# The Use of SMAP-Reflectometry in Science Applications: Calibration and Capabilities
## 3 Calibration of SMAP-R Signals
### SMAP-R Calibration Method  
The calibration of SMAP-R data follows Equation (1), with some specific constraints:  
* \(T_{i}=25\) ms for SMAP-R data, instead of the typical value of 1000 ms used in CYGNSS and TDS-1 missions;
* \(P_{tx}\) and \(G_{tx}\). Extract CYGNSS GPS transmitted parameters for the same GPS satellite on the same day and for the same incidence angle as the SMAP-R measurement being processed. We also generated look-up-tables, that in case there are no coincident GPS satellite for CYGNSS and SMAP-R for that day, we can use typical values;
* \(G_{rx}(\overrightarrow{\rho})\) is approximated by the gain of SMAP antenna at the specular point \(G_{rx_{op}}\) and computed from Equation (3) for the corresponding \(\theta_{i}\);
* \(R_{rx}(\overrightarrow{\rho})\) and \(R_{tx}(\overrightarrow{\rho})\) are both approximated to a constant value computed from the specular point, \(R_{rx_{op}}\) and \(R_{tx_{op}}\) respectively. \[\int\ \Lambda^{2}(\tau-\tau_{s}(\overrightarrow{\rho}))\left|S(f_{d}( \overrightarrow{\rho})-f)\right|^{2}\overrightarrow{d\rho}\] is the ambiguity function that includes the surface spreading in delay and Doppler for all pixels \[\overrightarrow{\rho}\] within the scattering area. This can be approximated by an effective surface scattering area in the \[(\tau,\ f_{d})\] domain, \[\overline{A}(\tau,\ f_{d})\]. In the case of SMAP-R, the antenna pattern projection acts as a spatial filter \[F(\overrightarrow{\rho})\] in the surface domain that multiplies the surface spreading in both delay and Doppler for all pixels falling within the antenna footprint. The filtered effective surface scattering is referred as \[\overline{B}(\tau,\ f_{d})\]. Figure 12 provides actual examples of \[\overline{B}(\tau,\ f_{d})\]  calculations for two cases (a) \[40^{\circ}\] and (b) \[42.5^{\circ}\]. Figure 11: Top: Delay-Doppler Maps of the scattered power measured by SMAP-R for an ocean surface for (**a**) \(42.5^{\circ}\), (**b**) \(40^{\circ}\), and (**c**) \(37.5^{\circ}\) incidence angles. Bottom: iso-delay (blue ellipses) and iso-Doppler lines (blue hyperbolae) computed from actual SMAP-R and GPS geometries and velocities with the antenna beam pattern (red) over-imposed to those isolines for the same (**d**) \(42.5^{\circ}\), (**e**) \(40^{\circ}\), and (**f**) \(37.5^{\circ}\) incidence angles. Note that these are not to scale: The scattering area represents 200 km major axis ellipse, while antenna beam pattern diameter is represented by the \(-3\) dB beamwidth, i.e., 40 km. The antenna boresight is shown as a red dot. In Figure 12a, delay-Doppler bins showing a value of 1 correspond to a negligible effect of the SMAP antenna, therefore the scattering area size contributing to that delay-Doppler bin is the same as if the measurement was performed with an omnidirectional antenna. Decaying delay-Doppler values shown in Figure 12, both a and b, correspond to non-negligible effect of the SMAP antenna, therefore the scattering area size contributing to that delay-Doppler bin is lower than that of a measurement acquired with an omnidirectional antenna. To compute \(\overline{B}(\tau,\ f_{d})\), we followed these steps:  
1. Convert the transmitter and receiver position and velocity coordinates from Earth Centered Earth Fixed (ECEF) to the specular frame;
2. Define a fine grid;
3. Calculate the delay and Doppler values over that grid;
4. Compute the size of the surface patches at each delay-Doppler grid set by SMAP-R DDM bins; 1. Compute those surface patch sizes using the SMAP antenna footprint (40 km diameter) located at variable distances from the specular point (incidence angle of 40\({}^{\circ}\) is centered with the specular point, incidence angles smaller or larger are not centered) = \(B(\tau,\ f_{d})\); 2. Compute the surface patch sizes with no filter (i.e., filtering effect compensation) = \(A(\tau,\ f_{d})\);
5. Compute the compensating factor for the filtered scattering surfaces as \(\overline{B}(\tau,\ f_{d})=\frac{B(\tau,\ f_{d})}{A(\tau,\ f_{d})}\).