), the vegetation data of these land covers are excluded by filling anomaly values. Besides, due to the strong influence of farming activities (e.g., irrigation, fertilization, and harvest) on the crop growth, vegetation data of farmland are excluded as well. We note that although Moran's index (Li et al., 2007) is widely employed to detect anomalies in vegetation data (Jia et al., 2011; Duan and Bastiaanssen, 2013), it is not used in this study for its inapplicability in large areas with continuous anomaly pixels (e.g., farmland). As such, we identify anomaly pixels simply by land-use type: pixels categorized as water, wetland, urban, cropland, snow/ice, and barren will be identified as anomalies. The detected anomaly pixels are excluded from the original NDVI dataset and then filled with interpolated values using the IDW method so as to generate an optimized NDVI dataset.

Based on the optimized NDVI dataset, the NDVI data at the gauge locations are retrieved with the neighbor-average method (i.e., the value of a certain grid is determined as the average of all its eight neighboring grids) and will be used for the precipitation–vegetation regression.

As far as we know, there is no widely accepted form of the precipitation–vegetation relationship. Therefore, the final regression form will be determined from several candidate relationships, including polynomial, exponential, logarithmic, and linear forms, according to the five metrics: correlation coefficient (R), coefficient of determination (R2), root-mean-square error (ERMS), mean relative error (EMR), and mean absolute relative error (EMAR), which are given as follows:

R=∑i=1n(Pi-P‾)(Oi-O‾)∑i=1n(Pi-P‾)2∑i=1n(Oi-O‾)2,R2=∑i=1n(Pi-Oi)2∑i=1n(Oi-O‾)2,ERMS=∑i=1n(Pi-Oi)2n,EMR=1n∑i=1nPi-Oi,EMAR=1n∑i=1n|Pi-Oi|Oi, where O‾ is the mean annual precipitation of all gauges, Oi the mean annual precipitation of gauge i, Pi the estimated precipitation at gauge i, and n the total number of gauges.

Also, considering the annual variability of precipitation, the regression model is further determined for two temporal scales: (1) the entire period covering all the study years and (2) the individual year of the entire study period. The regression models for the entire study period and for individual years are thus termed RME and RMI, respectively. RME can utilize the full knowledge of precipitation characteristics of the entire study period, whereas RMI implies the inter-annual variability. Besides, RME can reasonably reconstruct the precipitation series of the years when data gaps exist.

The calibration–validation procedure for each candidate model is conducted under three scenarios with different numbers of gauges and/or years: Scenario a

Fully random: a random number of gauges and a random number of years are independently used for calibration and validation;

With the optimized vegetation dataset and the precipitation–vegetation regression model, the RSBP product is then remapped over the study region. Thanks to the finer resolution of the NDVI dataset than the RSBP product and the accurate estimate of precipitation by gauges, the remapped RSBP product is expected to provide more detailed spatial characteristics of precipitation over mountainous areas.

The Nu-Salween basin (Fig. 2a), where 6 million people live, is one of the largest river basins in South Asia and spreads across three countries with an area of 324 000 km2. This study focuses on the Chinese part of the Nu-Salween basin (termed the Nu River basin hereafter), where the elevation ranges from 446 to 6134 m and the narrowest part is only 24 km. The annual precipitation of the Nu River basin ranges from 400 to 2000 mm with an average of 900 mm, and the mean annual runoff is 69 km3. The precipitation of the Nu River basin generally decreases from southwest to northeast and demonstrates high variability due to mountain weather systems (e.g., the difference in annual precipitation between the mountaintop and valley of Gongshan is larger than 1000 mm). Annual rainfall varies significantly across this region. Figure 2b shows the annual rainfall distributions of seven stations located in the upstream, middle, and downstream of the Nu River basin. The upstream and downstream have similar rainfall distributions, with larger rainfall occurring in summer compared to winter, while the middle part observes relatively large rainfall in winter and spring. Thanks to the adequate rainfall and minimal human perturbation, the Nu River basin has an extensive vegetation coverage, with the dominant types grassland in the Qinghai–Tibetan Plateau (upper basin) and mixed forest in Yunnan Province (lower basin). However, the dense vegetation cover increases the difficulty in conducting precipitation observations and only 13 gauges are very unevenly distributed over the whole basin of 142 479 km2, which makes it highly challenging to obtain the accurate spatial precipitation characteristics with traditional interpolation approaches. Although the RSBP products are available for this area, they are too coarse (usually with a spatial resolution of ∼ 50 km) to capture the high spatial variability of precipitation.

Considering the limited number of gauges (i.e., 13) in the Nu River basin, an enlarged area covering 23–33∘ N and 91–101∘ E is chosen for the application of the fusion framework, where 59 gauges are available and the climatic and topographic conditions are similar: both regions are characterized as mountainous areas under the subtropical climate influenced by the southeast and southwest monsoons. Besides, given no rain gauges are available outside of China in this study region, the non-Chinese region is excluded from the study area.

Based on the results of six evaluation metrics for different regression form candidates (Fig. 3a), the second-order polynomial is chosen as the regression model form in this study: p=aNDVI2+bNDVI+c, where p denotes the precipitation amount in millimeters, and a, b, and c are regression coefficients. The results of regression coefficients and evaluation metrics are given in Table 1, and the NDVI–precipitation relationships for the study period are demonstrated in Fig. 3b.

The best performance of the regression model is found within 0.2 < NDVI < 0.7 and 400 mm yr-1 < p < 1500 mm yr-1. Larger errors are found at pixels with NDVI larger than 0.7 or annual rainfall higher than 1500 mm, implying the water supply is no longer a determinant of vegetation growth as annual rainfall exceeds a certain threshold.

In general, the RMIs demonstrate better performance than RME, which can be attributable to the lower variability of precipitation in a single year than the whole study period. It is also noted that the R2 values of RMIs for drier years (2003, 2009, and 2011) are less than wetter years, indicating the weaker coupling effect between vegetation growth and precipitation.

The performance of regression models is assessed under three scenarios as described in Sect. 2.2. A total of 300 tests are conducted and performance metrics (i.e., R, R2, ERMS, and EMAR) are calculated accordingly (Fig. 4 and Table 2). The high R values (> 0.85) indicate a strong correlation between NDVI and precipitation independent of sampling method. Also, the regression models demonstrate good performance, with R2 larger than 0.75 and EMAR less than 20 %. In addition, the metrics of regression models fluctuate around that of the RME, with narrow inter-quartile ranges, indicating the regression models have remarkable consistency with the RME model.

Scenario a is designed to examine inter-annual stability in the performance of regression models, where the good performance indicates the acceptable ability of the RME model in estimating precipitation during periods when precipitation measurements are not available. Scenarios b and c investigate the impacts of spatial and temporal coverages of measurements, respectively. It is noteworthy that under Scenario b better performance in regression models is observed as compared with Scenario c, implying the greater importance of spatial coverage of measurements in conducting the regressions. In addition, the results of calibration are better than validation, as revealed by all metrics criteria, as expected. However, the differences between calibration and validation are not significant, implying the consistent performance of regression models under various scenarios.

The performance of RME is further assessed by comparing the estimates against observations (Fig. 5), and good agreement between estimates and observations is observed. It should be noted that the RME shows difficulty in estimating precipitation higher than 2000 mm (cf. the dashed line in Fig. 5), implying the limitation of the fusion framework inherited from the oversaturation effect of the vegetation index.

Elevation effect on the relationship between precipitation and NDVI is a concern to appreciate. An overall negative relationship is found between precipitation and elevation for the whole elevation range (i.e., 0–5000 m) with the R2 value of 0.62 (Fig. 6a), whereas there is only an unapparent/weak relationship at different elevation bands (Fig. 6b–f). Given the spatial heterogeneity of orographic effects on precipitation (Brunsdon et al., 2001; Daly et al., 2008) and the insufficient data of this study, a more thorough investigation of the relationship between precipitation and elevation needs to be conducted with more information that might be available in the future. Positive precipitation–NDVI relationships are found at different elevation bands (Fig. 7), with the best and worst fitness observed at elevation band 2000–3500 m with an R2 value of 0.94 and at elevation band 0–2000 m with an R2 value of 0.62, respectively. By comparing the three regressions at different bands with the global regression, we notice that more significant overestimates of precipitation are observed with the range of lower NDVI values (< 0.4) at band 0–2000 m than the other three regressions, whereas regression at band > 3500 m has a significant overestimation of precipitation than the other three regressions for higher NDVI values (> 0.5).

The spatial characteristics of the precipitation of the study area are investigated with RME for the whole study period (Fig. 8). Annual precipitation in the Nu River is observed to decrease from south to north and from west to east with prominent spatial variability. Two “hot-spot” regions, whose annual precipitation exceeds 1500 mm, can be identified in the study areas: one near the southern border and the other close to the southwestern mountain border. The eastern part of the Nu River basin featuring a dry and warm climate receives an average annual precipitation of 800 mm with large inter-annual variability. A precipitation product (DEMP) based on a precipitation–elevation relationship is used to compare with RME. There is no obvious distribution pattern of precipitation (Fig. 9a) and a smaller spatial variability compared to RME in the DEMP product, indicating the advantage of RME in representing the spatial variability of annual precipitation. And the overall underestimation of precipitation is observed in the DEMP product across the whole study area (Fig. 9b). In addition, the pixels in Fig. 8 with a value out of the valid range (i.e., 400 mm yr-1 < P < 1500 mm yr-1) may have a relatively large error as discussed in Sect. 4.1. As there is no justifiable method for such a correction and given the limited fraction of invalid pixels (10 % in the whole study area and 7 % in the Nu River basin), the figure can be used to demonstrate a full picture of the spatial precipitation pattern in the study area, but we note those pixels are of large uncertainties and should be interpreted with caution.

The performance between the IDW approach, the TRMM product and the fusion framework is compared in this section. IDW is one of the most popular methods for spatial interpolation of rainfall due to its easy implementation and flexibility in incorporating other auxiliary information (e.g., elevation). In general, the IDW approach is unable to demonstrate the high spatial variability, though it can capture the general spatial distribution of the whole basin (Fig. 10a), as TRMM (Fig. 10b). Due to the coarse spatial resolution, TRMM cannot capture the high variability in the river valley, where the elevation varies significantly. Although large rainfall (> 1800 mm) is observed in both our and TRMM products in the southwest of the study area region, our product gives lower rainfall compared to TRMM. As discussed above, the regression model tends to underestimate rainfall as the annual rainfall exceeds a certain threshold because the water supply is no longer a determinant of vegetation growth.

To demonstrate the advantage of the fusion framework, a cross-validation is conducted against the randomly sampled gauge observations by varying the number of samples (1–40). The cross-validation shows a higher ERMS for the IDW approach, followed by TMMM and RME (Fig. 11a). A higher mean EMR of 15 % is observed for TRMM than for IDW (8 %) and RME (5 %), while the differences in EMAR are minimal between TRMM and IDW. The results indicate an overestimated precipitation by TRMM as compared to gauge observations. Table 3 summarizes the maximum, minimum, and mean values of each method and shows the relative difference between RME and the other two methods. On average, the ERMS of RME is smaller than that of IDW and TRMM by 20.4 and 17.4 %, respectively. In general, the fusion framework demonstrates better performance than the other approaches.

To further evaluate the performance of RME, the annual averages of precipitation of five hydrological stations (Fig. 12a) and the whole basin estimated by the three approaches (IDW, RME, and TRMM) are compared. At the whole basin scale, the estimate by RME is 5.2 % higher than that of IDW but 7.9 % lower than TRMM. Although the difference between the three approaches is minimal at the basin scale, the difference at the sub-basin scale is remarkable. In the upstream region (i.e., the Gongshan sub-basin) located on the Tibetan Plateau, TRMM overestimates precipitation by 13.2 %, while IDW underestimates it by 7.6 % as compared with RME. In the other four downstream sub-basins, estimates by RME are larger than those by IDW and TRMM. In general, in the midstream and downstream regions with large variability in terrain height, RME gives larger estimates of precipitation than IDW and TRMM.

To validate the accuracy of different precipitation estimates, we utilize the monthly MODIS (MOD16) global ET (evapotranspiration) product with 1 km spatial resolution (Mu et al., 2011) (i.e., ET + R) and to compare it with five products, including RME, BandP (rainfall based on the precipitation–NDVI relationship with the consideration elevation band), DEMP, TRMM, and IDW (Fig. 12b). Although all five products underestimate the sub-basin-scale precipitation, RME and BandP give the closest estimates to the water-budget-based precipitation, indicating the effectiveness of the precipitation–NDVI relationship in precipitation remapping.

We also compared our products with the Multi-Source Weighted-Ensemble Precipitation (MSWEP) product. The dataset takes advantage of a wide range of data sources, including gauges, satellites, and atmospheric reanalysis models, to obtain the best possible precipitation estimates at the global scale with a high 3-hourly temporal and 0.25∘ spatial resolution (Beck et al., 2016). Comparison in the annual mean precipitation between the gauge measurements and predictions by the MSWEP and TRMM products (Fig. 13) shows acceptable performance of both MSWEP and TRMM in predicting the precipitation with an overall overestimation. The RMSE values for MSWEP, TRMM, and RME are 241, 196, and 174 mm, respectively, indicating that RME gives the best prediction among the three products. The possible reason why MSWEP shows no superiority over TRMM in predicting annual precipitation is that very few gauges are available in this region that might limit the applicability of the MSWEP methodology. However, the MSWEP methodology does provide insights into the production of high temporal resolution (3-hourly) rainfall, which we believe will be helpful to our future work.

Considering the possible degradation in model performance caused by oversaturation of NDVI in high biomass areas, another vegetation indicator, the enhanced vegetation index (EVI), is suggested as an alternative for estimating vegetation growth (Matsushita et al., 2007; Liao et al., 2015). As such, we also test the fusion framework with EVI in addition to NDVI and the results are assessed against the gauge observations.

Based on the chosen metrics, EVI is found to outperform NDVI with better regression quality (Table 4): the EVI-based regression model gives higher R2, and smaller ERMS and EMAR compared to the NDVI-based model. Also, a remarkable difference is observed in the precipitation estimates based on the two vegetation indices (Fig. 14). It is noted that the curvature of the EVI-based model is larger than the NDVI-based model, suggesting higher sensitivity of the EVI-based model in a humid environment. Although the EVI-based model demonstrates better performance than the NDVI-based one, it should be noted that NDVI is the most popular vegetation index used in operational applications among the available vegetation index products. Besides, NDVI has a relative longer temporal coverage compared to other vegetation index products. For instance, the AVHRR (Advanced Very High Resolution Radiometer) NDVI data have been available since 1982 with a global coverage. As such, under scenarios when EVI is unavailable, NDVI is a satisfactory index that can be used in the fusion framework.

One major assumption of the proposed framework is that precipitation is the only determinant of vegetation growth, and thus NDVI is regarded as a proxy for precipitation. However, other ambient factors, such as soil properties, solar radiation, air temperature, and elevation, may significantly influence the vegetation growth as well as NDVI values. Considering the data availability of various ambient factors, air temperature and elevation, in addition to NDVI, are adopted as extra determinants to establish the regression models, which are thus termed RME + T and RME + H for air temperature and elevation, respectively. We note that, for simplicity, the extra determinants are assumed to have a linear relationship with precipitation.

The differences in R2, ERMS, and EMAR between the three models are minimal, and the regression coefficients of the three models are very close to each other (Table 5). The negative regression coefficient of temperature in RME + T indicates inconsistent trends between precipitation and temperature. Since the temperature decreases with the increase in elevation, RME + T and RME + H essentially provide consistent estimates of precipitation which are also clearly shown in Fig. 15. It is also noted that the information added by extra determinants (i.e., air temperature and elevation) is in fact minimal. Overall there is little difference between RME and the other two products. As such, we consider the RME-only-based vegetation index to be a simple and efficient model for precipitation estimation.