# Multi-Sensor Precipitation Estimation from Space: Data Sources, Methods and Validation
## 3 Physical Basis of Multi-Sensor Precipitation Estimation
### Cloud-Motion-Based Method  
Given the accurate measurements of cloud-top motion provided by IR data, these observations can be effectively utilized for cloud system detection and tracking. The pioneering work by [PERSON] et al. (2016) introduced the Cloud Motion-based Precipitation Estimation (CMORPH) method to estimate global precipitation patterns. This approach assumes a linear evolution of precipitation between successive PWV images and considers constant precipitation intensity during both the forward and backward propagation of PMW estimates. By leveraging cloud movement, PMW-derived rainfall is advected or morphed across satellite overpasses, accounting for changes in intensity and shape through the inverse weighting of forward- and backward-propagated rainfall values. The computation process for obtaining morphed values is as follows:  
\[\text{value}_{(t+1/n\text{th})}=w1\times p_{forward(t+1/n\text{th})}+w2\times p_ {backward(t+1/n\text{th})} \tag{4}\]  
\[\text{value}_{(t+2*1/n\text{th})}=w2\times p_{forward(t+2*1/n\text{th})}+w1 \times p_{backward(t+2*1/n\text{th})} \tag{5}\]  
where \(w1\) and \(w2\) are weight coefficients, and \(1/n\) represents the temporal resolution of the retrieval. Instantaneous PMW rainfall (time = \(t\)) is propagated forward to produce analyses at \(t+1/n\text{th}\) (\(P_{forward(t+1/n\text{th})}\)) and \(t+2*1/n\text{th}\) (\(P_{forward(t+2*1/n\text{th})}\)) using the IR-derived cloud vector. Meanwhile, the instantaneous PMW rainfall (time = \(t+3*1/n\text{th}\)) is propagated backward to produce analyses at \(t+2*1/n\text{th}\) (\(P_{backward(t+2*1/n\text{th})}\)) and \(t+1/n\text{th}\) (\(P_{backward(t+2*1/n\text{th})}\)). It is noted that the 3 in the numerator inherently assumes a 3*1/n-hour time gap between PMW retrievals. In the CMORPH algorithm, \(w1\) and \(w2\) are 0.67 and 0.33, respectively, and 1/n is 0.5 h. The aryl CMORPH versions used the IR-derived cloud vector directly to propagate PMW precipitation. To correct advection rates, a speed- adjustment procedure was subsequently developed by first computing rainfall advection vectors using spatially lagging hourly U.S. NEXRAD stage-II radar rainfall data. The IR-derived cloud vectors were then synthesized to an hourly frequency to match that of the radar-derived rainfall vectors. A potential primary source of error in this algorithm lies in the fact that the advection analysis solely accounts for the temporal variation in the precipitation system. To address this, a Kalman filter (KF) was employed to enhance feedback information and more accurately represent the temporal variation in the precipitation system [110]. Consequently, a KF-model-based algorithm (GSMaP_MVK) was developed to refine the propagated precipitation rate based on cloud vectors. Specifically, a KF was applied to propagate rainy pixels, and the same propagation and KF procedure were utilized during backpropagation of these pixels. At time t, when the PMW sensor passes over, the rainy pixel undergoes repeated propagation, and the Kalman Filter (KF) updates the precipitation rate as \(R^{f}_{t,i}\) after \(i\) hours. The rainy pixels that are revisited undergo repeated backward propagation, while the precipitation rate is updated by the KF as \(R^{b}_{t,i}\), equal to the estimated precipitation rate j hours before the most recent PMW sensor overpass (t + hour time gap between PMW retrievals). In this case, the optimal estimate \(R\) is defined with the following equation:  
\[\hat{R}=R^{f}_{t,i}\frac{\sigma_{j}}{\sigma_{j}+\sigma_{i}}+R^{b}_{t,i}\frac{ \sigma_{i}}{\sigma_{j}+\sigma_{i}} \tag{6}\]  
where \(\sigma\) is the root mean square uncertainty in the estimates \(i\) or \(j\) hours after the PMW overpass. [PERSON] and [PERSON] [111] further enhanced the KF-based CMORPH algorithm, which incorporates both forward and backward propagation of PMW estimates from their observation times to the target analysis time (final retrieval time).