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The main limitations of standard nadir-looking radar altimeters have been known for long. They include the lack of coverage (intertrack distance of typically 150 km for the T/P / Jason tandem), and the spatial resolution (typically 2 km for T/P and Jason), expected to be a limiting factor for the determination of mesoscale phenomena in deep ocean. In this context, various solutions using off-nadir radar interferometry have been proposed by Rodriguez and al to give an answer to oceanographic mission objectives. This paper addresses the performances study of this new generation of instruments, and dedicated mission. A first approach is based on the Wide-Swath Ocean Altimeter (WSOA) intended to be implemented onboard Jason-2 in 2004 but now abandoned. Every error domain has been checked: the physics of the measurement, its geometry, the impact of the platform and external errors like the tropospheric and ionospheric delays. We have especially shown the strong need to move to a sun-synchronous orbit and the non-negligible impact of propagation media errors in the swath, reaching a few centimetres in the worst case. Some changes in the parameters of the instrument have also been discussed to improve the overall error budget. The outcomes have led to the definition and the optimization of such an instrument and its dedicated mission.

Topex/Poseidon, Jason and Envisat are currently orbiting, delivering data for ocean monitoring and forecasting. Addition of spatial radar altimetry observations to in-situ measurements and models has given evidence of success for the last decades (see [

A new and revolutionary altimetric measurement has been assessed for the past few years (see [

In the first section, we give an overview of the geometric layout of such concepts mixing altimetry and interferometry. We assess the derivation of the observed target features (position and height). Subsequently, based on the physics of the interferometric measurement for a given configuration (frequency, chirp bandwidth, antenna length, interferometric base, etc.), we set-up the instrumental error budgets leading to the performance estimation. Instrumental height error profiles are finally simulated for several processing configurations. The above-mentioned approach is primarily applied to the nominal profile of the WSOA onboard Jason-2.

In the second section, the impact of external error sources is evaluated: platform (attitude, atmospheric drag), and propagation media (ionospheric and tropospheric effects). Subsequently, we focus on the required quality of a specific ground processing procedure dedicated to the roll angle determination, which is shown to be a critical aspect of the concept.

A data simulation case is presented in the third section. It is based on real data acquired during the tsunami episode of December 26^{th}, 2004. Some of the possibilities offered by such concepts are illustrated.

Finally, the last section develops a new set of instrument characteristics and dedicated mission by taking into consideration the outcomes of the previous error budget studies. An overall satellite configuration (platform + payload) is proposed, which optimizes the existing technologies in order to much better answer the scientific objectives.

Before going into some details in the technique of wide swath interferometric altimetry and into the optimization of a proposed space mission to primarily observe ocean mesoscale features, let us recall one main requirement at mission level. Indeed, to observe ocean mesoscale features and assimilate observations in models usefully, it is now agreed that the lower the nadir altimeter instrument range noise, the better mesoscale features will be retrieved, assuming that resolution is provided thanks to multiple orbiting instruments in a well-phased configuration. The optimal range noise level is at the level of some 2 cm, which is a figure that is reached by the well known TOPEX, POSEIDON-1, POSEIDON-2, ENVISAT RA-2 instruments in typical situations characterized by a 2 meter significant waveheight and a 11 dB backscatter coefficient. As far as swath instruments are considered, estimating the measurement noise is a multi-parameter task that will be described in the coming sections. For the purpose of data assimilation in ocean models, it is important to have a good knowledge of the statistics of the error measurements over the entire swath. Previous use of wide-swath simulated data in a high-resolution ocean model have shown that a 4 to 5 cm overall noise level may be adequate to retrieve ocean signals at the same level as would be achieved by a constellation of 3 to 4 nadir instruments (see [

Both swaths are the results of an off-nadir observation by two passive antennas separated by a mast (see

The purpose of this subsection is to figure out how the real height and position of an observed target within one given swath is derived from the collected data. The determination of the altimetric measurement requires the precise knowledge of four main parameters: the altitude _{1}

As illustrated on

The height measurement _{real}_{real}

POD performances directly impact the knowledge of _{1}_{real}

The interferometric phase measurement quality is strongly related to the coherence value γ_{T} between the two receiving channels. This coherence is mainly affected by three factors: the instrument signal to noise ratio, the range migration within the swath, and the speckle (its effect is also called geometric decorrelation (see [

The interferometric phase statistics are distributed according to the Wishart distribution (see [

_{T} for a single look and

The phase variance can be numerically calculated from

This approximation strengthens the need for a large degree of coherence. The next subsection will confirm this behaviour when entering the details of the WSOA operating characteristics.

Therefore, the instrument SNR has to be studied carefully, as the main contributor to the decorrelation between the two channels. The analysis of the derived degree of coherence in the swath will then fix the multilooking strategy to be implemented. Indeed, the objective of such a strategy will be to reach the desired standard deviation of the estimated interferometric phase that keeps the height performances within the range of constraints specified by the targeted scientific objectives.

The signal to noise ratio for an extended target (a resolution cell) can be expressed as follows, where the different terms refer respectively to the range loss, the EIRP (Equivalent Isotropic Radiated Power), the effective antenna surface, the range resolution, the azimuth resolution, a global addition of different losses related to the instrument RF and the atmosphere, the reception channel noise, the range compression gain and the observed scene contribution through the backscattering coefficient (see [

Based on published specifications for WSOA ([

The 4 to 5 cm objective can be reached when the averaging process is applied over 16*14 km pixels (see

Another major objective is to get a quality of measurement as uniform as possible over the swath. However, degradations caused mainly by the range loss, the backscattering coefficient and the range resolution are far more important at the edge of the swath. The center of the antenna beam has then to be oriented toward the far range of the swath in order to maximize the gain there and compensate these losses.

The required instrumental error is so stringent that a large amount of multilooking is necessary. There is a modification that could be brought to the WSOA operating characteristics that could help get slightly better coherence, and allow us to reduce the number of looks and thus improve the intrinsic spatial resolution (see section 5).

However, the WSOA operating characteristics cannot be completely assessed and discussed without determining the impacts of external factors on the error budget. This is the purpose of the next section.

Only instrumental effects on the error budget have been reported yet. External effects need to be analysed and associated errors shall then be incorporated in the overall budget. The ones stemming from the errors on the POD and the radiometer measurement as well as the electromagnetic bias are already well known thanks to current missions. Their in-depth study is not the purpose of this section.

First of all, we will focus on the influence of the platform attitude. Two components of the three dimensional attitude angles are considered. On the one hand, yaw rotation is considered: indeed, as selected orbits can be sun-synchronous or not, the use of a yaw motion may be required to supply energy to the carrier satellite. It will be shown that a yaw rotation significantly degrades the spatial coverage; on the other hand, the impact of a roll motion is estimated, and will be shown to hugely amplify the height error.

Last, electromagnetic wave propagation perturbations through the ionosphere and the troposphere will be statistically reviewed. We will consider worst case scenarios as is usual when evaluating instrument performances and dimensioning systems.

The Jason-2 orbit, as the ones from Topex/Poseidon and Jason, has been constrained to be non sun-synchronous to cope with the tidal aliasing. However, this kind of orbit requires a strong attitude control onboard so as to keep the solar panels directed toward the Sun, by driving the yaw angle rotation around the axis directed toward the Earth, called yaw steering (see _{0}_{az}

Furthermore, these rotated configurations of the resolution cells deteriorate the averaging process and the formation of clean final pixels. If we only consider valid data at zero yaw angle, the global functioning time reduces to

Finally, this orbital configuration leads to the orientation of the solar panels perpendicular to the satellite velocity. The atmospheric drag has known effects causing the semi major axis to drift and then the ground track to move, as follows (see [

Sticking to classical repeat orbits that remain located within a +/- 1 km band from a nominal ground-track (to make easier the study of time-varying phenomena), we can then infer the frequency of maneuvers to maintain the orbit as it is required.

The required rotation of the panels around their symmetric axis extends the surface drag of the satellite, accelerating the frequency of maneuvers and then reducing the satellite life-time or degrading the mass and cost budgets. In addition, the rotation of the interferometer antennas around the yaw axis further increases the drag.

As far as the interferometric altimetry system is considered, flying a sun-synchronous orbit would eliminate many of the drawbacks of flying a non sun-synchronous one: (i) the observation geometry would be kept constant, (ii) the carrier platform could be made very stable and the sun-satellite relative geometry would be always optimal to provide energy to the satellite, avoiding negative impacts on the interferometric error budget. The overall impact of this change of orbit, and especially the problem of tidal aliasing will be considered in section 5.

The range value between the target in the swath and the master antenna _{1}

The ionospheric delay is mainly dependent of the Total Electronic Content (TEC) and the radar signal frequency. The nadir altimeter, which now commonly operates at two frequencies, has thus the ability to determine the TEC along its path and in the mean time the ionospheric delay. However, this measurement only applies in the nadir direction, and not in the off-nadir direction. Therefore, there is an error in applying the nadir ionospheric delay to the interferometer master antenna range.

The ionosphere is known to have variability with spatial wavelengths of the order of one thousand kilometers. However, in order to get an accurate error budget, the maximum amplitude of TEC variations over about 100 km (FR) must be determined (worst cases consideration). Dual-frequency ionospheric data have already been computed along nadir altimeter paths (see [

Statistics show that ionospheric errors (difference between the nadir and off-nadir delays) above 0.7 cm in the Middle Range (MR) and 1 cm in the Far Range (FR) are common, especially in the tropics. Then, the use of worldwide GPS data in the ground processing segment can be an appropriate solution to solve such an issue.

The case is similar for the tropospheric correction, which is derived from measurements of a three-frequency nadir radiometer (18.7, 23.8 and 34.0 GHz) that is a standard part of an optimal altimetry payload. The amplitude of the variations of the water vapor content of the troposphere have been looked at through the computation of SSMI (Special Sensor Microwave Imager) data (see [^{th} of September is clearly visible in the tropospheric error plots (see arrows).

The statistical distribution of the errors when going through a typical large atmospheric feature (a few hundred of kilometers wide) has been computed (see

The roll angle plays a major role in the height estimation process (see I.1): a roll error linearly impacts the height measurement thanks to the across track distance in the swath x according to:

An a priori knowledge of the roll angle spectral characteristics is provided by the PAD system onboard the satellite. However, the required accuracy of less than a tenth of an arc second is so strong that ground post processing procedures using crossover data need to be done. A crossover is an area where an ascending track crosses a descending track. In the case of WSOA, swath/swath, nadir/swath and nadir/nadir crosses shall be considered as calibration points (see

Rodriguez and al ([

The main idea of the crossover estimator is that the over flying time of the crossovers zones are small enough to consider a linear evolution of the roll angle with time (see

Four parameters (two for the ascending track, two for the descending track) are then to be estimated:

The interferometer noise and the external noises are derived from previous sections. The nadir altimeter noise is assumed to be 2 cm (see [

“A-D” (ascending minus descending) differences in _{i}, which can be represented using the following:

The histogram in

Processing crossovers may also enable us to correct the linear part of the ionospheric and tropospheric gradients errors in the swath, which would then be assimilated as an additional roll angle.

Results from the last two sections eventually will provide an overall performances budget (see ^{-6}.

On a worst case basis, the contribution of the instrumental error (see

A primary advantage of such a WSOA system is that data are collected in two dimensions, namely along-track and across-track. Then surface gradients and Laplacian can be computed, leading to computations of the geostrophic velocity and vorticity (see [

A way to illustrate the usefulness of such a system has been to simulate the observations that could have been done at the time of an extreme ocean event. The case of the December 2004 tsunami over the Indian Ocean is such an example. Indeed, the Topex/Poseidon (T/P) and Jason satellites flew over the area of interest a few hours after the beginning of the seism. However, the direction of the tsunami, as well as an estimation of its power could not be derived from the acquired data. We simulated observations of such a tsunami assuming that an interferometer is implemented on the two platforms, based on CEA (Comité à l'Energie Atomique) outputs of a tsunami propagation model (which has been well validated using the real T/P and Jason-1 altimeter data).

The tsunami started at 0 h 58 GMT at the top North of Sumatra Island. Jason went over the equator, at 85.7° East longitude 116 minutes later. Finally, Topex/Poseidon went over 7 minutes later, at 84.3° East longitude. For the purpose of the simulation, the SLA (Sea Level Anomaly), referring to the height of the water above the MSS (Mean Sea Surface), have been computed. Simulations were performed with post processing resolution of 16*4 km (21 pixels in each swath), which is the best compromise between quality of the height budget and minimal density of measurements to detect small-scale phenomena.

The simulation takes into account the overall instrumental and external factors studied, with the worst cases in propagation errors and attitude errors.

The direction of the tsunami wave can be easily determined from the height profiles derived from the simulated WSOA data on the Jason platform (see

It confirms the strong advantage of getting rid of any platform rotation by moving to a sun synchronous orbit.

Let us now assume that the scientific objectives remain the same as for WSOA onboard Jason-2. Some other applications can be considered over coasts and continental basins (typically some tens of square kilometers), but the weak azimuth resolution of several kilometers prevents from studying small scale surfaces. The study of these other potential applications is not the purpose of this paper. They do not provide us with major constraints on the following optimization of the instrument and the dedicated mission. The additional requirement of observing the entire ocean surface between -80 and 80 degrees latitude is brought up. A secondary objective is to maximise the use of technologies that have already been developed, like the mast and the TWTA (Traveling Wave Tube Amplifier).

We focused on areas between 0 and 60 degrees latitude (North and South), as higher latitude areas are well sampled and do not constrain the spatial and temporal coverage. The histogram in

The 9.9156 solar day repeat period of T/P and Jason was chosen to minimize the effects of tidal aliasing (see [

For a given interferometric phase uncertainty, the height error is proportional to the wavelength and to the inverse of the interferometric baseline (see

From an ionospheric perspective, the presence of a dual-frequency nadir altimeter onboard the platform is required as the interferometer uses signals whose carrier frequency is in Ku Band or below. Indeed, it is the only way to very accurately cope with the ionospheric correction. However, section II.2.a showed that the interferometer ground processing procedures could take benefit from the use of GPS data in some specific areas in order to extend the correction throughout the swath.

A three-frequency radiometer is of course required; however, using external data such as SSMI or numerical weather prediction analysis to improve the wet tropospheric correction may be considered, especially at low latitudes and in the middle of large storms where the errors can reach more than 2 cm more than 20 % of the time (see Section II.2.b).

The only interest in enlarging the chirp bandwidth would be to get a better resolution in the range direction, which would cost a lot in terms of link budget (leading to degraded performances), power budget and telemetry budget (see [

Conversely, the pulse length should be further checked. It was constrained on Jason-2 by the presence of the nadir altimeter and the telemetry budget. Keeping the nadir altimeter in the new proposed mission freezes the PRF and then constrains the pulse length as well. However, the telemetry budget can be improved and therefore frees a degree of freedom on the pulse length (see [

As it was done in Section 3.4, the worst cases external errors are brought up in the overall height budget related to the instrument operating characteristics, which can then be compared to the one on Jason-2 (see

By changing the instrument operating characteristics and the dedicated mission parameters, performances equivalent to a WSOA on a 16*14 km pixel are now likely to be obtained on 12*1 km pixel. The decreasing of the intrinsic surface area has been done by almost 95 %. This huge profit enables to get more hits to phenomena at Rossby radius of deformation scales. Furthermore, coastal observations are improved in the worst case of a perpendicular approach as now the limit distance is no more than 8.4 km. Finally, the 4-5 cm noise level specified in section I is almost reached throughout the whole swath for a 12*4 km pixel; FR data errors fill the constraint when averaging up to 6 km in the range direction (keeping 12 km in azimuth) or up to 13 km in the azimuth (keeping 1 km in range).

Wide Swath Altimeters must be part of the future of the overall altimetry observation system. They enable us to get a global coverage of the Earth between -80 and 80 degrees of latitude and an appropriate temporal sampling, required to observe mesoscale phenomena and improve ocean monitoring and forecasting to an unprecedented level. Though their error budget is greater than for a nadir altimeter, and the errors are partially correlated across-track, these new altimeters are the only ones capable of providing us with a global two-dimensional topography of the ocean. Unfortunately, the demonstration of such revolutionary systems could not be incorporated as part of the Jason-2 program. However, studies of such instruments have continued (see [

The studies performed and presented throughout this paper have led to a better knowledge of the performances of this interferometer and its interactions with all kinds of external factors, finally enabling us to optimize the instrument parameters as well as the definition of the overall mission. Different levels of accuracy associated with different pixel resolutions can be achieved. In addition, integrating overlapping passes over the whole mission lifetime will enable to get even better accuracy.

There have been a number of people whose assistance has made this research possible. V. Enjolras especially likes to thank CNES and ALCATEL for financial and technical support. He would also like to thank Bruno Cugny (CNES) and Brian D. Pollard (NASA/JPL) for fruitful discussions and comments.

Here is shown how to derive the onboard yaw angle and his effects on the interferometer swath.

Let's first define the frames in which we are working:

the local orbital frame ROL(t), with :

Z_{ROL} : in the direction Earth/Satellite, towards the Earth(- yaw axis)

Y_{ROL} : orthogonal to the orbital plane, opposite to the angular momentum of the satellite (- pitch axis)

X_{ROL} : to complete the trihedron (roll axis)

the satellite frame SAT(t), in rotation around-Z_{ROL} with an angle Ψ

In order to optimize the solar panels orientation, the platform first has to be oriented for –X_{target} to be directed toward the Sun. The following spherical triangle illustrates this:

The formula related to spherical triangles leads to:

There still remains a π uncertainty on the knowledge of the yaw angle. Another condition is going to help erase this ambiguity. –X_{target} has to face the sun, therefore we have:

In the local orbital frame, we can write:

Finally, a last condition needs to be taken into account: when β is less than 15° in absolute value, there is no yaw steering. Computing all that gives the plot on

The effects on the swath can then easily be calculated as followed (see

B | Interferometric baseline |

θ | View angle |

r / R | Range |

X | Position in the swath |

h | Height above the tangent plane at nadir |

α | Roll angle |

H | Altitude of the satellite |

Φ | Interferometric phase |

k | Wavenumber/Boltzman constant |

R_{n}/R_{n} |
Earth radius nadir/Earth radius target |

h_{real} |
Height in the local frame above the ellipsoid |

X_{real} |
Target position in the local frame |

A | Ellipsoid arc between nadir and target |

γ | Correlation between the two channels |

N | Number of looks |

P_{t} |
Transmitted power |

G | Antenna gain |

λ | Wavelength |

L_{az} |
Antenna length |

T_{pulse} |
Chirp length |

B_{d} |
Chirp bandwidth |

σ_{0} |
Backscattering coefficient |

L_{0}/L_{Ψ} |
Swath length |

R_{az} |
Azimuth resolution |

Ψ | Yaw angle |

β | Angle between the Sun and its projection on the orbital plane |

ν | Angle between the projection of the Sun on the orbital plane and the satellite |

f_{D} |
Doppler frequency |

V_{sat} |
Satellite velocity |

n | Orbit pulsation |

ρ | Atmospheric density |

C_{D} |
Atmospheric drag coefficient |

μ | Gravitational constant |

a | Semi major axis |

Ω_{T} |
Earth rotation velocity |

X_{A/D} |
Across track coordinate in the ascending/descending local frame |

Y_{A/D} |
Along track coordinate in the ascending/descending local frame |

N_{interferometer/nadir_altimeter/ocean/tropo_iono} |
Height error brought by the interferometer/nadir altimeter/ocean temporal decorrelation/tropospheric and ionospheric delays |

(a) Lay-out of the geometry of observation (b) Antenna and feeds configuration arbitrarily for the right swath; on the other side, feeds' polar are reversed (Emission/Reception H Feed and Reception. V Feed); a V-polarization pulse is emitted by one antenna, and the consecutive pulse, with H-polarization, is emitted by the second antenna; both polarizations are received by the two antennas.

(a) Geometry of the interferometric measurement (b) Local Frame Transformation.

Interferometric phase densities of probability versus coherence (a) and multi looking (b).

Statistics of the interferometric phase in multilooking context.

(a) WSOA signal coherence throughout the swath with and without the onboard correction of the geometric decorrelation (b) Single looking instrumental noise impact on the height budget.

(a) Number of looks associated with different incoherent average processing procedures in the range direction (b) Associated WSOA instrumental noise impact on the height error (the digitization step in the number of looks is related to the intrinsic range resolution, that results in discontinuity when considering 1 km pixels).

Antenna Pattern Orientation Optimization.

(a) Yaw steering onboard Jason-2 over a year (b) WSOA Swath Width over a year. Note the yaw and swath width vary between the minimum and maximum of their envelope within each orbital period.

(a) Evolution of the Swath over 13 orbits at the beginning of yaw steering (b) Width of the swath over 10 days in the worst case (-75°<β<-73°).

(a) Impact of the yaw steering onboard Jason-2 on the predicted observing time (b) Impact on the signal processing in terms of Doppler centroid.

Evolution of the different contributions to the overall satellite surface drag.

(a) Example of the spatial distribution of the FR ionospheric errors (differences between the nadir and the off-nadir delays) in an intermediate case at midnight in April 2000 (b) Maximum ionospheric errors in the swath (Middle Range and Far Range) in different cases.

(a) Zonal variations of the Far Range mean tropospheric error in October 2003 (b) Worst cases tropospheric errors during fall 2003 with evidences of the Hurricane Juan (orange arrows).

(a) Spatial differences in wet tropospheric delays over a large atmospheric feature (typically a few hundred kilometers wide) in mm (b) Statistical distributions of the tropospheric errors over the feature.

Geometry of crossovers measurements and definition of the appropriate local frame.

(a) Latitudinal Dependence of the interval between successive crossovers (b) WSOA Jason-2 crossovers characteristics: blue refers to ascending tracks first and red to descending tracks first; each rectangle corresponds to one crossover, its width defining how long it is over flown.

(a) Influence of the orbit characteristics on the quality of the processing (b) Influence of the presence of nadir altimeter on the quality of the processing.

Influence of the pixel range resolution on the quality of the estimation process.

Global error budget of WSOA in worst and best cases for 16*14 km

(a) Simulated sea level anomaly as would have been observed by an interferometer onboard the Jason platform over December 2004 tsunami and determination of the wave direction (yellow arrow) (b) 3D view of the observations (azimuth, range, height).

Simulated interferometer data on T/P platform with yaw steering and Poseidon 2 data to the right of it.

(a) Number of Hits per cycle of repeatability for a 815 km sun-synchronous orbit over the Indian Ocean (b) Temporal coverage of this orbit between 0 and 60 degrees latitude.

Tidal aliasing comparison between Jason and the proposed sun synchronous 14-day cycle.

New operating characteristics for improving the height budget.

Effects of the yaw angle on the interferometric swath.