# Impact of Satellite Attitude on Altimetry Calibration with Microwave Transponders
## 3 Satellite Attitude and Transponder Calibration
### Range and Dutation Bias  
The range bias of a satellite altimeter represents the systematic error in measuring distances. This bias is estimated by comparing the altimeter (or measured) range with the corresponding geometric distance (considered the \"true\" value) between the satellite CoG and the reflecting target (in this work the microwave transponder). The initial estimation of the measured range is typically performed by the altimeter tracker or by a retracking algorithm and is based on the return waveform. The shape of the return waveform is described by the Brown model [41] over ocean regions and by a point target response (PTR) over transponders. A profile map of the PFAC transponder sites and the Jason-3 return waveforms transitioning between a Brown model and a PTR is presented in Figure 3.  
In the context of calibration with a transponder, a number of corrections need to be applied to the measured and geometric ranges to correctly estimate the altimeter range bias [29]. The corrections to the initial measured range (i.e., tracker range) are applied partially from the processing Agencies and the Cal/Val teams. Conventional corrections applied by the processing Agencies account for the offset distance between the APC and satellite CoG, the ultra-stable oscillator (USO) drift, internal path delay and instrumental and system errors. The corrections applied by the Cal/Val teams are due to the effects of the ionosphere and troposphere (atmospheric delays), the transponder internal path delay and the Doppler effect as a consequence of the relative velocity of the altimeter satellite with respect to the transponder. Additional corrections are applied to the geometric range to account for displacements of the transponder position caused by tidal and non-tidal effects of atmosphere, solid Earth, pole and ocean loading. The offset between APC and CoG, also known as \"CoG correction\", is associated with satellite attitude. Given the objectives of this study, we isolate this parameter from the rest of the corrections and examine how its current implementation influences bias results. Based on the information given in the beginning of this section, the range bias \(B(t)\) is defined as follows:\[B(t)=r(t)-r_{0}(t), \tag{8}\]  
where \(r(t)\) and \(r_{0}(t)\) denote the corrected measured range and corrected geometric range, respectively, both of which refer to the satellite CoG. Since both ranges change as the satellite flies over the transponder, they are functions of time \(t\). The range \(r(t)\) is written in terms of the corrected measured range \(R(t)\) referred to the APC, as follows:  
\[r(t)=R(t)+\delta r, \tag{9}\]  
with \(\delta r\) being the CoG correction. The corrected geometric range \(r_{0}(t)\) is calculated by:  
\[r_{0}(t)=\sqrt{\left(X^{\text{TRP}}-X^{\text{CoG}}(t)\right)^{2}+\left(Y^{ \text{TRP}}-Y^{\text{CoG}}(t)\right)^{2}+\left(Z^{\text{TRP}}-Z^{\text{CoG}}(t )\right)^{2}}, \tag{10}\]  
where \((X,Y,Z)\) are the coordinates of the transponder (superscript TRP) and satellite CoG (superscript CoG) in the Earth-fixed reference system. In Equation (10), it is assumed that the transponder coordinates are constant for a specific cycle. A second type of bias estimated during the calibration of an altimetric system is the datation (or time-tagging) bias. This bias provides information regarding the error made on the reference time of the altimetric measurements [42]. The definition of datation bias is based on the time of closest approach (TCA), which represents the time at which the distance between transponder and satellite is minimized. Two types of TCA are associated with the calculation of datation bias. The first one is the time at which the corrected measured range is minimized, and the second one is the time at which the corrected geometric range is minimized. Since the histories of both ranges form a parabolic curve over a transponder (Figure 3), the two TCAs denote the vertex position of the corresponding parabolas.