10.2.1 GEOMETRIC PARAMETERS

Since a specific site of the Moon-based SAR is yet to be determined, to proceed with analyzing the coverage on the Earth’s surface, the position of the Moon-based SAR is assumed to be the same as that of the Moon because of the extremely long distance from the Earth to the Moon. The relevant parameters for investigating the coverage are listed in Table 10.1.

The geocentric angle, defined as the angle between the nadir point of the Moon-based SAR and the ground target referring to the Earth’s center, is

${\theta }_e={\cos }^{-1}(\cos {\delta }_m\cos {\delta }_g\cos \alpha +\sin {\delta }_m\sin {\delta }_g)$(10.22)

$\alpha =a_g-a_m.$(10.23)

${\theta }_{el}={\cos }^{-1}\{[R_{SAR}-R_E(\cos {\delta }_m\cos {\delta }_g\cos \alpha +\sin {\delta }_m\sin {\delta }_g)]/R_c\}.$(10.24)

$\psi ={\sin }^{-1}\{[R_{SAR}(\cos {\delta }_m\cos {\delta }_{\mathrm{g}}\cos \alpha +\sin {\delta }_m\sin {\delta }_g)-R_E]/R_c\}.$(10.25)

$\varphi ={\cos }^{-1}\{\cos {\delta }_{m}sin\alpha {[1-{(\cos {\delta }_g\cos {\delta }_m\cos \alpha +\sin {\delta }_g\sin {\delta }_m)}^2]}^{-0.5}\}.$(10.26)

$R_c=\sqrt{R_{SAR}^2+R_E^2-2R_{SAR}{R}_E(\cos {\delta }_m\cos {\delta }_g\cos \alpha +\sin {\delta }_m\sin {\delta }_g)}.$(10.27)

Detailed derivations of Equations 10.22–10.25 are given in [19].

The geocentric and azimuthal angles, together with the grazing angle or elevation angle, determine the spatial coverage of a Moon-based SAR. Here, we choose the geocentric, azimuthal, and grazing angles to examine the Moon-based SAR’s spatial coverage.

From Figure 10.3, the grazing angle should be bounded by

${\psi }_{far}\le \psi \le {\psi }_{near}.$(10.28)

where ${\psi }_{far}$ and ${\psi }_{near}$ are the lower and upper bounds of the grazing angle, respectively.

Substituting Equation 10.25 into Equation 10.22 and imposing Equation 10.28, Equation 10.22 becomes

$\cos {\theta }_e({\delta }_m,{\delta }_g,\alpha )=\sin \psi \cdot R_c/R_{SAR}+R_E/R_{SAR}.$(10.29)

As far as the Moon-based SAR’s spatial coverage is concerned, the slant range in Equation 10.27 can be further approximated to

$R_{ca}\approx R_{SAR}\{1+[0.5R_E^2-R_{SAR}{R}_E\cos {\theta }_e({\delta }_m,{\delta }_g,\alpha )]/R_{SAR}^2\}.$(10.30)

$\delta R=(R_c-R_{ca})/R_{SAR}.$(10.31)

Figure 10.4 presents the relative errors at perigee and apogee of the lunar orbit in the $\alpha -{\delta }_g$ domain. For the sake of simplicity but without loss of generality, the latitude of the Moon-based SAR’s nadir point is set to 0°, and the grazing angle is ranged from 0° to 90°. As detailed in Figure 10.4, the maximum value of the relative error $\delta R$ is far smaller than 10^‒3, which appears at the nadir point regardless of the distance from the Earth to the Moon-based SAR. This suggests that the approximation of the slant range in Equation 10.30 has little bearing on the dimensionless quantity, $\delta R$. In other words, the error induced by the approximation of the slant range can be reasonably ignored in determining the spatial coverage of a Moon-based SAR. However, we must compensate for this error given the rise in the image focusing. Accordingly, Equation 10.29 can be approximated by taking account of Equation 10.30 to

$\cos {\theta }_e({\delta }_m,{\delta }_g,\alpha )\approx \kappa (\psi ,R_{SAR}),$(10.32)

$\kappa (\psi ,R_{SAR})=\frac{\sin \psi +0.5R_E/R_{SAR}}{\sin \psi \times R_E/R_{SAR}+1}+0.5R_E/R_{SAR}.$(10.33)

Note that the ratio of Earth’s radius to Earth–Moon-based SAR distance is spatiotemporally varying as follows

$R_E/R_{SAR}\approx R_E^A/R_{SAR}^A\times (1-\Delta R_{SAR}^T/R_{SAR}^A+\Delta R_E^S/R_E^A).$(10.34)

where $\Delta R_E^S$ is the spatially varying component of the Earth’s radius at an arbitrary position on the Earth’s surface for the average Earth’s radius, $R_E^A$; $\Delta R_{SAR}^T$ is the temporally varying part of the Earth–Moon-based SAR distance at any time referring to the average distance from the Earth to the Moon-based SAR, $R_{SAR}^A$. It can be calculated that $\Delta R_{SAR}^T/R_{SAR}^A$ is approximately 30 times larger than $\Delta R_E^S/R_E^A$ during one period of the lunar revolution. Thus, the spatially varying component of the Earth’s radius contributes little to Equation 10.34 (i.e., the ratio of Earth’s radius to Earth–Moon-based SAR distance). Hence, it is reasonably to accept a constant Earth’s radius (6371.0 km) here.

10.2.2 EFFECTIVE RANGE

Generally, a single antenna can observe a limited portion of the Earth’s surface in a given time frame. However, multi-antennas or antenna array can achieve the maximum extent of the spatial coverage by adjusting the pointing direction of each antenna. For example, the half-power beam width of a single antenna is [37]

$\beta \approx k_{xy}\lambda /\ell ,$(10.35)

${\psi }_{far}\le {\sin }^{-1}\left\{\frac{\sin \left|{\delta }_m\right|-R_E/R_{SAR}}{{(1+R_E^2/R_{SAR}^2-2\sin \left|{\delta }_m\right|\cdot R_E/R_{SAR})}^{0.5}}\right\}.$(10.36)

${\psi }_{near}\ge {\sin }^{-1}\left\{\frac{R_E/R_{SAR}-\cos {\delta }_m}{\cos {\delta }_m\cdot R_E/R_{SAR}-0.5\cdot R_E^2/R_{SAR}^2-1}\right\}.$(10.37)

The inclination of the lunar orbit to the Earth’s equatorial plane stands at between 18.3° and 28.6° [20]; in other words, the maximum scale of the nadir point’s latitude varies from 18.3° to 28.6°. In Figure 10.5(a), we plot the bounds of the far grazing angle at apogee and perigee of the lunar orbit versus nadir point’s latitude. Then, bounds of the near grazing angle at apogee and perigee of the lunar orbit for the magnitude of the nadir point’s latitude are plotted in Figure 10.5(b). The range of latitude for the nadir point of the Moon-based SAR is 18.3° to 28.6°.

Even though the spatial coverage is bounded by the near and far grazing angles, the image quality of the Moon-based SAR is still an issue. The Moon-based SAR system usually works in the squint mode when observing the bounded regions, which would lead to pixel skewing on the ground and raise a significant challenge to the image quality. Under the skewing effect, satisfactory imaging is only available when the iso-range and iso-Doppler profiles are relatively far from being parallel. Generally, the quasi-parallel regions of the iso-range and iso-Doppler contours occur at where the azimuthal angles are close to 0° or 180°. Therefore, it is practical to examine the spatial coverage of the Moon-based SAR by limiting the azimuthal angle, namely

$\left|\cos \varphi \right|\le \cos {\varphi }_b,$(10.38)

In Figures 10.6(a–d) we plot the iso-range and iso-Doppler profiles in the $\alpha -{\delta }_g$ domain within the boundary set by Equation 10.38 with the Moon-based SAR locating at different positions listed in Table 10.2. The beam center crossing time is set to 00:00:00 without loss of generality.

From Figure 10.6, the iso-Doppler and iso-range contours cut across each other wherever the Moon-based SAR is located. This suggests that the image quality is acceptable even when the Moon-based SAR system operates in a squint mode. Thus, the skewing effect exerts a little impact on the spatial coverage of the Moon-based SAR within the scope from Equation 10.38.

Finally, we can obtain the bounds of the geocentric angle that determines the Moon-based SAR’s spatial coverage as

${\theta }_e^{low}\le {\theta }_e({\delta }_m,{\delta }_g,\alpha )\le {\theta }_e^{up},$(10.39)

${\theta }_e^{up}={\cos }^{-1}[\kappa ({\psi }_{far},R_{SAR})],$(10.40)

${\theta }_e^{low}=\{\begin{array}{l}{\cos }^{-1}[\kappa ({\psi }_{near},R_{SAR})],\left|\alpha \right|\le {\alpha }_{th}\\ {\cos }^{-1}[\cos {\varphi }_b{({\cos }^2{\varphi }_b-{\cos }^2{\delta }_{m}si{n}^2\alpha )}^{0.5}],\left|\alpha \right|\ge {\alpha }_{th}\end{array}$(10.41)

${\alpha }_{th}={\sin }^{-1}\{\cos {\varphi }_b\cdot {\cos }^{-1}{\delta }_m\cdot {[1-{\kappa }^2({\psi }_{near},R_{SAR})]}^{0.5}\}$(10.42)

10.2.3 MOON-BASED SAR’s SPATIAL COVERAGE

Now, the spatial coverage of the Moon-based SAR can be examined in terms of the coverage area and ground coverage. To begin with, the coverage area of the Moon-based SAR within the bound of Equation 10.39 is detailed below.

Once the spatial coverage is limited to the range in Equation 10.39, the Moon-based SAR’s coverage area on the Earth’s surface can be approximated by [19]

$S_c=4R_E^2\{{\cos }^{-1}[\Re ({\varphi }_b)\cdot \kappa ({\psi }_{far},R_{SAR})\cdot \zeta ({\psi }_{far},R_{SAR})]-{\cos }^{-1}[\Re ({\varphi }_b)\cdot \kappa ({\psi }_{far},R_{SAR})\cdot \zeta ({\psi }_{near},R_{SAR})]+{\sin }^{-1}[\zeta ({\psi }_{far},R_{SAR})]-{\sin }^{-1}[\zeta ({\psi }_{near},R_{SAR})]\},$(10.43)

$\{\begin{array}{l}\Re ({\varphi }_b)=\cos {\varphi }_b-\sin {\varphi }_b,\\ \zeta ({\psi }_{far},R_{SAR})={[1+{\kappa }^2({\psi }_{far},R_{SAR})]}^{-0.5},\\ \zeta ({\psi }_{near},R_{SAR})={[1+{\kappa }^2({\psi }_{near},R_{SAR})]}^{-0.5}.\end{array}$(10.44)

$r_c=4R_E^2\{{\cos }^{-1}[\Re ({\varphi }_b)\cdot \kappa ({\psi }_{far},R_{SAR})\cdot \zeta ({\psi }_{far},R_{SAR})]-{\cos }^{-1}[\Re ({\varphi }_b)\cdot \kappa ({\psi }_{far},R_{SAR})\cdot \zeta ({\psi }_{near},R_{SAR})]+{\sin }^{-1}[\zeta ({\psi }_{far},R_{SAR})]-{\sin }^{-1}[\zeta ({\psi }_{near},R_{SAR})]\}/\pi \times 100\%.$(10.45)

Detailed derivations of Equations 10.43 and 10.45 can be found in [19].

It follows that the coverage area on the Earth’s surface is related to the ratio of the Earth’s radius to the Earth–Moon-based SAR distance, the near and far grazing angles, and bound of the azimuthal angle. As the bounds of the grazing and azimuthal angles are ascertained in the preceding analysis, the coverage area of the Moon-based SAR depends only on the ratio of the Earth’s radius to the distance from the Earth to the Moon-based SAR.

Given the imaging geometry of the Moon-based SAR, the maximum ground coverage’s latitude occurs at 0° of the longitudinal deviation. Then, the maximum and minimum latitudes of the ground coverage can be obtained from Equations 10.39 and 10.40, as given by

${\delta }_g=\pm {\cos }^{-1}[\kappa ({\psi }_{far},R_{EM})]+{\delta }_m.$(10.46)

${\delta }_{mth}=90°-{\cos }^{-1}[\kappa ({\psi }_{far},R_{EM})].$(10.47)

In Figure 10.7, we plot the ground coverage of the Moon-based SAR at various latitudes of the nadir point (0°, −10°, 18°, and −28°) in the $\alpha -{\delta }_g$ domain. The Earth–Moon-based SAR distance is set to 385,000 km, the average distance from the Earth to the Moon-based SAR; it can be calculated that the threshold ${\delta }_{mth}$ for this distance is to 15.916°. As illustrated in Figure 10.7, there exist blind regions within the ground coverage bounded by the far grazing angle due to the effects of the near grazing angle and bound of the azimuthal angle. Besides, the longitudinal ground coverage is rotationally symmetric about the axis of $\alpha =0°$, whereas the latitudinal ground coverage is asymmetric except the nadir point being at the Earth’s equator. Moreover, when the nadir point’s latitude is smaller than the threshold of ${\delta }_{mth}$, neither of the Earth’s Poles is covered by the Moon-based SAR. Meanwhile, the maximum size of the longitudinal deviation is always smaller than 90°. Regarding the Moon-based SAR’s nadir point whose latitude is larger than ${\delta }_{mth}$, such as in Figure 10.7(c), the North Pole is covered by the Moon-based SAR. However, the high latitudes in the southern hemisphere of the Earth cannot be observed. On the contrary, if the nadir point’s latitude is smaller than $-{\delta }_{mth}$, the South Pole can be covered by the Moon-based SAR, but the high latitudes in the northern hemisphere are not. Under both conditions, the maximum longitudinal deviation can reach as large as 180°.

10.2.4 TEMPORAL VARIATIONS IN THE SPATIAL COVERAGE

The spatial coverage of the Moon-based SAR is determined by the Earth–Moon-based SAR distance, bounds of grazing and azimuthal angles, and position of the nadir point at a specified time. The nadir point, which temporally varies, effectively induces the temporal variation in the spatial coverage. In what follows, the geographic coordinates of the nadir point, in light of the temporal variations are discussed.

The nadir point’s latitude and longitude are provided by Equations 10.20 and 10.21, both of which are a function of time and can be numerically obtained when the lunar ephemeris is applied. Now, the temporal variations in the latitude and longitude of the nadir point during one cycle of the lunar revolution are plotted in Figures 10.8(a) and (b), respectively.

We see that the latitude of the nadir point is mainly determined by the lunar revolution. By comparison, the nadir point’s longitude is primarily decided by the Earth’s self-rotation. Furthermore, the nadir point’s longitude differs from that at the initial time after one period of the lunar revolution. This phenomenon is attributed to the Earth’s self-rotation and lunar revolution around the Earth. The temporal variation of the nadir point’s geographical coordinate is a continuous process. However, the temporal variation of the spatial coverage is not exactly equivalent to that of the nadir point. A time interval related to synthetic aperture, namely the synthetic aperture time, still needs to be considered in the imaging formation of the coverage region. A synthetic aperture time of 200 s can realize an azimuth resolution around 10 m, which is suitable for the Earth observation from the Moon-based SAR. Nevertheless, such a synthetic aperture time requires a large antenna with an aperture length in azimuth larger than 1000 m, which poses a significant challenge to build in practice. The sub-aperture imaging technique provides a feasible method to implement the desired synthetic aperture time without utilizing the large antenna [21]. Thus, the synthetic aperture time of the Moon-based SAR may be set to 200 s.

Figure 10.9 displays the track of the Moon-based SAR’s nadir point within one week, from which we observe the daily change of the nadir point’s latitude is relatively small (around 3°). The temporal variation of the longitude of the nadir point within one day can exceed 340°. Hence, the temporally varying nadir point, together with the extensive spatial coverage, leads the revisit time to be less than one day for most of the regions covered.

10.2.5 NUMERICAL ILLUSTRATION OF SPATIOTEMPORAL COVERAGE

Based on the lunar ephemeris from the DE430, numerical simulations are performed to visualize the spatiotemporal coverage of the Moon-based SAR. We begin by discussing the hourly variations in the spatial coverage of the Moon-based SAR within one day.

10.2.5.1 Hourly Variations

Assume that the initial visit time is 00:00:00 on March 20, 2024, when the nadir point’s latitude is relatively large. The hourly variations in the Moon-based SAR’s spatial coverage (eight-hour time intervals) within one day are plotted in Figure 10.10, where we see that most of the northern hemisphere of the Earth is measured by the Moon-based SAR, while the southern hemisphere of the Earth is less observed. Besides, it is understood that the spatial coverage moves from east to west and the revisit time of the covered region is commonly less than one day, implying that Earth’s self-rotation is a dominant factor in determining the temporal variation of the spatial coverage. Comparing Figure 10.10(d) with (a), we see that after a one-day lapse, the spatial coverage has moved to the east. This phenomenon is caused by the lunar revolution around the Earth. Moreover, it can be observed a large proportion of the Earth surfaces are viewed (e.g., South America is almost entirely covered by the Moon-based SAR at 00:00:00). Nevertheless, there are still blind regions within the coverage bounded by the far grazing angle. This suggests that the effects of the near grazing angle and bound of the azimuthal angle on the spatial coverage are significant. Hence, the spatial coverage is jointly determined by the near and far grazing angles and bound of the azimuthal angle.

For comparison, the hourly variations in the spatial coverage of the Moon-based SAR with a small size of the nadir point’s latitude are presented in Figure 10.11. The visit time is reset to 00:00:00 on March 25, 2024, when the latitude of the nadir point approaches to 0°. The spatial coverage of the Moon-based SAR within one day is still significant when the size of the nadir point’s latitude is small, but neither the North Pole nor the South Pole of the Earth can be observed. Also, the revisit time of the observed region is less than one day as a result of the extensive spatial coverage and Earth’s self-rotation. Moreover, the effects of the lunar revolution result in the eastward motion of the spatial coverage after a one-day lapse, as well. Furthermore, a blind region of considerable extent appears on the region bounded by the far grazing angle, implying the effects of the near grazing angle and the bound of  the azimuthal angle are indeed not negligible even when the size of the nadir point’s latitude is small.

10.2.5.2 Global Accumulated Visible Time within Different Periods

To exploit the spatiotemporal variation in the Moon-based SAR’s ground coverage for a specified period, we analyze the accumulated visible time that is the accumulated time for a specific area covered by the Moon-based SAR within a specified period. The global daily-accumulated visible time of the Moon-based SAR on March 20, March 25, April 01, and April 07 of 2024 are shown in Figures 10.12(a–d), respectively. As is seen, the majority of the world can be covered by the Moon-based SAR within one day, and those regions are observed continuously, ranging from several hours to tens of hours. Besides, there is a conspicuous stripe that lies along the track of the nadir point, indicating that the daily-accumulated visible time is affected by the near grazing angle. Also, the longitudinal accumulated visible time approximates to uniformly distribute. By contrast, the distribution of the latitudinal accumulated visible time depends on the nadir point’s latitude. When the nadir point’s latitude is in the northern hemisphere of the Earth, the maximum latitudinal accumulated visible time appears at the region near the North Pole whereas the fully Antarctica cannot be observed. The trend reverses when the nadir point lies in the southern hemisphere of the Earth. Once the nadir point’s latitude approaches 0°, the latitudinal accumulated visible time trends to be symmetrically distributed. In such a situation, neither of the Earth’s poles could be observed by the Moon-based SAR within one day.

In general, there is not distinct regularity in the lunar revolution for one Julian year (365.25 days). Thus, the accumulated visible time over one Julian year cannot certainly reflect the tendency of the spatiotemporal coverage of the Moon-based SAR. As a result, the accumulated visible time of the Moon-based SAR within one Julian year is not presented. However, the inclination angle of the lunar orbit with respect to the Earth’s equator periodically varies with a period of 18.6 years [18,22]. Hence, the range of the latitude of the nadir point reaches a maximum or minimum every 18.6 years. Hence, it is of interest to illustrate the accumulated visible time of the Moon-based SAR over 18.6 years. By setting the initial visit time to 00:00:00 on March 20, 2024, the global accumulated visible time of the Moon-based SAR over 18.6 years is presented in Figure 10.13. The global accumulated visible time of the Moon-based SAR over 18.6 years is on the order of several years. Also, the longitudinal accumulated visible time trends uniformly distributed. In comparison, the latitudinal accumulated visible time is symmetrically distributed about the Earth’s equator. What is more, the peak values of the latitudinal accumulated visible time appear at ±40° of latitudes, whereas the valleys occur at high latitudes with a size around 80°. Interestingly, there are apparent stripes above and below the Earth’s equator in global accumulated visible time within an 18.6-year period, which is given rise by the effect of the near grazing angle. This implies the near grazing angle exerts a considerable impact on the spatiotemporal coverage of the Moon-based SAR with a period of 18.6 years.

Typically, the accumulated visible time over a period of one nodical month or one Julian year is different from that over the next period due to the temporal variation in the lunar orbital inclination. Thus, the accumulated visible time within one nodical month or one Julian year is not suitable for setting a Moon-based SAR. In contrast, the lunar orbital inclination is periodically varying every 18.6 years, thus the accumulated visible time of the Moon-based SAR within a period of 18.6 years is approximately the same as that in the next period of 18.6 years.

For the effective range-limited grazing and azimuthal angles, the far grazing angle plays a dominant role in determining the spatiotemporal coverage of the Moon-based SAR. The effects of the near grazing angle and bound of azimuthal angle are also profoundly significant to affect the spatiotemporal coverage. Regarding the optimal site selection, the site near the lunar North Pole is preferred for observing the Arctic of the Earth, while the site near the lunar South Pole may be a good choice for observing Antarctica. For other regions on the Earth, the site of the Moon-based SAR has no bearing on the spatiotemporal coverage of the Moon-based SAR over a long period. Consequently, the Moon-based SAR can perform long-term, continuous Earth observations on a global scale to enhance our capabilities of understanding the planet.

10.3.1 PHASE ERROR DUE TO TEMPORAL-SPATIAL VARYING BACKGROUND IONOSPHERE

The synthetic aperture of the Moon-based SAR is realized mainly by the Earth’s rotation. The long slant range history and relative slow Earth’s rotating velocity bring about an extremely long synthetic aperture time, further resulting in the temporal-variation characteristics of the background ionosphere. For a rigorous study, the relationship between the synthetic aperture time and the azimuthal resolution of the Moon-based SAR is now examined.

The synthetic aperture time, the aperture length in azimuth, can be accurately obtained by

$T_{sar}={\eta }_{end}-{\eta }_{star}$(10.48)

${\cos }^{-1}\left(\frac{{\mathbf{R}}_{\mathbf{s}\mathbf{t}\mathbf{a}\mathbf{r}}\cdot {\mathbf{R}}_{\mathbf{c}}}{\left|{\mathbf{R}}_{\mathbf{s}\mathbf{t}\mathbf{a}\mathbf{r}}\right|\cdot \left|{\mathbf{R}}_{\mathbf{c}}\right|}\right)=\frac{{\theta }_b}{2},$(10.49)

${\cos }^{-1}\left(\frac{{\mathbf{R}}_{\mathbf{e}\mathbf{n}\mathbf{d}}\cdot {\mathbf{R}}_{\mathbf{c}}}{\left|{\mathbf{R}}_{\mathbf{e}\mathbf{n}\mathbf{d}}\right|\cdot \left|{\mathbf{R}}_{\mathbf{c}}\right|}\right)=-\frac{{\theta }_b}{2}.$(10.50)

${\theta }_b=\lambda /{\mathit{\ell }}_a$(10.51)

${\mathit{\rho }}_a=\frac{{\mathit{\ell }}_a}{2}\frac{R_E}{R_{EM}}\frac{\cos {\delta }_g}{\cos {\delta }_m}\cos \alpha$(10.52)

where $R_E$ is the Earth’s radius, and $R_{EM}$ is the distance between the Earth and Moon-based SAR (see Figure 10.1); ${\delta }_g$ is the latitude of the Earth’s target, ${\delta }_m$ is the declination of the Moon-based SAR.$\alpha$ is the angular difference between the Moon-based SAR’s ascension and the longitude of the Earth’s target at beam center crossing time. As can be seen, three extra terms: $R_E/R_{EM}$, $\cos {\delta }_g/\cos {\delta }_m$ and $\cos \alpha$ are presented to modify the ideal azimuth resolution [12].

By combining Equations 10.48–10.50, the relationship between the synthetic aperture time and azimuthal resolution is plotted in Figure 10.14, with, as an example, the declination of the Moon-based SAR set to be 28.5° and latitude of the Earth’s target assumed to be 0°.

As can be seen from Figure 10.14, the synthetic aperture time of the Moon-based SAR can be up to hundreds or even thousands of seconds. In contrast, the synthetic aperture time of the LEOSAR is usually limited to 1–2 s. As a result, it seems that the ionospheric freezing assumption for LEOSAR loses its effect in the Moon-based SAR. To catch a better notion, in Figure 10.15, we plot the measured vertical TEC at Guangzhou, China (113.23°E, 23.16°N), with a synthetic aperture time of 1800 s, a typical value of the Moon-based SAR. The TEC data used in the schematic diagram for the temporal variation of background ionosphere is acquired in October 2016 and reported by International Reference Ionosphere 2012 (IRI 2012) [29].

It is clear that the variation of the vertical TEC over a period during the synthetic aperture time exceeds 3 TECU, which seems quite appreciable, and the TEC temporal variation is nonlinear. Consequently, the temporal-varying background ionosphere should be taken into consideration, thus the temporal-varying ionospheric TEC is supposed to express in the form of slow time:

$TE{C}_t(\eta )=TE{C}_{t0}+k_1\eta +k_2{\eta }^2+k_3{\eta }^3+k_4{\eta }^4+\mathcal{O}({\eta }^5)$(10.53)

Because the Moon-based SAR is observing the Earth on a global scale, the spatial-varying ionospheric TEC at different positions within the imaging swath should be taken account, as is evident from the strongly varying TEC with relative positions of the ground target illustrated in Figure 10.16.

Assume that the spatial variation of the TEC along the azimuth direction is time-invariant, the spatial-varying TEC can be written as:

$TE{C}_s(x)=TE{C}_{s0}+s_{1}x+s_2{x}^2+s_3{x}^3+s_4{x}^4+\mathit{O}(x^5)$(10.54)

$x=V_{iono}\eta$(10.55)

$V_{iono}=\left(R_E+h_{iono}\right)\cos {\delta }_g(\chi {\omega }_E)$(10.56)

$TEC(\eta )=TE{C}_0+t_{g1}\eta +t_{g2}{\eta }^2+t_{g3}{\eta }^3+t_{g4}{\eta }^4+\mathit{O}({\eta }^5)$(10.57)

When the SAR signal propagates through the ionosphere, the phase-path length will be changed under the impact of the background ionosphere [31]. The phase error induced by the change of the phase-path length in the round-trip can be defined as

$\Delta {\phi }_{iono}=\frac{-4\pi A}{c(f_{\tau }+f_c)}TEC$(10.58)

It turns out that the phase error induced by the temporal-spatial varying background ionosphere should be modified by substituting Equation 10.57 into Equation 10.58:

$\Delta {\phi }_{iono}(\eta )=\frac{-4\pi A}{c(f_{\tau }+f_c)}TEC(\eta )$(10.59)

It shows in Equation 10.59 that the spatial-varying ionospheric TEC, together with the temporal-varying ionospheric TEC, further makes the background ionospheric effects even more complicated. It may be drawn that the spatial-variation of background ionosphere within an image scene is approximately time-invariant, implying that the ionospheric TEC in Equation 10.57 potentially can be used to probe the temporal-spatial background ionospheric effects [32].

10.3.2 SLANT RANGE IN THE CONTEXT OF BACKGROUND IONOSPHERIC EFFECTS

To derive an expression of the slant range, a right-handed geocentric inertial reference frame is defined, with Z-axis towards the North Pole, and X-axis pointing to the true equinox of the date (referring to Figure 10.1). Without loss of generality, let’s initialize the azimuth time to zero for the shortest distance between the ground target and the Moon-based SAR. The slant range between the Moon-based SAR and the ground target at the time $t=\eta$ takes the following expression:

$R(\eta )=\{R_E^2+R_{EM}^2-2X_m(\eta )R_E\cos {\delta }_g\cos a_g(\eta )-2Y_m(\eta )R_E\cos {\delta }_g\sin a_g(\eta )-2Z_m(\eta )R_E\sin {\delta }_g\},$(10.60)

$X_m(\eta )=R_{EM}\cos ({\delta }_m+{\omega }_M\eta \sin {\vartheta }_S)\cos (a_m+{\omega }_M\eta \cos {\vartheta }_S)$(10.61)

$Y_m(\eta )=R_{EM}\cos ({\delta }_m+{\omega }_M\eta \sin {\vartheta }_S)\sin (a_m+{\omega }_M\eta \cos {\vartheta }_S)$(10.62)

$Z_m(\eta )=R_{EM}\sin ({\delta }_m+{\omega }_M\eta \sin {\vartheta }_S)$(10.63)

Recalled that the propagation distance of the Moon-based SAR is over hundreds of times longer than that of the LEOSAR, thus the “stop-and-go” assumption used in the LEOSAR is no longer applicable for the Moon-based SAR [16]. Now let’s calibrate the error in “stop-and-go” assumption: Suppose that the transmitted signal arrives to the ground target at time T₁, where $T_1=R(\eta )/c$. In an epoch of propagation delay T_D, the Moon-based SAR has moved forward for a certain distance to the new position, so now the slant range between the ground target and radar is $R(\eta +T_D)$. The backscattering signal is received at a time $T_2=R(\eta +T_D)/c$. The total time delay of wave propagation is the sum of T₁ and T₂:

$T_D=T_1+T_2,$(10.64)

where T_D is typically between 2.3s and 2.7s [16]. Notice that the time delay given rise by the atmosphere is far smaller than T_D; therefore, it can be reasonably ignored in removing the “stop-and-go” assumption. The Moon-based SAR observation geometry under “non-stop-and-go” assumption shown in Figure 10.17 depicts a scenario whereas the transmitted signal scatters back, the radar moves a distance within an epoch of T_D. Thus, the Moon-based SAR can no longer be regarded as operating in monostatic, but instead in bistatic mode.

Finally, the slant range for the Moon-based SAR under the equivalent bistatic mode could be written as:

$R_S(\eta )=\left[R(\eta )+R(\eta +T_D)\right]/2$(10.65)

It is seen that the 4^th order Taylor series range equation is capable of dealing with the curved trajectory of the Moon-based SAR’s signal during the long synthetic aperture time, which is most suitable for Moon-based SAR imaging [16]. Consequently, the slant range can be expanded into Taylor series up about $\eta =0$ to 4^th order,

$R_S\left(\eta \right)=R_0+R_1\eta +R_2{\eta }^2+R_3{\eta }^3+R_4{\eta }^4+\mathcal{O}({\eta }^5)$(10.66)

where R₀ is the shortest slant range. The derivation of expansion coefficients is straightforward but tedious. Complete expressions for R_i are given in [16].

10.3.3 SAR SIGNAL IN THE CONTEXT OF BACKGROUND IONOSPHERIC EFFECTS

We shall establish a Moon-based SAR signal model in considering of background ionospheric effects, based on the curved trajectory, in which the relative positions between the Earth target and the Moon-based SAR are accounted for. The received signal of the Moon-based SAR system from the point of interest can be written as [33]

$s_r(\tau ,\eta )=w_r[\tau -2R_S(\eta )/c]w_a(\eta )\exp \{-j4\pi f_c{R}_S(\eta )/c\}\cdot \exp \left\{j\pi K_r{\left[\tau -2R_S(\eta )/c\right]}^2\right\}$(10.67)

Fourier transform in range direction is first applied to Equation 10.67 by using the principle of stationary phase (POSP):

$s_r(f_{\tau },\eta )=w_r(f_{\tau })w_a\left(\eta \right)\exp (-j\pi f_{\tau }^2/K_r)\cdot \exp [-j4\pi R_S(\eta )(f_{\tau }+f_c)/c]$(10.68)

Consequently, the signal under the effects of the temporal-varying background ionosphere can be obtained by including the phase shift Equation 10.59 into the signal Equation 10.68:

$s_{r.iono}(f_{\tau },\eta )=w_r(f_{\tau })w_a(\eta )\exp (-j\pi f_{\tau }^2/K_r)\exp [-j4\pi R_S(\eta )(f_{\tau }+f_c)/c+j\Delta {\phi }_{iono}(\eta )]$(10.69)

Taking Fourier transform along azimuth direction in Equation 10.69 by virtue of the POSP and the method of series reversion (MSR) [33], we have

$s_{r.iono}(f_{\tau },f_{\eta })=w_r(f_{\tau })w_a(f_{\eta })\exp \{j\Psi (f_{\tau },f_{\eta })\},$(10.70)

$\Psi (f_{\tau },f_{\eta })=-\pi \frac{f_{\tau }^2}{K_r}-\frac{2\pi }{c}{a}_0(f_c+f_{\tau })-\frac{2\pi }{c}\frac{2A\cdot TE{C}_0}{(f_c+f_{\tau })}+2\pi \frac{1}{4P_2}\left(\frac{c}{f_c+f_{\tau }}\right){\left(f_{\eta }+P_1\frac{(f_c+f_{\tau })}{c}\right)}^2+2\pi \frac{P_3}{8P_2^3}{\left(\frac{c}{f_c+f_{\tau }}\right)}^2{\left(f_{\eta }+P_1\frac{(f_c+f_{\tau })}{c}\right)}^3+2\pi \frac{9P_3^2-4P_2{P}_4}{64P_2^5}{\left(\frac{c}{f_c+f_{\tau }}\right)}^3{\left(f_{\eta }+P_1\frac{(f_c+f_{\tau })}{c}\right)}^4$(10.71)

From Equation 10.71, we see that the range and azimuth frequency are highly coupled in the phase term. To process the signal and to identify the background ionospheric effects on the Moon-based SAR imaging, Equation 10.71 is further expanded as follows:

$\Psi (f_{\tau },f_{\eta })={\Psi }_r(f_{\tau })+{\Psi }_a(f_{\eta })+{\Psi }_{rcm}(f_{\tau },f_{\eta })+{\Psi }_{src}(f_{\tau },f_{\eta })+{\Psi }_{res}$(10.72)

${\Psi }_r(f_{\tau })=2\pi [(-\frac{a_0}{c}+\frac{2ATE{C}_0}{c{f}_c^2}+\frac{P_1^2}{4P_{2}c}+\frac{P_1^3{P}_3}{8P_2^{3}c}+\frac{9P_3^2-4P_2{P}_4}{64P_2^{5}c}{P}_1^4)f_{\tau }+\left(-\frac{1}{2K_r}-\frac{2ATE{C}_0}{c{f}_c^3}\right)f_{\tau }^2+\frac{2ATE{C}_0}{c{f}_c^4}{f}_{\tau }^3-\frac{2ATE{C}_0}{c{f}_c^5}{f}_{\tau }^4]$(10.73)

$\begin{array}{c}{\Psi }_a(f_{\eta })=2\pi [\left(\frac{P_1}{2P_2}+\frac{3P_1^2{P}_3}{8P_2^3}+\frac{9P_3^2-4P_2{P}_4}{16P_2^5}{P}_1^3\right)f_{\eta }+\left(\frac{c}{4P_2{f}_c}+\frac{3P_1{P}_{3}c}{8P_2^3{f}_c}+\frac{9P_3^2-4P_2{P}_4}{32P_2^5{f}_c}3P_1^{2}c\right)f_{\eta }^2\\ +\left(\frac{P_3{c}^2}{8P_2^3{f}_c^2}+\frac{9P_3^2-4P_2{P}_4}{16P_2^5{f}_c^2}{P}_1{c}^2\right)f_{\eta }^3+\frac{9P_3^2-4P_2{P}_4}{64P_2^5{f}_c^3}{c}^3{f}_{\eta }^4]\end{array}$(10.74)