Analysis of Temporal-Spatial Variant Atmospheric Effects on GEO SAR

Abstract

Due to the ultra-long integration time and large coverage characteristics of geosynchronous synthetic aperture radar (GEO SAR), the atmospheric frozen model in the traditional low Earth orbit SAR (LEO SAR) imaging fails in GEO SAR. The temporal-spatial variation of troposphere and ionosphere should be taken into account for the GEO SAR imaging. Based on the accurate GEO SAR signal model, the two-dimensional spectrum of GEO SAR signal in the context of temporal-spatial variant troposphere and background ionosphere are derived, and then the two-dimensional image shift and defocusing are investigated. The boundary conditions of relevant effects are analyzed and summarized which are related to the status of troposphere and background ionosphere, the GEO SAR imaging geometry and the integration time dependent on the resolution requirement. GEO SAR is also sensitive to ionospheric scintillation which causes the amplitude and phase fluctuations of signals. The corresponding degradation will have a different pattern from LEO SAR. The azimuth point spread function considering the scintillation sampling model is constructed. Then, based on the measurable statistical parameters of ionospheric scintillation, performance is quantitatively analyzed. The analysis suggests that in GEO SAR imaging the azimuth integrated side lobe ratio deteriorates severely, while the degradations of the azimuth resolution and azimuth peak-to-sidelobe ratio are negligible when scintillation occurs.

© 2017 IEEE. Reprinted, with permission, from Performance Analysis of L-Band Geosynchronous SAR Imaging in the Presence of Ionospheric Scintillation, IEEE Transactions on Geoscience and Remote Sensing, vol.55, no. 1, pp. 159–172, 2017 and Background Ionosphere Effects on Geosynchronous SAR Focusing: Theoretical Analysis and Verification Based on the BeiDou Navigation Satellite System (BDS), IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 9, no. 3, pp. 1143–1162, 2016.

Appendices

Appendix 1: Derivation of the Two-Dimensional Spectrum of the GEO SAR Signal Under the Effects of Temporal-Spatial Variant Background Ionosphere

Through the azimuth Fourier transform of the echo signal in (4.33), it is able to derive the analytical expression of the two-dimensional GEO SAR signal spectrum considering the background ionosphere. Since both the GEO SAR accurate slant range and the TEC variations are expressed in the form of high-order polynomials, it will be difficult to use the principle of stationary phase (POSP) to obtain the stationary phase point and then derive the two-dimensional spectrum. Herein, the series inversion [33] method is used to derive the two-dimensional spectrum of the GEO SAR signal in the context of the background ionosphere.

Firstly, the echo signal in (4.33) can be rewritten as

$$\begin{aligned} S\left( {f_{r} ,t_{a} } \right) & = A_{r} \left( {f_{r} } \right) \cdot A_{a} \left( {t_{a} } \right) \cdot \,\exp \left( { - j\frac{{\pi f_{r}^{2} }}{{k_{r} }}} \right) \\ & \cdot \,\exp \left[ { - j\frac{{4\pi \left( {f_{r} + f_{0} } \right)}}{c}\left( {r\left( {t_{a} } \right) + \frac{{40.3 \cdot TEC\left( {t_{a} } \right)}}{{\left( {f_{r} + f_{0} } \right)^{2} }}} \right)} \right] \\ \end{aligned}$$

(4.95)

Assuming $R\left( {t_{a} } \right) = r\left( {t_{a} } \right) + \Delta r_{iono} \left( {t_{a} } \right) = r\left( {t_{a} } \right) + {{40.3 \cdot TEC\left( {t_{a} } \right)} \mathord{\left/ {\vphantom {{40.3 \cdot TEC\left( {t_{a} } \right)} {\left( {f_{0} + f_{r} } \right)^{2} }}} \right. \kern-0pt} {\left( {f_{0} + f_{r} } \right)^{2} }}$, wherein, $r\left( {t_{a} } \right)$ is the slant range history in the absence of ionosphere, and $\Delta r_{iono} \left( {t_{a} } \right) = {{40.3 \cdot TEC\left( {t_{a} } \right)} \mathord{\left/ {\vphantom {{40.3 \cdot TEC\left( {t_{a} } \right)} {\left( {f_{0} + f_{r} } \right)^{2} }}} \right. \kern-0pt} {\left( {f_{0} + f_{r} } \right)^{2} }}$ is the slant range difference introduced by the ionosphere. The specific expressions of $r\left( {t_{a} } \right)$ and $\Delta r_{iono} \left( {t_{a} } \right)$ are written as follows

$$\begin{aligned} r\left( {t_{a} } \right) & = = r_{0} + q_{1} \cdot t_{a} + q_{2} \cdot t_{a}^{2} + q_{3} \cdot t_{a}^{3} + \cdots \\ \Delta r_{iono} \left( {t_{a} } \right) & = \frac{{40.3 \cdot TEC_{0} }}{{\left( {f_{0} + f_{r} } \right)^{2} }} + \frac{{40.3 \cdot k_{1} }}{{\left( {f_{0} + f_{r} } \right)^{2} }} \cdot t_{a} + \frac{{40.3 \cdot k_{2} }}{{\left( {f_{0} + f_{r} } \right)^{2} }} \cdot t_{a}^{2} + \frac{{40.3 \cdot k_{3} }}{{\left( {f_{0} + f_{r} } \right)^{2} }} \cdot t_{a}^{3} + \cdots \\ \end{aligned}$$

(4.96)

According to the properties of Taylor series expansion, then

$$\frac{1}{{\left( {f_{r} + f_{0} } \right)^{2} }} = \frac{1}{{f_{0}^{2} }} - \frac{2}{{f_{0}^{3} }} \cdot f_{r} + \frac{3}{{f_{0}^{4} }} \cdot f_{r}^{2} - \frac{4}{{f_{0}^{5} }} \cdot f_{r}^{3} + \cdots$$

(4.97)

As the first-order, second-order and third-order terms of $\Delta r_{iono} \left( {t_{a} } \right)$ are relatively small, the constant term in (4.97) for those three were approximated. Hence, $\Delta r_{iono} \left( {t_{a} } \right)$ was simplified as

$$\Delta r_{iono} \left( {t_{a} } \right) = \frac{{40.3 \cdot TEC_{0} }}{{\left( {f_{0} + f_{r} } \right)^{2} }} + \frac{{40.3 \cdot k_{1} }}{{f_{0}^{2} }} \cdot t_{a} + \frac{{40.3 \cdot k_{2} }}{{f_{0}^{2} }} \cdot t_{a}^{2} + \frac{{40.3 \cdot k_{3} }}{{f_{0}^{2} }} \cdot t_{a}^{3} + \cdots$$

(4.98)

$R\left( {t_{a} } \right)$ can be denoted in the form as follows

$$R\left( {t_{a} } \right) = R_{0} + Q_{1} \cdot t_{a} + Q_{2} \cdot t_{a}^{2} + Q_{3} \cdot t_{a}^{3} + \cdots$$

(4.99)

where,

$$\begin{aligned} R_{0} & = r_{0} + \frac{{40.3 \cdot TEC_{0} }}{{\left( {f_{0} + f_{r} } \right)^{2} }}\quad Q_{1} = q_{1} + \frac{{40.3 \cdot k_{1} }}{{f_{0}^{2} }} \\ Q_{2} & = q_{2} + \frac{{40.3 \cdot k_{2} }}{{f_{0}^{2} }}\quad Q_{3} = q_{3} + \frac{{40.3 \cdot k_{3} }}{{f_{0}^{2} }} \\ \end{aligned}$$

(4.100)

Next, the series inversion theorem was utilized to derive the accurate two-dimensional spectrum of the GEO SAR signal in the context of the background ionosphere. In order to use the series inversion, $s\left( {f_{r} ,t_{a} } \right)$ should be re-expressed as

$$s\left( {f_{r} ,t_{a} } \right){ = }s_{1} \left( {f_{r} ,t_{a} } \right) \cdot \,\exp \left( { - j\frac{{4 \cdot \pi \cdot \left( {f_{r} + f_{c} } \right)}}{c} \cdot Q_{1} \cdot t_{a} } \right)$$

(4.101)

In the subsequent derivation, it needs to firstly derive the two-dimensional spectrum of $s_{1} \left( {f_{r} ,t_{a} } \right)$, and then use the following relations to calculate the two-dimensional spectrum of $s\left( {f_{r} ,t_{a} } \right)$

$$s_{1} \left( {f_{r} ,t_{a} } \right) \Leftrightarrow S_{1} \left( {f_{r} ,f_{a} } \right)$$

(4.102)

$$s_{1} \left( {f_{r} ,t_{a} } \right) \cdot \exp \left( { - j\frac{{4 \cdot \pi \cdot \left( {f_{r} + f_{c} } \right)}}{c} \cdot Q_{1} \cdot t_{a} } \right) \Leftrightarrow S_{1} \left( {f_{r} ,f_{a} + \frac{{2 \cdot \left( {f_{r} + f_{c} } \right)}}{c} \cdot Q_{1} } \right)$$

(4.103)

$$s\left( {f_{r} ,t_{a} } \right) \Leftrightarrow S\left( {f_{r} ,f_{a} } \right) = S_{1} \left( {f_{r} ,f_{a} + \frac{{2 \cdot \left( {f_{r} + f_{c} } \right)}}{c} \cdot Q_{1} } \right)$$

(4.104)

First of all, $s_{1} \left( {f_{r} ,t_{a} } \right)$ was Fourier transformed, and its integral phase expression was

$$\psi = - \pi \cdot \frac{{f_{r}^{2} }}{\beta } - \frac{{4 \cdot \pi \cdot \left( {f_{r} + f_{c} } \right)}}{c} \cdot \left( {R_{0} + Q_{2} \cdot t_{a}^{2} + Q_{3} \cdot t_{a}^{3} + \cdots } \right) - 2\pi \cdot f_{a} \cdot t_{a}$$

(4.105)

Assuming $\frac{\partial \psi }{{\partial t_{a} }} = 0$, then

$$\left( { - \frac{c}{{2\left( {f_{r} + f_{0} } \right)}}} \right) \cdot f_{a} = 2Q_{2} \cdot t_{a} + 3Q_{3} \cdot t_{a}^{2} + \cdots$$

(4.106)

Through the series inversion, it can get

$$t_{a} \left( {f_{a} } \right) = A_{1} \cdot \left( { - \frac{{c \cdot f_{a} }}{{2\left( {f_{r} + f_{0} } \right)}}} \right) + A_{2} \cdot \left( { - \frac{{c \cdot f_{a} }}{{2\left( {f_{r} + f_{0} } \right)}}} \right)^{2} + A_{3} \cdot \left( { - \frac{{c \cdot f_{a} }}{{2\left( {f_{r} + f_{0} } \right)}}} \right)^{3} + \cdots$$

(4.107)

where,

$$A_{1} = \frac{1}{{2Q_{2} }}\quad A_{2} = - \frac{{3Q_{3} }}{{8Q_{2}^{3} }}\quad A_{3} = \frac{{18Q_{3}^{2} - 8Q_{2} Q_{4} }}{{32Q_{2}^{5} }}$$

(4.108)

Substituting (4.108) into (4.107), and we can get the two-dimensional spectrum of $s_{1} \left( {f_{r} ,t_{a} } \right)$

$$\begin{aligned} S_{1} \left( {f_{a} ,f_{r} } \right) & = \sigma A_{a} \left( {f_{a} } \right)A_{r} \left( {f_{r} } \right) \cdot \,\exp \left( { - j \cdot \pi \frac{{f_{r}^{2} }}{\beta }} \right) \\ {\kern 1pt} & \quad \times \exp \left( { - j \cdot 2\pi \cdot \frac{{2\left( {f_{r} + f_{0} } \right)}}{c} \cdot \left( {r_{0} + \frac{{40.3 \cdot TEC_{0} }}{{\left( {f_{0} + f_{r} } \right)^{2} }}} \right)} \right) \\ & \quad \times \exp \left( {j \cdot 2\pi \cdot \frac{1}{{4 \cdot \left( {q_{2} + \frac{{40.3 \cdot k_{2} }}{{f_{0}^{2} }}} \right)}} \cdot \frac{c}{{2\left( {f_{r} + f_{0} } \right)}} \cdot f_{a}^{2} } \right) \\ & {\kern 1pt} \quad \times \exp \left( {j \cdot 2\pi \cdot \frac{{q_{3} + \frac{{40.3 \cdot k_{3} }}{{f_{0}^{2} }}}}{{8 \cdot \left( {q_{2} + \frac{{40.3 \cdot k_{2} }}{{f_{0}^{2} }}} \right)^{3} }} \cdot \left( {\frac{c}{{2\left( {f_{r} + f_{0} } \right)}}} \right)^{2} \cdot f_{a}^{3} } \right) \\ \end{aligned}$$

(4.109)

According to (4.104), it can derive the final two-dimensional spectrum expression of $s\left( {f_{r} ,t_{a} } \right)$ as shown in (4.34).

Appendix 2: Klobuchar Model and TEC Calculation

In this chapter, Klobuchar Model [16] was used to calculate the ionospheric TEC values on the BD IGSO satellite propagation path. This model is particularly suitable for United States, China, Western Europe and other countries and regions at mid-latitudes. The eight ionospheric delay parameters and Klobuchar model were used to calculate the vertical ionospheric delay $I_{z}^{{\prime }} \left( t \right)$ of the IGSO satellite signal in seconds, specifically expressed as follows

$$I^{\prime}_{z} (t) = \left\{ \begin{aligned} & 5 \times 10^{ - 9} + A_{2} \cos \left[ {\frac{2\pi (t - 50400)}{{A_{4} }}} \right],\quad \left| {t - 50400} \right| < A_{4} /4 \hfill \\ & 5 \times 10^{ - 9} \, ,\quad \left| {t - 50400} \right| \ge A_{4} /4 \hfill \\ \end{aligned} \right.$$

(4.110)

where t is the local time of the ionosphere at the IPP and its value is in the range of 0–86,400 in seconds. $A_{2}$ is the magnitude of Klobuchar cosine curve in the day time, which is dependent on the $\alpha_{1} \sim\alpha_{4}$ and the latitude of the IPP. $A_{4}$ is the period of cosine curve in seconds, which is dependent on the $\beta_{1} \sim\beta_{4}$ and the latitude of the IPP. The geographic latitude of the IPP is also determined by the user geographic latitude, satellite azimuth and geocentric aperture angle.

As shown in Fig. 4.31, the IPP at the time of t₁ was located in Point A, and the satellite was moving forward at the time of t₂, while the IPP was located in Point B. Then, the segment AB was the length of the IPP of the IGSO satellite.

Fig. 4.31

Relations between the local time and the satellite motion within the Klobuchar model

In the light of the equation of $I_{B1I} (t) = \frac{1}{{\sqrt {1 - \left( {\frac{R}{R + h}\cos E} \right)^{2} } }}I_{z}^{\prime } (t)$, it is able to convert the vertical ionospheric delay $I_{z}^{\prime } (t)$ into the ionospheric delay $I_{B1I} (t)$ along the signal propagation path.

In accordance with the relations between the ionospheric delay $I_{B1I} (t)$ along the signal propagation path and TEC, it is able to calculate the TEC values on the propagation path at the time of t by the following equation

$$TEC\left( t \right) = \frac{{I_{B1I} (t) \cdot cf^{2} }}{40.26}$$

(4.111)

Appendix 3: Derivation of (4.64)

Here we provide the derivation of (4.64). First, we assume $A$ fits Nakagami-m distribution and its variance can be calculated as

$$\begin{aligned} \sigma_{A}^{2} & = E\left( {A^{2} } \right) - \left[ {E\left( A \right)} \right]^{2} \\ & =\Omega - \left[ {E\left( A \right)} \right]^{2} \\ \end{aligned}$$

(4.112)

where $\Omega = E\left( {A^{2} } \right)$.

According to the definition of mathematical expectation, $E\left( A \right)$ can be written as

$$\begin{aligned} E\left( A \right) & = \int\limits_{ - \infty }^{ + \infty } {xf(x,m,\Omega )dx} \\ & = \int\limits_{ - \infty }^{ + \infty } {x\left[ {\frac{{2m^{m} }}{{\Gamma \left( m \right)\Omega ^{m} }}x^{2m - 1} \exp \left( { - \frac{m}{\Omega }x^{2} } \right)} \right]dx} \\ \end{aligned}$$

(4.113)

where $f(x,m,\Omega )$ is the probability density function of Nakagami-m distribution, $x$ is the variable and $m = {1 \mathord{\left/ {\vphantom {1 {S_{4}^{2} }}} \right. \kern-0pt} {S_{4}^{2} }}$.

With a change of variable, (4.113) may be re-written as

$$E\left( A \right) = \sqrt {\frac{\Omega }{m}} \frac{1}{{\Gamma \left( m \right)}}2\int\limits_{ - \infty }^{ + \infty } {\left( {\sqrt {\frac{m}{\Omega }} x} \right)^{{\left( {2m + 1} \right) - 1}} \exp \left( { - \left[ {\sqrt {\frac{m}{\Omega }} x} \right]^{2} } \right)d\left( {\sqrt {\frac{m}{\Omega }} x} \right)}$$

(4.114)

Considering the definition of Gamma function, which is shown as

$$\Gamma \left( n \right) = 2\int\limits_{0}^{ + \infty } {t^{2n - 1} } \exp \left( { - t^{{^{2} }} } \right)dt$$

(4.115)

Based on (4.115) and $x > 0$, (4.114) can be written as

$$E\left( A \right) = \sqrt {\frac{\Omega }{m}} \frac{{\Gamma \left( {m + \frac{1}{2}} \right)}}{{\Gamma \left( m \right)}}$$

(4.116)

We consider (4.116) in (4.112) and then simplify it. Finally, we obtain (4.64).