SCIENCE CHINA Information Sciences, Volume 60 , Issue 6 : 060307(2017) https://doi.org/10.1007/s11432-016-9065-1

## Modelling of tropospheric delays in geosynchronous synthetic aperture radar

Dexin LI 1,3,*,
• AcceptedMar 28, 2017
• PublishedMay 19, 2017
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### Abstract

As a direct consequence of the orbital height, the integration time in geosynchronous synthetic aperture radar (GEO SAR) with metric or decimetric azimuth resolutions is in the order of several hundreds or even thousands of seconds. With such long integration time, the compensation of residual tropospheric propagation terms poses one of the fundamental challenges associated with GEO SAR missions. In order to better characterise the impact of the propagation errors on GEO SAR imaging, we put forward a model for the simulation of the tropospheric delay appropriate for the accurate simulation of GEO SAR surveys. The suggested model, with a deterministic background component and a random turbulent one, incorporates some of the most recent meteorological data for the characterization of the troposphere. To illustrate the relevance of the derivation, the suggested model is used for performance estimation and raw data simulation of GEO SAR raw data. Substantialconclusion on the system impulse response and the associatedcalibration requirements is also drawn from the analysis.

### Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant No. 41271459). The authors would like to thank Paco López-Dekker for technical advice and the anonymous reviewers for their constructive comments.

• Figure 1

(Color online) (a) Tropospheric delay propagation model; (b) block diagram for the generation of the slant tropospheric delay including the meteorological and geometrical models.

• Figure 2

(Color online) Global pressure computed using the GPT2w model for February 11th (a) and August 12th (b).

• Figure 3

(Color online) Global temperature computed using the GPT2w model for February 11th (a) and August protectłinebreak 12th (b).

• Figure 4

(Color online) Global water vapour pressure computed using the GPT2w model for February 11th (a) and August 12th (b).

• Figure 5

(Color online) (a) Saastamoinen zenith hydrostatic delay as a function of the pressure and temperature at the Equator, with $h=222$ m, $m_\mathrm{\scriptscriptstyle{ T} }= 0.006$ K$\cdot$m$^{-1}$; (b) Askne zenith wet delay as a function of the temperature and the water vapour pressure at the Equator for a reference height of 222 m, $T_\mathrm{\scriptscriptstyle{ T} }= 270K$, and $m_\mathrm{\scriptscriptstyle{ T} }\approx 2.775$.

• Figure 6

(Color online) Global zenith hydrostatic delay distribution as predicted by the Saastamoinen model for February 11th (a) and August 12th (b) using the values derived from the GPT2w model. Note the plots have been computed using an ellipsoidal Earth (i.e., $h\approx 0$) and may not be representative in areas with large topography.

• Figure 7

(Color online) Global zenith wet delay distribution as predicted by the Askne model [21] for February 11th (a) and August 12th (b) using the values derived from the GPT2w model. Note the plots have been computed using an ellipsoidal Earth (i.e., $h\approx 0$) and may not be representative in areas with large topography.

• Figure 8

Illustration of the angular variables in (eq:~angles) within the GEO SAR observation geometry.

• Figure 9

(Color online) Vienna mapping functions (i.e., zenith to slant in black) for February 11th, around the Equator and with a reference height of 222 m for the hydrostatic (a) and wet (b) delays as a function of the incident angle in LOS. The values of $a_\mathrm{\scriptscriptstyle{ H} }$ and $a_\mathrm{\scriptscriptstyle{ H} }$ are approximately 0.001232 and 0.0005565, respectively. The red part of the plots show the percentages of the total delay which are due to the bending of the ray paths caused by the varying refractive index.

• Figure 10

(Color online) Maximum angular variation of the Vienna mapping function as a function of the incident angle. The variable can be interpreted as a phase error amplification factor, which shows an opposite trend as in the LEO case.

• Figure 11

(Color online) Required accuracy in the knowledge of the zenith tropospheric delay to avoid a phase error at the edge of the synthetic aperture higher than 45$^{\circ}$. The plots show the results for 4 different carrier frequencies for 20$^{\circ}$ protectłinebreak (a) and 60$^{\circ}$ (b) incident angles.

• Figure 12

(Color online) Diurnal variations of the hydrostatic (a) and wet (b) tropospheric delays as predicted by [29-31].

• Figure 13

(Color online) Effective hydrostatic (a) and wet (b) tropospheric delays responsible for defocussing introduced by the diurnal variation of the slant wet tropospheric delay shown in Figure 12.

• Figure 14

(Color online) (a) Comparison of Kolmogorov-based segmented power spectral density suggested in [10]with three different Matérn-based power spectral densities with $v$ equal to 1/3, 0.4, and 5/6. The parameters have been selected to contain the different segments of the Kolmogorov-based spectral density. (b) Variance of the realizations of turbulent Matérn-based slant tropospheric delay as a function of the factor $C^2_{\rm M}$. The fluctuations in the curves are due to numerical instabilities in the computation.

• Figure 15

(Color online) Normalized power spectral densities (top) and corresponding normalized tropospheric delay maps (bottom) of a common seed. The Figure shows the anisotropy both in the spectral and the spatial domains introducing local orientation in the simulated turbulence.

• Figure 16

(Color online) Exemplary normalised space and time (from top left to bottom right) variation of the turbulent tropospheric delay.

• Figure 17

(Color online) Azimuth impulse responses (gold) obtained via a Monte Carlo simulation of a 300 s GEO SAR L-band survey for 1 cm$^2$ (a) and 3 cm$^2$ (b) turbulent tropospheres. The red curve shows the ideal impulse response, and the blue curve shows the incoherent average of the different realizations.

• Figure 18

(Color online) Azimuth impulse responses (gold) obtained via a Monte Carlo simulation of a 1 mm$^2$ turbulent troposphere observed from a GEO SAR X-band survey for 60 s (a) and 200 s (b) integration time.

• Figure 19

(Color online) Peak power (a), resolution (b), PSLR (c), and ISLR (d) losses (mean value) for a GEO SAR X-band survey as a function of the power of the turbulences for an integration time of 60 s (top) and 200 s (bottom).

• Figure 20

(Color online) Mean value of the peak power (a), resolution (b), PSLR (c), and ISLR (d) losses for a GEO SAR L-band survey as a function of the power of the turbulences for 300 seconds integration times.

• Figure 21

(Color online) Initial (a) and final (b) states of the slant tropospheric delay used in the raw data simulation.

• Figure 22

(Color online) Refocused intensity image (a) and (interferometric) phase error (b) of the refocused image. Note the tropospheric phase screen is an averaged version of the errors introduced by the simulated realizations in Figure 21after incorporation of the tropospheric propagation effects. No significant defocussing is visible in this result.

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