## 2 Materials and Methods
### Downscaling SMAP TB
#### 2.2.2 Calculating \(\beta\)  
With more than five years of coinciding CYGNSS and SMAP data, there are sufficient collocated time-series data to calculate \(\beta\) statistically. As described above, \(\beta\) is the spatially varying slope determined via linear regression between SMAP emissivity and CYGNSS reflectivity values. We chose to use emissivity, rather than TB, because TB is affected by surface temperature, which fluctuates seasonally. The \(\beta\) calculation is very important for the success and accuracy of the TB algorithm. \(\beta\) is essentially the scaling factor that adjusts \(TB_{F}\) to be higher or lower than \(TB_{C}\), so an accurate representation of spatial variability depends on the estimated \(\beta\) value. Since the dielectric constant of soil varies based on water content, which affects the microwave emissivity of the soil [23], SMAP emissivity decreases as NSSM increases and CYGNSS reflectivity increases as NSSM increases. Therefore, \(\beta\) values should theoretically be negative, and positive \(\beta\) values are unrealistic. Lower, or more negative, \(\beta\) values can produce larger differences between \(TB_{F}\) and \(TB_{C}\), increasing spatial heterogeneity. As we show below, calculating \(\beta\) is challenging in some situations. We experimented with a variety of options before determining an optimal \(\beta\) calculation method. Due to the -8-14 day repeat period for CYGNSS at a 3 km scale [12], the CYGNSS spatial coverage within a 33 km coarse-scale box on any given day is sparse. Therefore,  
Figure 3: All displayed data falls within the 33 km box centered on the 9 km grid cell at [31.2N, 98.7W]. In (**b–d**), the black borders are the 33 km box and the red squares are the 9 km grid cell. (**a**): A month-long timeseries of collocated SMAP and CYGNSS data, where each temporal collocation period is \(\pm\)half the time between successive SMAP observations. The black lines denote the time of each SMAP observation, and the alternating gray and white shaded regions show the temporal collocation periods. The blue shaded region shows the collocation period depicted in (**b–d**). The green squares are SMAP observations. The blue dots are CYGNSS observations. The red diamond is the \(\Gamma_{C}\) value shown in (**c**). (**b**): The \(\Gamma_{F}\) map shows all the CYGNSS reflectivity values collocated with the SMAP observation on 17 February 2020. (**c**): \(\Gamma_{C}\) is the median of all \(\Gamma_{F}\) and is used to calculate \(TB_{F}\) within the central 9 km grid cell. (**d**) The \(TB_{F}\) map shows the 3 km SMAP/CYGNSS TBs calculated using the displayed \(\Gamma_{C}\) and \(\Gamma_{F}\) values. daily CYGNSS observations within a 33 km box are not fully representative of its spatial heterogeneity. To account for this, we aggregated SMAP and CYGNSS data over 45-day periods. While longer periods provide slightly better correlation statistics, we chose this aggregation interval as a compromise to balance (1) the need for full CYGNSS coverage within 33 km boxes, (2) the need for ample data pairs for each linear regression, and (3) the ability to capture seasonal variations in soil moisture. To calculate \(\beta\), we located all CYGNSS reflectivity values within each 33 km box over 45-day periods and calculated the mean. Using the water mask described above, we excluded all CYGNSS observations within 3 \(\times\) 3 km grid cells that exceeded 5% semi-permanent open water from our 45-day CYGNSS reflectivity means. We also calculated the mean of all SMAP emissivity values for each 33 km box over 45-day periods. Whereby, we calculated a single pair of SMAP and CYGNSS values for each 33 km box for each 45-day period. Using a calibration period of 1 April 2017 through 31 December 2020, this creates up to 29 data pairs for each linear regression. SMAP emissivity and CYGNSS reflectivity data pairs and their corresponding \(\beta\) values are vastly different from location to location (Figure 4).