Accuracy analysis of multi-base InSAR altimeter in ocean surface relative elevation measurement

: A new sea-level elevation measurement system was introduced, which uses a distributed multi-base forward- scattering mechanism. After introducing the working principle of the system, this study analyses the main influencing factors of relative elevation measurement in detail. It is proved through quantitative expression that the interferometric phase error is the main error source affecting relative elevation measurement. In order to evaluate the system capability and prove that the multi-base InSAR ( MbInSAR ) can meet the application requirements of gravity field inversion through relative ocean surface elevation measurement, three aspects of decorrelation, geometric decorrelation, volumetric decorrelation, and SNR decorrelation, are analysed as the interferometric phase error sources.


Introduction
Satellite radar altimeter can be used for sea-level measurement and can obtain the global ocean geoid. It is one of the main methods for data acquisition in marine dynamics and marine environmental studies. The geoid is the equipotential surface of the gravitational field that is closest to the global average sea surface, which reflects the information on the structure, density, and material structure inside the earth, and has a certain correlation with the topography of the seabed [1,2].
Precision altimetry missions such as TOPEX/Poseidon, Jason-1, OSTM/Jason2, and Jason3 have provided unprecedented observations of the ocean surface topography at scales larger than about 200 km, which is normally used to study large-scale (>300 km) oceanic processes [3,4]. However, the coarse cross-track sampling and measurement precision have prevented resolving scales <100 km, the sub-mesoscales that are important for understanding the dynamics of the ocean kinetic energy and the vertical transfer processes in the ocean that account for 50% of the exchange of water properties (nutrients, dissolved CO2, heat) between the upper and the deep ocean [5]. The SAR altimeter, with the launch of Cryosat-2, also became mature [6]. Compared with traditional radar pulse altimeters, SAR altimeters have higher spatial resolution in orbital direction, and their accuracy has been improved. However, they still do not solve the problem of low accuracy of the east-west gravity field products and cannot achieve both high temporal resolution and spatial resolution simultaneously.
The interferometric synthetic aperture radar altimeter, represented by the SWOT system, can use image interference to achieve surface topography of wide swathes. With a better accuracy and efficiency, InSAR radar altimeter has gained widespread attention [7][8][9].
The interferometric imaging radar altimeter (InIRA) on board Chinese Tiangong-II space laboratory is capable of measuring the dynamic sea surface height with wide swath. InIRA is the second space-borne InSAR that can perform single-pass interferometry after SRTM. The two antennas work at Ku-band and form a baseline of 2.3 m. The theoretical accuracy of the system can reach 7 cm. However, due to the effect of residual calibration error and data processing-induced error, the final measurement accuracy is nearly 40 cm [10,11].
Limited by the size of the Tiangong II capsule, the baseline length of InIRA is only 2.3 m. In order to improve the altimetric precision, a long baseline is needed. Dual/multi-base InSAR (MbInSAR) can realise a long baseline easily. A new MbInSAR altimeter using the forward-scattering is been designed. Unlike other altimeters, MbInSAR altimeter is specially designed for marine gravity field detection, which means it only concerns about the ocean surface relative elevation. This paper will perform a systematic analysis of the relative elevation precision based on the MbInSAR altimeter system. This paper is organised as follows: Section 2 gives a brief introduction to the MbInSAR altimeter system. Section 3 analyses the absolute elevation measurement principle of the system. Then, a quantitative analysis of the relative elevation measurement accuracy is performed. In Section 4, the source of the interferometric phase error which is the main influencing factor of the accuracy of relative elevation measurement is analysed. Finally, the performance of the system is evaluated and summarised.

MbInSAR altimeter system
The MbInSAR ocean surface elevation measurement system adopts a forward-scattering and a three-satellite formation flying scheme with one-transmit-dual-receive operation mode to obtain sea surface elevation information. The geometry of the system is demonstrated in Fig. 1.
In Fig. 1, the red hexagon represents the transmitting satellite and the two blue hexagons stand for the receiving satellites. The direction of the three satellites is perpendicular to the paper surface, the transmitting satellite is far away from the receiving satellites, and the two receiving satellites are close to each other, forming a one-transmit-dual-receive working mode.
In the SAR geometry working for an ocean scene, most of the energy escapes from specular scattering. The backscattering can only obtain a small part of the energy. Theoretically, the backscattering coefficient of the calm water surface is zero. Therefore, the forward scattering system can make a good use of the radiation energy of the transmitter.
In order to meet the high-accuracy ocean surface elevation and wide-swath measurement requirements, the transmitter emits microwave signals to the left and right sides at an angle of incidence of 10° to 20°. After being forward-scattered from the sea surface, it is received by two receiving satellites which formed a cross-track effective baseline about 1 km. The transmitted signal pulses cross-cover both sides of the specular reflect point. The width of both sides is 63 km and HH and VV polarisations are adopted, respectively. The SAR image has an azimuth resolution of 13 m and a slant range resolution of 0.5 m. The range resolution is improved along with the distance from the specular reflect point to the imaging range cell, ranging from 28 to 2 m, and the relative elevation measurement accuracy is designed to be better than 1 cm.

Absolute elevation measurement principle
The interferometric phase used by the InSAR is essentially determined by the two-way path difference of the radar signal. The MbInSAR altimetry system has only one transmitter, so the path difference comes from the reflection path. Fig. 2 shows the working geometry of the bistatic receiving system.
In Fig. 2, coordinate system R S1 − x′y′z′ is the centroid orbit coordinate system of satellite R S1 . The y′-axis points to the opposite direction of flight. The reverse extension of the z′-axis passes through the centre of the earth O. H presents the distance between R S1 and O. P is a scatter within the swath, which is at the beamcentre of the transmit antenna. After scattered by P, the microwave signal is received by R S1 and R S2 , respectively, and the reflected wave paths are r 1 and r 2 correspondingly. The length of the baseline between R S2 and R S1 is B. The azimuth and elevation angles of the baseline B defined in the R S1 − x′y′z′ coordinate system are α and β, respectively. The squint angle from R S1 to target P is θ. ψ is the azimuth dihedral between the R S1 OP plane and the R S1 z′x′ plane (z′ − x′ plane). In the coordinate system defined above, the difference Δr = r 2 − r 1 can be expressed as: in which, A = 1 − cos 2 βsin 2 α + ψ , and sin β^= sin β/ A. Interference phase can be expressed as The distance between P and Earth centre is h = H 2 + r 1 2 − 2Hr 1 cos β^α, β, ψ − arcsin λΔϕ 2π A α, β, ψ B Projecting the interference baseline can yield three important baseline components, namely the range effective baseline B ⊥ r , the azimuth effective baseline B ⊥ a and the horizontal baseline B ∥ . B ⊥ r is the length of the projection of the baseline vector to the R S1 antenna distance-height plane and then to the r 1 perpendicular direction. [12] B ⊥ r = ABcos β^− θ Defining the azimuth vertical effective baseline B ⊥ a as the component of the spatial baseline perpendicular to the R S1 antenna distance-height plane, we have The horizontal baseline B ∥ is the length of the projection of the baseline vector to the R S1 antenna distance-height plane and then to the r 1 direction.

Elevation measurement accuracy
This system is concerned about the relative elevation measurement accuracy. Suppose there are two pixel cells E and F in a small imaging area, as shown in the following Fig. 3. The distance from the E to the R S1 antenna is r E , the squint angle corresponding to the R S1 antenna is θ E , and the ψ E is the azimuth dihedral angle between the R S1 OE plane and the R S1 z′x′ plane. [13] Target F is similar and satisfying For pixel cell E, the phase difference of the two-antenna interferometry can be expressed as which is ψ stands for the beam-orientation azimuth angle. Similarly, the interference phase produced by the pixel cell F can be expressed as The difference of the interference phase between pixels E and F is When acquiring SAR images, any pixels in the same range distance experience the same beam coverage history. The obtained imaging results indicate the phase of the target pixel covered by the centre of the azimuth beam. As a result, there is no difference between ψ E and ψ F . Therefore, Δψ does not appear in the Δϕ F and Δϕ EF expression. For a small image area, r E ≈ r F . The relative elevation of pixels E and F can be expressed as: then, we have in order to separate the variable parameters, the baseline will be handled in a unified manner, which is B ⊥ r . It can be seen from the above equation that the relative elevation of the pixels E and F is related with the incidence angles θ E , the range effective baseline length B ⊥ r , the interference phase difference Δϕ EF , and the distance r E . All these relevant quantities which affect the relative height measurement accuracy can be analysed by the error transfer coefficients.
The error transfer coefficients can be expressed as Compared with the expression of the absolute elevation of the transmission coefficients of relative height error, all the error transfer coefficients are multiplied by a coefficient of λ B ⊥ r except for Δϕ EF , resulting in a significant decrease in the error conductance of other errors on relative elevation measurement.
In summary, for a small block of imaging area, except for the interference phase error Δϕ ε (which is express as Δϕ EF above), the influence of other errors on the two receiving paths is basically the same and can be considered as a common error. Therefore, the relative height error h ε is determined by Δϕ ε only, i.e.

Source of the interferometric phase error
The interferometric phase error Δϕ ε is closely related to signal-tonoise ratio (SNR), baseline decorrelation, and volumetric decorrelation [8,11]. The coherence coefficient γ can be expressed as in which, γ SNR indicates the system noise decoherent, γ v indicates the volumetric decorrelation, and γ G indicates geometric decoherence. The specific expressions are in which, SNR represents the signal-to-noise ratio; σ H represents the standard height distribution of the sea surface, which is related to the significant wave height (SWH) (SWH = 2 m is indicated in the SWOT scientific demand) [3,5], usually we use SWH = 4σ h ; r denotes the average distance of the transmitting and receiving path; ρ r denotes the distance resolution. The interference phase error Δϕ ε can be approximated by the coherence coefficient γ of the two receiving satellite images: N represents the multi-look number. The spatial resolution of sea surface elevation measurement is generally in the order of kilometre, and the spatial resolution of MbInSAR altimeter based on SAR imaging principle is about 10 m. It can be seen from (17) that the interferometric phase error can be reduced by 1/ N through two-dimensional multi-look processing, and the interferometric phase error can be reduced by 21 to 25 dB under the MbInSAR system parameters, which greatly improves the relative elevation measurement accuracy.

Conclusions
This paper introduced the multi-base forward scatter sea-level elevation measurement system and analysed the accuracy of the elevation measurement, especially the relative elevation measurement accuracy, as the MbInSAR altimeter is designed for the gravitational field detection. The results show that the main factor affecting the relative height measurement accuracy is the interference phase error. Compared to traditional terrain 3D mapping of InSAR, the effects of other error sources such as baseline length errors are significantly reduced. This work has important implications for guiding subsequent system design.