## 3 Construction of Coherent Integration Time Optimization Model
### The Reconstruction of Altimetric Precision Model  
According to [18], the estimated altimetric precision can be converted by the uncertainty of the incoherent average power waveform at SP. \[\sigma_{h}^{\text{m}}(\tau)=\frac{c}{2\times\cos(i)}\frac{\sigma_{Z}(\tau)}{ \overline{Z}^{\prime}(\tau)}=\frac{1}{2\times\cos(i)}\frac{1}{S_{h}(\tau)} \frac{\sigma_{Z}(\tau)}{\overline{Z}(\tau)} \tag{7}\]  
where \(i\) is the incident angle, \(c\) is the speed of light in vacuum, \(\overline{Z}(\tau)=\left\langle Z(t,\tau)\right\rangle\) is the average of the power waveform, and the derivative of which corresponds to \(\overline{Z}^{\prime}(\tau)\). \(S_{h}=\overline{Z}^{\prime}(\tau)/c\overline{Z}(\tau)\) indicates the logarithmic derivative of the power waveform at SP which is defined as the altimetric sensitivity. \(\sigma_{Z}(\tau)/\overline{Z}(\tau)\) is the ratio of the standard deviation to power amplitude which is defined as the effective independent incoherent average number [21]. Since \(\sigma_{Z}(\tau)\) depends on the correlation between the waveforms, the ratio is derived analytically in this section. As introduced in Section 2, the complex waveform after coherent integration of direct and reflected signals can be represented by a discrete array \(y(nT_{c},\tau)\):  
\[y(nT_{c},\tau)=[y(nT_{c},\tau_{1}),y(nT_{c},\tau_{2})\cdot\ldots\ldots y(nT_{ c},\tau_{n})] \tag{8}\]  
where \(nT_{c}\) is the nth coherent integration process, \(T_{c}\) is the coherent integration time which is usually in milliseconds, and \(\tau_{n}\) is the code delay whose resolution is inversely proportional to the signal sampling rate. The square of each complex waveform results in a one-shot power waveform expressed as [21]:  
\[z(nT_{c},\tau)=y(nT_{c},\tau)y^{*}(nT_{c},\tau) \tag{9}\]  
In the case of an incoherent average of \(N_{\rm I}\) one-shot power waveforms, the power waveform can be expressed as:  
\[Z(\tau)=\frac{1}{N_{\rm I}}\sum_{n=0}^{N_{\rm I}-1}z(nT_{c},\tau)=\langle z( nT_{c},\tau)\rangle \tag{10}\]  
where \(\langle\rangle\) stands for the ensemble average. A power waveform is usually processed in seconds, which can be expressed as:  
\[T_{\rm I}=T_{c}\times N_{\rm I} \tag{11}\]  
Due to the relative motion of the transmitter and the receiver, the correlation between two complex waveforms separated by \(\vec{n}T_{c}\) in time is determined by the coherence of the signals from two ocean contribution areas. These regions are elliptical, and the length of the semi-axis is related to \(\tau_{n}\). The correlation between complex waveforms is defined as [19]:  
\[C_{y}(\vec{n}T_{c},\tau)=\langle y(nT_{c},\tau)y^{*}(nT_{c}+\vec{n}T_{c},\tau)\rangle \tag{12}\]  
where \(\vec{n}T_{c}\) is the time interval between complex waveforms. The spatial geometry of correlation function is illustrated in Figure 3. It can be seen that the correlation between the two sea surface scattering signals gradually decreases as \(\vec{n}T_{c}\) increases. An analogous correlation between one-shot power waveforms can also be derived as:  
\[C_{z}(\vec{n}T_{c},\tau)=\langle[z(nT_{c},\tau)-\langle z(nT_{c},\tau)\rangle ][z(nT_{c}+\vec{n}T_{c},\tau)-\langle z(nT_{c}+\vec{n}T_{c},\tau)\rangle]\rangle \tag{13}\]  
The correlation time of the signal is usually measured in milliseconds, while the time of incoherent averaging can reach the order of seconds. Thus, it can be assumed that there  
Figure 3: Sea surface contribution area corresponding to \(\langle y(nT_{c},\tau)y^{*}(nT_{c}+\vec{n}T_{c},\tau)\rangle\). is no correlation between the incoherent average power waveforms, i.e., \(C_{Z}(\widetilde{t},\tau)=0\widetilde{t}\
eq 0\). The variance of the power waveform can be expressed as [21]:  
\[\sigma_{Z}^{2}(\tau)=C_{Z}(0,\tau)=\left\langle\left[Z(t,\tau)-\left\langle Z(t,\tau)\right\rangle\right]^{2}\right\rangle \tag{14}\]  
The effective incoherent average number can be reconstructed following Appendix A as [21]:  
\[y(nT_{c},\tau)=[y(nT_{c},\tau_{1}),y(nT_{c},\tau_{2})\cdot\ldots\ldots y(nT_{ c},\tau_{n})] \tag{8}\]0