The GOCE mission concept has been conceived and designed to provide the most accurate, global and high-resolution map of the gravity field and its correspond-ing geoid surface, takcorrespond-ing into account the research objectives discussed in the previous section jointly with advanced technological solutions. It will combine the satellite gradiometry (delivering medium to short wavelength observations) and the satellite-to-satellite high-low tracking (SST-hl, delivering long to medium wavelength observations) techniques that have been found to be optimum for providing the required high-quality, high-resolution staticgravityfield.
The quantitative requirements for the different multidisciplinary scientific goals are summarized and expressed in terms ofgeoid heightand gravity anomaly ac-curacies in Table II, together with the corresponding spatial resolution to which they apply (expressed in half-wavelengths). The possibility to recover thegravity anomaliesandgeoid heightto these accuracies is further highlighted in this section based on closed-loop and end-to-end simulations (Oberndorfer et al., 1999; Sünkel
TABLE II
The requirements (discussed in Sections 2 and 3) expressed in terms of geoid height and gravity anomaly accuracies (Orbits∗: 1 cm radial orbit error for altimetric satellites)
Application Accuracy Spatial resolution
Geoid Gravity (have wavelength, (cm) (mGal) D, in km) Solid earth
Lithosphere and upper-mantle density structure 1–2 100 Continental lithosphere:
• Sedimentary basins 1–2 50–100
• Rifts 1–2 20–100
• Tectonic motions 1–2 100–500
Seismic hazards 1 100
Ocean lithosphere and interaction with asthenosphere 0.5–1 100–200 Oceanography
– Short 1–2 100
0.2 200
– Basin scale ∼0.1 1000
Geodesy
– Levelling by GPS 1 100–1000
– Unification of world-wide height systems 1 100–20,000
– Inertial navigation system ∼1–5 100–1000
– Orbits∗ ∼1–3 100–1000
Sea-level change Many of the above applications,
with their specific requirements, are relevant to studies of sea-level change.
et al., 2000). These simulations, which include thetime-wise approach (Balmino et al., 1999), reveal the precision of the spacecraft position recovered from SST, the precision of the gradients obtained from the gradiometer, and subsequently the accuracy of thegravitymodel that will be derived under the baseline mission and instrument characteristics (Touboul et al., 1999).
The analysis of the mission performance consists of two major stages and is supported by a mature, although complicated, overall procedure of data processing (CIGAR, 1996; Sünkel et al., 2000). First, the reliability and quality of the satellite measurement system is established. This is achieved by a comprehensive analysis and simulation of the gradiometer and SST-hl receiver characteristics themselves, and of their interaction with the satellite and the satellite environment. The raw data (Level 0) contain the readouts from the instruments, together with calibration,
374 J. A. JOHANNESSEN ET AL.
time, attitude and orbit control system and drag-free control data, and other house-keeping information such as temperatures. The data are pre-processed (Level 1a) and used together with calibrated and correctedgravity gradients, linear accelera-tions, thruster activity parameters, and attitude, angular velocity and acceleration information (Level 1b data).
Secondly, this, together with a first estimate of the GOCE orbit, make it possible to refer thegravity gradients to an Earth-fixed reference frame. In so doing, use is moreover made of accurately defined reference systems and the precise eph-emerides of the GPS satellites (as computed on a routine basis in the framework of the International GPS Service). Once processed, the Level 1a and Level 1b data will be further transformed and processed to the about 90000 globalgravityfield andgeoidmodel parameters (Level 2). Then the stability of the computation and the propagation of the error characteristics from the data along the orbit at altitude to thegravityfield parameters on the Earth’s surface will be assessed. The Level 2 products will in turn be used as a starting point for scientific analyses (Level 3) as already addressed in Section 3.
The mission and instrument characteristics are summarized in Table III. As already mentioned, the reference GOCE orbit is a Sun-synchronous, circular, dawn-dusk low Earth orbit with an inclination of 96.5◦ at a design altitude of 250 km. The mission lifetime is 20 months for a total measurement duration of 12 months (3 months commissioning phase, 5 months hibernation due to eclipse).
Using a drifting orbit (no exact repeat) a dense coverage with a pattern of inter-ground track distances not larger than 85 km at the equator can be obtained after 30 days. This may be made denser with time depending on the altitude window se-lected. The smallest recoverable wavelength (as shown by the simulation) requires a measurement sampling interval of 4 seconds. A one-second sampling time is recommended, which is not demanding on the overall system. Drag compensation, which is essential to enhance the mission capabilities, is assumed. Further informa-tion about the mission preparainforma-tion, including status of the instrument development can be found at the web site http://www.esa.int/livingplanet/goce.
The fundamental observables that will be used for the reconstruction of the gravityfield by means of thegravitygradiometry technique are the three diagonal components of thegravity gradienttensor expressed in Eötvös (1E =10−9s−2).
Their measurement errors, as affected by instrument errors, instrument-satellite coupling errors, satellite errors and post-flight on-ground processing errors, in the measurement bandwidth (frequency range) from 5 mHz to 100 mHz, are found to fall below 5 mE/ Hz1/2. Both the gradiometer and the satellite performance have been optimised to minimize the errors of thegravity gradients(i.e., Equation (2.2.6)) in the measurement bandwidth.
The recovery of thespherical harmonic coefficients, geoidandgravity anomaly error spectra have been simulated using covariance propagation analysis (ESA, 1999). Stabilisation of the resulting system of linear equations has been achieved making use of error degree variances of a recent global model (satellite only
solu-TABLE III
Baseline mission and instrument characteristics. The errors in position are obtained by including the errors of real GPS phase measurements, of the propagation of dynamic model errors and of the IGS orbits (Visser and IJssel, 2000). The noise characteristics of the accelerometer take account of most recent instrumental studies and develop-ments (Touboul et al., 1999) for spacecraft (attitude, self gravity) and system error sources (Aguirre-Martinez and Cesare, 1999). The orbit error values are r.m.s. values derived from a coloured error power spectral density
Measurement duration 12 months
Orbit almost circular, 250 km, 96.5◦(Sun-synchronous) Position errors in monitoring x(along-track) 2 cm/Hz1/2
the SST high-low y(cross-track) 1 cm/Hz1/2 z(radial) 3 cm/Hz1/2 Gravitygradiometer Full diagonal instrument(Vxx, Vyy, Vzz) error spectrum frequencyf <10−3Hz:f−1coloured noise
5×10−3Hz <f < 10−1Hz, white noise < 5×10−3E/Hz1/2 Measurement sampling time 1 second
tion) and signal degree variances ofKaula’s rule. The maximum spherical degree and order that has been resolved isL= 300. This corresponds to a half-wavelength ofD = 65 km. The results of these computations, i.e., the errors on thegravity field coefficients, thegeoid heightand thegravity anomaly, represent the expected scientific performance of the GOCE mission. Table IV contains an extract of these results, while the full triangular error spectrum ofspherical harmonic coefficients is displayed in Figure 17.
In comparison with the specific measurement goals, the simulated r.m.s. errors of the recovered geoid height and gravity anomalies at the 100 km length scale reach values of about 0.25 cm and 0.1 mGal, respectively.
These results can also be visualized by mapping the retrieval errors as a function of colatitude (90◦ – latitude) by propagating the covariance error characteristics onto the sphere for thegeoidand gravity anomalies. This is shown in Figure 18.
The errors show some weak latitude dependency down to about 7◦below which the errors increase by up to about one order of magnitude due to gaps in the two polar regions. Elsewhere, the level of the r.m.s. curves is consistent with the cumulative errors up to the specified maximum degreeL.
The distortions in the spherical harmonic error spectrum caused by a non-polar orbit are of little concern because it is clear from Figure 18 that these distortions are perfectly mapped back to the polar regions. A strict lack of data over the pole would therefore not degrade the use of the GOCE results for the main applications.
On the other hand, this situation may be improved by including the results from
376 J. A. JOHANNESSEN ET AL.
TABLE IV
Expected r.m.s. errors in the geoid height and gravity anomalies at given resol-utions, as derived from the GOCE baseline mission simulations. At resolutions where the regularisation of the inverse problem plays a role (i.e., at scales finer than 100 km) the numbers are approximate
Spatial resolution Maximum degreeL Geoid height Gravity anomaly (half-wavelength) (corresponding toD) (mm) (mGal)
1000 km 20 0.4 0.0006
400 km 50 0.5 0.001
200 km 100 0.6 0.03
100 km 200 2.5 0.08
65 km 300 ∼45 ∼2
Figure 17.GOCE spherical harmonic error spectrum. The vertical axis of the triangle refers to the spherical harmonic degreel(or to the corresponding spatial resolutionD), the horizontal axis to the ordermof the coefficients, with theClmcoefficients on the right and theSlmcoefficients on the left.
Thus the coefficientC00would be located at the top of the triangle, whereas the coefficients with increasing degree and order refer to smaller and smaller scales of the gravity field. The colour code refers to the number of decimal digits (significant digits) to which the individual coefficients can be resolved. For example, two significant digits mean a determination of coefficients with only 1%
uncertainty.
Figure 18.Accumulated point error on the geoid [left] and gravity anomaly [right], for series ex-pansions up to degree and orderL= 150 (D= 135 km),L= 200 (D= 100 km),L= 250 (D= 80 km),L= 300 (D= 65 km), as a function of colatitude for the northern hemisphere (the results are symmetrical with respect to the equator), from the GOCE baseline mission simulation. The errors decrease when going towards the pole due to the accumulation of satellite ground tracks, and then increase due to the polar gap.
more aero-gravimetric campaigns over the Arctic and Antarctic; these have been carried out in the Arctic region and are planned for the Antarctic area in the years to come, and will help to close the two gaps. In any event, thegravity anomaly error at, for instance, 100 km length scale (curve with maximum degreeL= 200 in Figure 18) remains below 7 mGal close to the poles and below 1 mGal about 5◦of colatitude away, which is quite an astonishing and useful result over those remote areas.
Finally, it should be emphasized that the gradiometer is three-dimensional, i.e., it simultaneously measures the gravitational field in all three spatial directions.
Consequently the errors of the resulting gravity parameters (gravity anomalies orgeoid heights) exhibit no preferred direction. Apart from the redundancy, the errors are independent and isotropic. This isotropy of the error structure is of great advantage, in particular when the directional structure of the gravitational field is of importance, as is the case in oceanography where slopes of dynamic topography are to be derived.
The results of the performance simulation consequently confirm that the GOCE baseline mission will be unique. The recovery of the best possible set of spherical harmonic coefficients of the Earth’sgravity field and precision orbit will ensure that the derived products (i.e.,geoid height model, gravity anomaly model, and gravity gradients) will clearly fulfil the observation requirements in solid Earth physics, oceanography and geodesy as listed in Table II and quantified by the impact analyses in Section 3.
378 J. A. JOHANNESSEN ET AL.
Figure 19.Simulated GOCE retrieval accuracy compared with present knowledge mapped in a dia-gram of the required accuracy (estimated at approximately 10% of the corresponding signal strength) as a function of spatial scales for a variety of features, process and phenomena occurring in (a) oceanography, (b) solid Earth physics and (c) geodesy.
5. Summary
The GOCE mission, planned for launch in 2006, will combine satellite gradiometry and SST-hl to produce a new model of thegravityfield of unprecedented accuracy and spatial resolution. Its main products will consist of sets ofspherical harmonic coefficientsup to a maximum degree and order 300, which will describe the global gravityfield andgeoid. In addition, a range of detailed global and regionalgravity anomaly, gravity gradients, geoidand geoidslope maps or gridded data sets will become available to the scientific community. All of these primary products will be accompanied by estimates of their error standard deviations and correlations.
Although these products will clearly become available only after the mission, it is already anticipated that the real error estimates will be very similar to those obtained from the full simulation of the system performance from sensor level to the retrievals of the gravityfield andgeoid signals. For instance, up to degree 200 (100 km resolution half-wavelength) GOCE yields ageoid error at the 2.5 mm level, and agravity anomaly error just below 0.08 mGal, for the static field.
Using these simulated error estimates, it has been possible to assess reliably the impact which GOCE is expected to make in the fields of science and applications as discussed in this paper in the context of solid Earth physics, oceanography and geodesy, and schematically summarized in Figure 19.
Solid Earth.In solid Earth physics the production of the GOCEgravitymodel is not in itself the primary goal; rather it is the provision of a detailed three-dimensional image of density variations in the lithosphere and upper mantle, derived from a combination ofgravity, seismic tomography, lithospheric magnetic anomaly in-formation and topographic models. This density image is well constrained by knowledge of thegravityfield. The density information then allows precise quantit-ative modelling of sedimentary basins, rifts, tectonic motions and sea/land vertical changes.
Absolute Ocean Circulation. With the mean dynamic ocean topography derived from the GOCEgeoidin combination with precise altimetry andin-situ observa-tions (i.e., such as by the planned deployment of the array of up to 3000 Argo profiling floats (see http://www.argo.ucsd.edu and http://www.coriolis.eu.org) by 2004 (Roemmich and Owens, 2000), practically all ocean current systems from the strongest (Gulf Stream, Kuroshio, Antarctic Circumpolar Current) through to weaker deep-ocean and coastal current systems should be determined in terms of location and amplitude. The high-spatial-resolution geoidafforded by GOCE has been demonstrated to reduce the uncertainties in mass and heat transport by a factor of 2 in the upper layers, with significant reductions throughout the water column.
Clear benefits are also expected in high-resolution ocean forecasting (Le Provost et al., 1999).
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Geodesy.GOCE data will serve four major challenges in geodesy. It will, in com-bination with GPS, help to produce a worldwide unification of height systems.
Moreover, it will have a practical impact in reducing the source of error for inertial navigation, and it will lead to significant improvement in orbit computations for Earth-orbiting satellites.
Finally, there are fields of application where the benefits of GOCE enter via several directions. For example, the study of sea-level changes spans research into changes in ocean circulation, steric changes (expansion and/or contraction of a body of sea water induced by variation in temperature and salinity) and changes in ocean volume, vertical land movements, ice mass changes, height systems and satellite orbits. GOCE data will facilitate more comprehensive investigation of complex topics including, for example, a comparison of sea-level and sea-level changes in the North Sea with those in the Mediterranean.
In addition to complementing classical altimetric missions, the scientific results from GOCE will be perfectly complemented with observations from other grav-ity missions such as CHAMP and GRACE, and from new dedicated missions to observe the cryosphere such as ICESat (2002) and CryoSat (2004). In particular, the different technical concept of GRACE, launched in March 2002, will recover the gravity field with very high precision for the long and medium spatial scales and thus support GOCE data processing for the recovery of the accurate short scale gravity anomalies. For instance, Wahr et al. (2002) suggest that methods could be developed to infer changes in deep ocean currents from GRACE measurements of time-varying gravity at temporal and spatial scales of around 30 days and 500 km.
In conclusion, we are convinced that the expected increasing availability of data from these dedicated gravity field missions within the next 5 years will introduce a new era for multi-disciplinary research and application in solid Earth phys-ics, oceanography and geodesy. (The website http://www.esa.int/livingplanet/goce/
gives further details and regular updates concerning the evolution of the GOCE mission.)
6. Glossary
Argo:An autonomous globalin-situ observing system based on an array of up to 3000 profiling floats (when fully implemented) which provides temperature and salinity measurements, typically at depths from 0 to 2000 meters every 10 days.
The data are transmitted via ARGOS. In between the 10 days regular ascents they descend back to about 2000 meters where they drift passively with the deep current.
Equipotential surface: A surface where the gravity potential W is a constant.
Points on one such surface may be determined regionally with tide-gauges, which define the regional mean sea-level.
Geoid:Equipotential surface which over the ocean approximates the global mean sea-level, i.e., a global set of tide-gauges and levelling benchmarks, after
sub-traction of the dynamic components. It can be considered as the surface of the hypothetical ocean at rest.
Geoid height:The height,N, of a point on the geoid with respect to the ellipsoid (positive upward). The values range from−105 to+95 metres, and are associated with long wavelength features (several thousand kilometres). In comparison, vari-ations of shorter scale (tens to hundreds of kilometres) have a magnitude from a few centimetres to one or a few metres.
Geostrophic surface current:The surface current maintained by the balance that is established between the oppositely directed Coriolis force (due to the Earth’s rotation) and the surface pressure gradient force. The current is oriented along the isobars, with the high pressure to the right (left) in northern (southern) hemisphere.
Global (geopotential) model:A model of the Earth’s gravitational potential in the form of a set of spherical harmonic coefficients, truncated at a maximum degree and orderL, so that the maximum resolved half-wavelength,D= 20,000 km/L.
Gravitational acceleration:The acceleration of a test mass due to the action on it of the gravitational force (mass attraction). Close to the Earth the dominant effect of the gravitational force is due to the mass of the Earth itself. In vacuum the acceleration of a free falling test mass is purely gravitational.
Gravitational field: This is the mathematical function expressing the variation of the action of gravitation in space (or on a surface). In the case of the gravita-tion force or acceleragravita-tion vector it is vector field, in the case of the gravitagravita-tional potential it is a scalar field.
Gravitational potential (V):Potential associated with the attraction of masses.
Gravity:The magnitude,g, of the gradient ofW at the Earth’s surface and ofV in space. It may be observed by an absolute technique (e.g., in a free fall experiment) or relatively (as a difference) by a spring gravimeter. Gravity is expressed in m/s2 or in milliGal (1 mGal = 10−5 m/s2; 1 Gal = 10−2 m/s2). The mean value of the Earth’sgravityat the surface is about 981000 mGal (the well-known 9.81 m/s2);
it varies from 978100 mGal at the equator to 983200 mGal to the pole due to the Earth’s flattening and rotation.
Gravity anomaly:At any point of given latitude and orthometric height, the grav-ity anomalyg is the value derived by subtracting measured and normal gravity (g = g− γ ). The normal gravity γ is calculated at a point with the ellips-oidal height put equal to the orthometric height. Gravity anomalies due to density inhomogeneities, mountain ridges, etc., range from tens to hundreds of milliGals.
Gravity gradients:Derivatives of the gravity vector, i.e., second-order derivatives ofW at the Earth’s surface and of V in space. Certain linear combinations may be measured by a torsion-balance at Earth’s surface, and by forming differences of adjacent accelerometer measurements in space. Gravity gradients are expressed inEötvös (1E = 10−9s−2). The largest component is the vertical gravity gradi-ent, being about 3000E on Earth (gravity changes by 3 ×10−6 m/s2 per metre of elevation). The horizontal components are approximately half this size; mixed
Gravity gradients:Derivatives of the gravity vector, i.e., second-order derivatives ofW at the Earth’s surface and of V in space. Certain linear combinations may be measured by a torsion-balance at Earth’s surface, and by forming differences of adjacent accelerometer measurements in space. Gravity gradients are expressed inEötvös (1E = 10−9s−2). The largest component is the vertical gravity gradi-ent, being about 3000E on Earth (gravity changes by 3 ×10−6 m/s2 per metre of elevation). The horizontal components are approximately half this size; mixed