Research on Error Analysis and Correction Technique of Atmospheric Refraction for InSAR Measurement with Distributed Satellites

SAR interferometry with distributed satellites is a technique based on the exploitation of the interference pattern of two SAR images acquired synchronously. The interferogram contains geometric, atmospheric, topographic and land defomation. This paper focuses on atmospheric effects on SAR interferometry, which shows theoretically that the relationship among ionosphere TEC and troposphere parameters such as temperature, relative humitdity and pressure with respect to slant rang changes. An atmospheric correction method is given in the end.


Introduction
With the development and the maturity of SAR Interferometry technology, its accuracy is increasing largely and it is significant to study the factors which can lower the accuracy of SAR Interferometry technology [1] [2].SAR interferometry system with distributed satellites is based on the exploitation of the interference pattern of two SAR images acquired synchronously, from which useful information is extracted.Atmospheric refraction is one of the main errors because that the temporal and spatial uncertainty of atmospheric change causes different delay of radar signal [3] [4] [5] [6].The paper discusses the fundamental theory of electromagnetic wave signal's atmospheric propagation delay and the main model of Troposphere delay correction and Ionosphere delay correction, establishing the relationship between tropospheric parameters and the distance change.Then the magnitude that Ionosphere and Tropospheric parameters exert influence on the ranging accuracy is analysed.The method to inhibit or eliminate the influence is discussed in the end.

Troposphere Delay
According to the Fermat principle and Snell's law in the basic theory of light propagation, the optical range ρ from satellite to ground point is formed as where r 0 is the center distance for a ground point (Figure 1), r1 is the center distance for the top of neutral atmosphere, z is the view zenith distance of Any point at signal propagation path, θ is the view height Angle of Any point at signal propagation path, and n is atmospheric refractivity [7] [8] [9].Define N for the difference of atmospheric refractive index: In a spherically symmetric layered medium, according to Snell law, a formula is given by  , , n r z are the corresponding values of ground measuring site.The path delay is given by the product of the tropospheric zenith path delay and the function of satellite-zenith distance z (the so-called mapping function).Only considering the first order effect, mapping function f(z) is approximate to 1 cos z .

( ) ( ) ( ) cos
According to Saastamoinen model (1972) of the atmospheric refraction, the zenith delay of atmospheric refraction can be given by where is the dry air delay of the zenith (dry item), is the water vapor delay of the zenith (wet item), P is the surface atmospheric pressure (Pa), H is the atmospheric high (km) of ground point, T is the surface temperature (˚C), Φ is the geographic latitude of measuring station, E is the vapor pressure (hPa) [10] [11].
According to Magnus' empirical formula, the relationship between vapor pressure and relative humidity at the condition of air temperature T (˚C) can be expressed as

Ionosphere Delay
Due to variable insolation of the Sun the spatial distribution of the layers varies during the day.The impact of the state of the ionosphere on the propagation of waves is characterized by the Total Electron Content (E), where The integral contains the total number of electrons that are included in a column with a cross-sectional area of 1 m 2 , counted along the signal path ρ between the ground point P0 and the satellite S. The unit of measurement is the TECU (Total Electron Content Unit): 1TECU = 1 × 10 16 el/m 2 .
The ionosphere is a dispersive medium for radio waves.For the formula of dispersion (e.g.Davies, 1990) of the index of refraction, rearranging and neglecting higher order terms gives [12] [13] [14]: where the parameter a is a constant.According to Equations ( 7) and ( 8), the ionosphere delay along the signal propagation path is given by ( ) That is to say, if we can get the E(TEC) along the signal propagation path, we can calculate the propagation delay directly.

Analysis of Influence on Ranging Accuracy Exerted by
Atmospheric Refraction

Troposphere Atmospheric Refraction Error
The atmospheric refractive effect of Troposphere is independent of the frequency but is decided by the atmospheric temperature, pressure and humidity.Atmospheric refractive error of Troposphere includes the dry item and the wet item.Refraction error of Tropospheric atmosphere in zenith direction is about 2.0 -2.7 m, which is computed approximately (Figure 2).Among them, the dry item of distance error caused by the refraction is about 2.4 m and the wet item is about 0.05 -0.05 m.
The inland distribution of distance error caused by Tropospheric atmosphere refraction, which is obtained by using the historical meteorological sounding data of 10 years, is shown in Figure 3.
From the above research on the inland distribution of distance error, we can get some conclusions as follow: 1) When the elevation of the satellite is above 30˚ (viewed from the ground), the distance error caused by Tropospheric atmosphere refraction is mostly less than 5.5 m.
2) The dry item of refractive error can be estimated accurately from the ground meteorological parameters due to its regular and steady changes.The wet item of refractive error is the main part of the residual error for refractive error correction due to its irregular changing with time and space.
3) The proportion changes of distance error caused by the wet item of the refraction vary with month and quadrillage obviously but do not vary with elevation.

Ionosphere Atmospheric Refraction Error
The refractive error of Ionospheric atmosphere has a close relationship to systemic working frequency, degree of solar activity, etc.To take Haiko as an illustration, the changing of distance error caused by Ionosphere atmospheric refraction with solar activity and working frequency is shown in Figure 4, which is achieved by using the reference Ionosphere model.By the analysis, we can get the following conclusion: when the frequency is in the X band and the elevation of the satellite is above 30˚ (viewed from the ground), the maximal distance error caused by Ionosphere atmospheric refraction is about 0.5 m, even in high solar activity years.

Error Correction Technique for Atmospheric Refraction
Distance errors caused by atmospheric refraction must be corrected to meet the accuracy demand of slant ranging in InSAR system with distributed satellites.

The Overall Technical Process
The overall technical process of distance error correction for atmospheric refraction is shown in Figure 5.
The key technoique mainly includes the following three aspects: 1) Methods of radio wave refraction correction.
2) Researches on profile model of global tropospheric refractive index.

Research on profile model of global tropospheric refractive index and distance error correction for atmospheric refraction in zenith direction
Measurement and analysis on system error by using calibration field

Real distance from satellite to ground
3) Methods of system error calibration.

1) Refraction correction method of mapping function
In view of the high elevation of the system, we can use the refraction correction method of mapping function under the premise of the accuracy being not affected.
The method adopts a mapping function as the following form. where is the distance error of refraction in zenith direction, 0 θ is exterior angle, , and the feature height 2) Profile model of global tropospheric refractive index To achieve the global distance error correction for refraction, it is necessary to establish the profile model of global tropospheric refractive index.Firstly, a global range of refractive index profile model of statistical data is established by using a great deal of historical meteorological sounding data obtained from the global observation stations.Secondly, the statistical model is corrected afterwards to improve the precision of the profile model by using the measured refractive index of global.
3) System error calibration Microwave radiometer, which has a good real-time performance, can observe automatically and continuously on atmospheric environment in all-weather conditions.And it can directly calculate the radio wave refraction error of the propagation path by measuring the atmospheric radiation brightness temperature of the propagation path.It is shown in the study that the accuracy of refraction correction can reach 2 cm in the elevation of above 100˚.Accordingly, the refractive error which calculated from the microwave radiometer measurement data can be considered as the total error caused by the radio environment.
The famous Marcor technique, which measures the additional time-delay integral of atmosphere on the path of radio wave propagation with ground-based microwave radiometer, gives distance error correction directly [15].
Due to the fact that the elevation of the satellite with distributed SAR is more than 30˚, ray bending effects can be ignored.Therefore, the distance error caused by atmospheric refraction can be formulated as follow.

Figure 2 .
Figure 2. The global distribution of refraction error from ITU-R P.834.

Figure 3 .
Figure 3.The inland distribution of distance error caused by Tropospheric atmosphere refraction.

Figure 4 .
Figure 4.The change trend of the refraction error caused by Ionospheric atmosphere.(a) Change trend with frequency; (b) Change trend with solar activity.

Figure 5 .
Figure 5.The overall technical process.
d N is the dry item of atmospheric refractive index, w N is the wet item of atmospheric refractive index, d R ∆ is the distance refractive error caused by the dry item, and w R ∆is the distance refractive error caused by the wet item.