Tingting Fu, Yuanhao Li, Zhiyang Chen, Zheng Wu, Cheng Hu

Abstract: The near-Earth asteroid collisions could cause catastrophic disasters to humanity and the Earth, so it is crucial to monitor asteroids.Ground-based synthetic aperture radar (SAR) is an observation technique for high resolution imaging of asteroids.The ground-based SAR requires a long integration time to achieve a large synthetic aperture, and the echo signal will be seriously affected by temporal-spatial variant troposphere.Traditional spatiotemporal freezing tropospheric models are ineffective.To cope with this, this paper models and analyses the impacts of temporalspatial variant troposphere on ground-based SAR imaging of asteroids.For the background troposphere, a temporal-spatial variant ray tracing method is proposed to trace the 4D (3D spatial +temporal) refractive index network provided by the numerical weather model, and calculate the error of the background troposphere.For the tropospheric turbulence, the Andrew power spectral model is used in conjunction with multiphase screen theory, and varying errors are obtained by tracking the changing position of the pierce point on the phase screen.Through simulation, the impact of temporal-spatial variant tropospheric errors on image quality is analyzed, and the simulation results show that the X-band echo signal is seriously affected by the troposphere and the echo signal must be compensated.

Keywords: near-Earth asteroids; ground-based SAR; troposphere; ray tracing

1Int roduction

The orbits of near-Earth asteroids are close to Earth’s orbit, and there is a possibility of collision with the Earth, which poses a great threat to humanity.Coping with the risk of near-Earth asteroid impact is a major long-term challenge,so it is of great significance to monitor asteroids [1].

Ground-based radar is an effective technique for near-Earth asteroids observation.It actively emits electromagnetic waves and receives the reflection of asteroids.This allows for active observation throughout the day and in all weather conditions[2, 3].

Near-Earth asteroids are very far away and small in size, requiring high-resolution observation, which can be achieved by two technical approaches: one is to form a distributed radar with large diameter and high power by using multiple unit radars; the other is to mount a radar with a moving platform, such as a vehicle,to form a large synthetic aperture, which is ground-based synthetic aperture radar (SAR) [4].The second technology has strong flexibility, low cost and easy implementation.

Compared with spaceborne SAR system, the platform of ground-based SAR system moves at a slower pace, so the integration time for obtaining high-resolution images is much longer, typically, several hours.Besides, ground-based SAR observes aerial targets, so the signal propagation path passes through a much wider area of the troposphere (hundreds of kilometres).The long integration time and synthetic aperture will cause spatial-temporal variant tropospheric impacts on asteroids imaging [5].The troposphere is generally modeled into two parts: the background troposphere (slowly varying portion which introduces signal propagation delays[6])and tropospheric turbulence ( rapidly varying portion which causes random fluctuations in signal amplitude and phase [7]).

The tropospheric impacts on spaceborne SAR imaging have been widely studied [8].The modeling methods of background troposphere are mainly based on traditional ray-tracing and mapping functions methods [9].The ray-tracing methods are to calculate tropospheric delay through path integration of refractivity [10].The mapping functions methods directly convert zenith delay into slant delay and are primarily used for high elevation angle conditions [11].These methods are often combined with empirical tropospheric models, such as Hopfield model [12],Saastamoinen model [13], Egnos model [14], etc.For the tropospheric turbulence, statistical turbulence theory, i.e., turbulence power spectrum model, is primarily to model turbulence phase screens.Typical power spectrum models include the Kolmogorov spectrum [7], Tatarskii spectrum [15], Andrews spectrum [16].The above models are generally combined with multi-phase screen theory to model amplitude phase errors caused by turbulence [17].

Since the integration time of spaceborne SAR is only a few minutes or less, and the signal path spans far less than a hundred kilometres in the troposphere in the horizonal direction, these studies are mainly based on the assumption of a spatiotemporal freezing model, i.e., the troposphere remains constant in time, and they only consider the change of the troposphere at different heights, neglecting horizontal tropospheric variations.Even in the research of geosynchronous (GEO) SAR systems with a long integration time up to 1 000 s, the tropospheric modeling is based on these two methods[18, 19].The integration time of GEO SAR is much less than ground-based SAR, and the location of the pierce point in the troposphere is nearly unchanged.Therefore, the above modeling methods cannot simulate the temporal-spatial variant troposphere at the scale of ground-based SAR.

To solve the above problem, this paper proposes a method for modeling and analyzing the impact of temporal-spatial variant troposphere on ground-based SAR imaging of asteroids.For large-scale background troposphere, we propose the temporal-spatial variant ray tracing method to trace the 4D refractive index network calculated by numerical weather models (NWM) and estimate the propagation path delay accurately to obtain the error model.For small-scale tropospheric turbulence, use the Andrew power spectrum model in conjunction with the multi-phase screen theory to model the effects, according to the change in the position of the signal through the turbulent phase screen to obtain the varying errors.Finally, the influence of tropospheric effect is analysed theoretically and simulated.

The structure of this paper is as follows.In Section 2, we give the accurate signal model of ground-based SAR under the influence of troposphere.In Sections 3 and 4, the errors of background troposphere and tropospheric turbulence are modeled, respectively.In Section 5, the effect of troposphere on SAR imaging is simulated.Finally, conclusions are drawn in Section 6.

2 Ground-Based SAR Signal Model

Ground-based SAR can obtain an equivalent large aperture through the movement of the platform, enabling the long-range detection of asteroids.The integration time can reach a few hours,and the synthetic aperture trajectory spans over thousands of kilometres.The geometric structure diagram of ground-based SAR imaging is shown in Fig.1.

Fig.1 Geometric structure of ground-based SAR imaging

Assumingrs(ta) represents the position vector of the radar at timetaandrris the position vector of the target， the instantaneous slant range of the SAR signal at timetacan be represented as

wheretais the slow time，r0is the slant range at the aperture center time, andqiis thei-th order derivative with respect to slow time.

Considering the troposphere effects, the accurate echo signal of the ground-based SAR can be expressed as

wheretis the fast time,Ar(·) andAa(·) are the envelope function in range and azimuth, respectively,Kris the range frequency modulation rate,andλis wavelength.And Δφatmos(p,ta) is the total tropospheric phase error, andpis the different locations.At different moments and locations,the tropospheric conditions vary obviously.The temporal-spatial variant troposphere results in the varying degree of tropospheric impact on the signals of different synthetic aperture time.However, traditional empirical models cannot describe the temporal-spatial variant troposphere.Therefore, it is essential to combine 4D meteorological data provided by NWM to accurately estimate the tropospheric effects.

3 Modeling and Impacts Analysis of Background Troposphere

The core idea of the proposed method is to calculate refractivity based on 4D meteorological data provided by NWM, which can reflect temporalspatial variant troposphere.Then the 4D grid of refractive index is tracked by the proposed algorithm called temporal-spatial variant ray tracing method to estimate the propagation path of the signal in the troposphere.The temporal-spatial variant error model can be obtained by the path integral of refractive index.

3.1 Refractive Index Calculation Using NWM

The impact of the troposphere on the propagation of electromagnetic waves can be typically represented by the refractive indexN.The spatial and temporal distribution of the refractive index is related to the distribution of three meteorological factors: temperatureT, water vapor pressuree, and tropospheric pressureP.The refractive indexNis categorized into dry itemNdand wet itemNw.Ncan be expressed as

where the atmospheric pressure of dry air isPd=P-e(hPa) , withTrepresenting temperature in Kelvin(K).Thayer provides reference values fork1=77.604±0.014,k2=64.79±0.08, andk3=377 600±400, respectively[20].

Using the High-precision ECMWF (European Centre for Medium-Range Weather Forecasts) ERA5 numerical weather model to model errors.The ERA5 model can provide meteorological parameters such as specific humidity, temperature, and pressure with a spatial resolution of 0.25°×0.25° at hourly intervals and a vertical resolution of 37 layers.

The value of water vapor pressure is difficult to measure and can be obtained from specific humidityq:

Because the ERA5 model is based on a geopotential height system, while the radar system is the geodetic height system.It is necessary to convert geopotential height to geodetic height.The method for converting the height system can be found in reference [21].After standardizing the height system, tropospheric refractive index for each layer can be calculated using the meteorological para- meters for each layer.

3.2 Path Delay Calculation by Temporal-Spatial Variant Ray Tracing

The main idea of the temporal-spatial variant ray tracing method is to use a 4D refractive index network based on ERA5 model, estimating the signal's propagation path through the troposphere.The precise delay is obtained by the path integral of the refractive index.The specific steps of the proposed algorithm are as follows:

1)Step 1：Resample to obtain a 2D refractive index grid containing the signal propagation path.

Firstly, to calculate the refractive index in the troposphere and the heights of each layer based on the ERA5 model to obtain the temporal-spatial variant 4D refractive index network.Selecting the 3D refractive index network of the current synthetic aperture time, with the signal propagation path projected horizontally on the ground as the horizontal direction and zenith direction as the vertical direction.Then, resampling to obtain a two-dimensional refractive index grid containing the signal propagation path, where the grid cells are uniform in the horizontal direction, and the heights in the vertical direction are measured in terms of actual distances of the horizontal grid.

2) Step 2: Propagation path estimation.

Based on the angle of incidenceθand the coordinate of entry into this layer, determine whether the signal propagation path crosses the top boundary of the cell.The cell height is defined as the height difference Δhbetween adjacent layers provided by the ERA5 model.Due to the slow variation of the refractive index in the horizontal direction, the change in angles caused by refraction between the same altitude layers is not considered.The entry point into the current layer is denoted as (xin,yin), and the coordinates of exit from this layer are determined based on the incident angle (xout,yout), as shown in Fig2.

Ifxoutis greater than「xout■(the symbol「■represents rounding up to the nearest integer.), it indicates that the signal did not cross the top of the cell.In this case, calculate the coordinates(xn,yn)of exiting this cell and proceed to calculate the next cell.Otherwise,xoutis less than「xout■, it indicates a crossing, and the calculation for the next layer is performed.

Fig.2 Signal passing through refractive index grids

Calculate the slant range path Δsiwithin the corresponding cell based on the coordinate of the entry point, to integrate and obtain the slant range delay, and to add it to the delay for this layer.To continue iterating until the ray reaches the top layer.

3) Step 3: Update the angle of incidence.

Due to the troposphere causing the bending of signal propagation paths, changes in the angle of incidence require to be considered.Refractive index changes slowly in the horizontal direction,so only changes in angles between different height layers are considered, ignoring variations in angles when crossing different grids at the same height layer.Based on Snell’s law, under the assumption of a spherical layered model, the product of refractive index at different heights,Earth's radius, and cosine of the elevation angle remains constant.

wherenlandθlrepresent the refractive index and angle of incidence when passing through thel-th layer of the grid.nl+1andθl+1represent the refractive index and angle of incidence when entering the (l+1)-th layer of the grid.Ris the Earth’s radius.

4) Step 4: Calculate the delay from the model top to an altitude of 60 km.

The ERA5 model provides meteorological data up to approximately 47 km in altitude.However, above the top layer, there is minimal water vapor influence.The top tropospheric delay Δrtopfrom the top layer to 60 km is calculated using the Saastamoinen dry delay model,and the total delay Δris finally obtained.

wheref(φ,h) is the correction due to the Earth's rotation-induced gravitational acceleration,φis the latitude,his the height (kilometres).

5) Step 5: Iteratively compute delays for all synthetic aperture time.

Update the position of radar and repeat the steps from Step 1 to Step 4 to obtain the tropospheric delays corresponding to the entire synthetic aperture time.

6) Step 6: Modeling the temporal-spatial variant phase error.

Considering the bidirectional propagation of the SAR signal, the temporal-spatial variant phase error model is modeled based on the calculated time delay error Δr(p,ta) can be expressed as

wheretais the slow time，qiis thei-th order derivative with respect to slow time, andpdenotes the different locations.

The overall workflow of this algorithm is shown in Fig.3.

The troposphere is a non-dispersive medium,and its impact on signals at different frequencies is consistent, which does not affect range imaging.Therefore, for ground-based SAR azimuth signals, phase error modeling takes the form of a series expansion that varies slowly with time.Considering the background tropospheric effects,the azimuth signal can be expressed as

Fig.3 Temporal-spatial variant ray tracing method flow chart

whereTais the integration time, andfdris the azimuth frequency modulation rate.

The time rate of change of tropospheric delay affects the image quality, and the linearly changing part of the delay with timeq1is related to the image offset.

wherevbfis the beam-foot velocity.

The nonlinear changing partq2causes quadratic phase errorφa2, leading to the widening of the main lobe and the elevation of side lobes.

The nonlinear changing partq3leads to cubic phase errorφa3, resulting in asymmetric side lobes and potentially causing azimuth defocusing [18].

4 Modeling and Impacts Analysis of Tropospheric Turbulence

Tropospheric turbulence occurs due to irregular and random variations of meteorological parameters in both time and space, especially under certain sudden and extreme weather conditions.This leads to random fluctuations in signal amplitude and phase, making it challenging to accurately and comprehensively describe these variations using mathematical expressions.Therefore, statistical methods are employed to characterize the changes in meteorological parameters,as well as amplitude and phase fluctuations.

The refractive index power spectral density of tropospheric turbulence follows a power-law spectral distribution.Modeling using the Andrews power spectral model [16], which is capable of describing the power spectral distribution of turbulence in all wavenumber domains.

The power spectrumΦs(κ) of the random phase introduced by tropospheric turbulence can be obtained from the refractive index power spectrumΦn(κ):

wherek=2π/λ, and Δxis the thickness of each layer.

According to the change of the position of the signal passing through the phase screen during the integration time, to obtain the corresponding temporal-spatial variant turbulent phase errors.Considering that the time correlation of turbulence is only a few minutes, the time update threshold is set according to its correlation.When the integration time exceeds this threshold, the turbulence phase screen is regenerated.

Using the multi-phase screen theory to analyze the amplitude and phase fluctuations caused by tropospheric turbulence.Firstly, the power spectrum of random phase is used to construct a filter to filter the complex Gaussian random number sequence.And then the phase random fluctuation is obtained by inverse discrete Fourier transform:

wherermis a Hermitian complex Gaussian random variable with zero mean and unit variance.

According to the multi-phase screen theory,the amplitude fluctuationIIFand phase fluctuationφTFof the signal can be obtained by calculating the tropospheric transfer functionDTF.

Due to the SAR signal bidirectional propagation, the tropospheric turbulence transfer function can be expressed as

Turbulence will cause fluctuations in signal magnitude and phase, resulting in a degradation of the imaging quality of ground-based SAR.The azimuthal signal of SAR affected by tropospheric turbulence can be expressed as

5 Simulations

5.1 Background Troposphere Effects Analysis

Simulation analysis of the background tropospheric impact on ground-based SAR azimuth imaging is conducted based on meteorological data provided by NWM.The temporal-spatial variant ray tracing method is employed in conjunction with high-precision ECMWF ERA-5 reanalysis model to simulate tropospheric delay errors.

Assuming the vehicle is the moving carrier of the radar and the Beijing-Shanghai Expressway as the synthetic aperture trajectory of groundbased SAR for simulation.The trajectory of the Beijing-Shanghai Expressway is shown in Fig.4.The trajectory length is approximately 1 220 km,and the time is about 12 hours.Selecting ERA5 data for 37 pressure levels, temperature, specific humidity, and surface geopotential from June 4th, 2023, 8:00 AM to 8:00 PM.The observed space target is a small celestial body located approximately 100 000 km away from Earth.

Fig.4 Synthetic aperture trajectory

Using the temporal-spatial variant ray tracing method, to calculate the variation in tropospheric delay error within the synthetic aperture and compare it with the results calculated using the tropospheric empirical model called the Hopfield model [11].The variation in delay error is shown in Fig.5, and the variation in phase error within the synthetic aperture time is obtained based on Eq.(12) and shown in Fig.6.

Fig.5 Variation in tropospheric delay within the synthetic aperture time

Fig.6 Phase error corresponding to the X-band

From Fig.5 and Fig.6, it is evident that the time delay errors computed by both methods vary with synthetic aperture time.The proposed algorithm’s delay ranges from a minimum of 3.09 m to a maximum of 3.26 m, while the delay calculated based on the Hopfield model ranges from 3 m to 3.27 m.The delay data calculated by the two methods are consistent, and the correlation coefficient is 86.74%.However, the Bias and RMSE reach 8.1 cm and 9.5 cm, respectively.The variations in delay obtained by the Hopfield model are primarily due to changes in the geometric structure of signal propagation, without accounting for the temporal-spatial variant atmospheric environment.Therefore, it can only provide a rough estimation of atmospheric delay and cannot describe its detailed variations.On the other hand, the numerical weather model, combined with spatiotemporal ray tracing considering the spatiotemporal characteristics of the tropospheric environment, enables precise calculation of spatiotemporal variations in time delay errors during synthetic aperture time.

Based on the simulated imaging geometry,to assess the influence of troposphere on imaging results at different azimuthal resolutions.To achieve a specific azimuthal resolution, the required integration time is provided in Tab.1 calculating the corresponding peak sideLobe ratio(PSLR) and integral SsideLobe ratio (ISLR) to evaluate the imaging results, as shown in the Fig.7.

Tab.1 The integration time required to achieve different resolutions

Fig.7 Assessment of impacts on SAR imaging of troposphere at different resolutions: (a).PSLR; (b) LSLR

From Fig.7 , it can be observed that the impact of the troposphere on SAR imaging varies with different resolution requirements.The higher the resolution requirement, the longer the integration time, and the more significant the influence of troposphere.

In the subsequent simulation analysis, the integration time is assumed to be 5 h for all cases.Adding the phase errors caused by background troposphere to the ground-based azimuthal echo signal as shown in Eq.(13), simulating and analysing the impact of temporal-spatial variant troposphere on ground-based SAR azimuthal imaging, obtaining the azimuthal profile as shown in Fig.8.Quantitatively analyse the impact of temporal rate of tropospheric delay variations on image quality, and calculate the PSLR and ISLR to obtain the imaging evaluation results shown in Tab.2.

Fig.8 The azimuthal profile

Tab.2 Orbital elements

According to SAR imaging theory, when the offset, second-order phase error, and third-order phase error are each smaller than one resolution cell, π /4 rad, and π /8 rad, respectively, their impact can be disregarded.Simulation results indicate that the linear variation of tropospheric delay results in significant offset in the SAR image, while the nonlinear component leads to pronounced defocusing in SAR azimuthal imaging.

Therefore, the quality of ground-based SAR imaging is significantly degraded by temporalspatial variant background tropos-phere.

5.2 Tropospheric Turbulence Effects Analysis

In the case of weak turbulence, the turbulence phase screen is simulated based on the Andrews power spectrum model, as shown in Fig.9.

Fig.9 Turbulence phase screen

Using the multiphase screen theory to simulate signal magnitude and phase variations caused by turbulence, assuming a time correlation of turbulence in the order of minutes, we assume that the turbulence phase screen remains unchanged within a 5-minute interval, and a new turbulence phase screen is generated every five minutes.During this period, only changes in position through the phase screen due to radar movement are considered.The signal magnitude and phase variations due to turbulence within the 5-minute interval are shown in Fig.10 and Fig.11,indicating random fluctuations in signal magnitude and phase.

Based on the Eq.(22) of the azimuthal signal model under the impact of turbulence in SAR signal, simulating the effect of tropospheric turbulence on azimuth imaging under the synthetic aperture time of 5h, the azimuth profile is obtained, as shown in Fig.12.The PSLR and ISLR are -13.64 dB and -1.78 dB, respectively.It can be observed that the azimuthal imaging result has defocusing obviously, so the effect of small-scale turbulence should also be considered.

Fig.10 Signal amplitude variation

Fig.11 Signal phase variation

Fig.12 Turbulence phase screen

5.3 Analysis of Two-Dimensional Imaging Effects on Point Targets

Combining the temporal-spatial variant errors of large-scale background tropospheric and smallscale tropospheric turbulence, an analysis of the two-dimensional imaging effects on point targets is conducted based on the ground-based SAR signal model in Eq.(2).The result is shown in Fig.13.

Fig.13 Single point imaging

According to the imaging result, the azimuthal PSLR and ISLR are - 1.29 dB and-0.81 dB, respectively.Simulation result shows that in the high-frequency X-band, temporal-spatial variant tropospheric errors can lead to severe defocusing in ground-based SAR azimuthal images.Therefore, to achieve high-resolution imaging in ground-based SAR, precise modeling and compensation of tropospheric errors are essential.

6 Conclusion

The prevention of asteroid impact is a major challenge for mankind.The ground-based SAR is an important technology for active observation of asteroids, and has a long integration time the synthetic aperture trajectory can span over a thousand kilometres.The temporal-spatial variant troposphere can lead to a degradation in the imaging performance of SAR, even causing defocusing.This paper models and analyses the troposphere at different scales.For slowly varying background troposphere, a high-precision numerical weather model is used in combination with temporal-spatial variant ray tracing for modeling.Tropospheric turbulence is modeled using a power-law spectral model combined with multiphase screen theory.Then, analysing the impact of the temporal-spatial variant troposphere on ground-based SAR imaging based on the SAR echo signal model.Simulation results indicate that in the X-band, the SAR signal is significantly affected by the troposphere and requires compensation.Background troposphere causes azimuthal image offset and defocusing, while tropospheric turbulence leads to random phase error of the signal, thereby affecting azimuthal imaging.