Abstract
In Chapter 3, we have shown that the Earth’s ionosphere exerts an adverse effect on SAR imaging. It is due to the mismatch between the actual radar signal affected by the dispersion of radio waves in the ionosphere and the matched filter used for signal processing. Accordingly, to improve the image one should correct the filter. This requires knowledge of the total electron content in the ionosphere, as well as of another parameter that characterizes the azimuthal variation of the electron number density (see Section 3.9). These quantities can be reconstructed by probing the ionosphere on two distinct carrier frequencies and exploiting the resulting redundancy in the data (see Section 3.10).
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Notes
- 1.
Polarimetric SAR sensors can transmit and record signals with different linear polarization. Typically, the waves with vertical (V) and horizontal (H) polarization are emitted and received, which creates four SAR imaging channels altogether: VV, HH, VH, and HV.
- 2.
In this case, the number densities of electrons and ions can be considered equal to one another, N e = N i, due to the overall quasi-neutrality of the ionospheric plasma.
- 3.
- 4.
- 5.
We do not discuss how the presence of FR may affect the dual carrier method that was developed in Section 3.10 for the case of an isotropic ionosphere and scalar interrogating field. The scalar treatment adopted in Chapter 3 may, in particular, remain valid if instead of the linearly polarized radar signals we consider circular polarization that is not subject to FR.
- 6.
Alternatively, one can use point scatterers: \(\nu (\boldsymbol{z}) =\sum _{m}\nu _{m}\delta (\boldsymbol{z} -\boldsymbol{ z}_{m})\) so that (5.82) yields: \(I_{\text{F}}(\boldsymbol{y}) =\sum _{m}\nu _{m}W_{\text{F}}(\boldsymbol{y},\boldsymbol{z}_{m})\). Considering \(I_{\text{F}}(\boldsymbol{y})\) at sufficiently many reference locations \(\boldsymbol{y}\) as given data, and taking into account that w p and w q in (5.79) are known analytically, one can obtain an overdetermined system of equations and solve it with respect to p and q in the weak sense. In practice, however, the dominant point scatterers may not always be available. That’s why we subsequently focus on the approach suitable for distributed scatterers.
- 7.
- 8.
A detailed analysis is provided in Section 7.2
- 9.
Finite integration limits in formula (5.94) effectively imply that the quantity g(h) will depend not only on the shift h but also on the location or, rather, area (patch) within the image, across which the integration is performed. This, in turn, means that the quantity σ 2 on the right-hand side of formula (5.88) can also depend on the patch instead of being interpreted as a constant for the entire image.
- 10.
- 11.
- 12.
- 13.
Actually, regularization of (5.105) is not needed if zeros of \(a(t - t_{\boldsymbol{y}})\) happen to be outside the intersection of supports of \(A(t - t_{\boldsymbol{y}})\) and \(A(t - t_{\boldsymbol{z}})\). The analysis in this section takes into account only the support of \(A(t - t_{\boldsymbol{y}})\) though. Hence, for \(t_{\boldsymbol{y}}\neq t_{\boldsymbol{z}}\) the condition that the interval of φ F given by (5.108) does not contain \(2\pi n \pm \frac{\pi } {2}\) is sufficient but not necessary for the absence of singularities in (5.105).
- 14.
- 15.
- 16.
This value corresponds to Bτ = 200. Besides setting a finite upper limit for the integral in the numerator of (5.118), the value of Bτ also enters into the argument of the \(\mathop{\mathrm{sinc}}\nolimits\) function: \(W_{\text{R}}(\xi ) \sim \mathop{\mathrm{sinc}}\nolimits (\xi -2\xi \vert \xi \vert /(B\tau ))\), see (2.61). In the limit Bτ → ∞, with the help of the anti-derivative \(\mathop{\mathrm{sinc}}\nolimits ^{2}\xi =\Big (\text{Si}(2\xi ) -\sin \xi \mathop{\mathrm{sinc}}\nolimits \xi \Big)^{{\prime}}\), where \(\text{Si}(\xi ) =\int _{ 0}^{\xi }\mathop{ \mathrm{sinc}}\nolimits \zeta \, d\zeta\) is the sine integral, expression (5.118) for the case of no FR evaluates to \(10\log _{10}[( \frac{\pi }{2} -\text{Si}(2\pi ))/\text{Si}(2\pi )] \approx -9.68dB\).
- 17.
The improvement of resolution provided by the weighted matched filter as compared to the baseline case should be considered insignificant.
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Gilman, M., Smith, E., Tsynkov, S. (2017). The effect of ionospheric anisotropy. In: Transionospheric Synthetic Aperture Imaging. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-52127-5_5
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