Keywords

1 Introduction

The difference between the sea surface height (SSH) above a reference ellipsoid and the geoid is the ocean’s dynamic topography. The accurate knowledge of its steady-state part, the mean dynamic ocean topography (MDT) is crucial for both oceanographers (e.g. Wunsch and Gaposchkin 1980; Fu 2014; Wunsch and Stammer 1998), as it gives valuable information about the ocean’s circulation and geostrophic surface currents, and geodesists (Rummel 2001), as it permits the unification of independent local vertical datums. The geodetic estimation of the MDT can be represented, under the concept of signal de-convolution, as the separation of the Mean Sea Surface (MSS) height into the MDT and the geoid height. For the separation process, additional independent information about the geoid, the MDT or both is required. When omitting any kind of oceanographic input data (ocean salinity, temperature, pressure), instead relying only on satellite derived geoid and SSH data, the resulting MDT estimate is called geodetic. A remaining scientific challenge is the spectral inconsistency of the involved data sets (e.g. Albertella et al 2008; Woodworth et al 2015).

Several approaches were developed and applied to determine a MDT from altimetric measurements and geoid information at the global or regional scale. Basis are the altimetric SSH measurements \(h_{\text{SSH}} = h_{\text{orb}} - h_{\text{alt}} - c + o + e\), which result from the difference between the altitude of the satellite (\(h_{\text{orb}}\)) and the raw altimeter range (\(h_{\text{alt}}\)) which have to be corrected due to environmental conditions (c). These include instrument and sea state bias, atmospheric, tidal and inverted barometer corrections (Aviso 2020). Additionally, the measurements contain random and systematic errors e and a mission specific bias o. Due to the highly complex signal structure of the SSH observations, containing a multitude of individual constituents, most of the MDT estimation approaches use preprocessed MSS products (e.g. Andersen et al 2015; Schaeffler et al 2016) instead of the original along-track SSH measurements. These result from a spatial and temporal gridding and averaging of multi-mission along-track observations, utilizing either deterministic or stochastic approaches and taking care of data homogenization, removal of the temporal ocean variability (OV) and a reduction of errors. The MSS products are provided as fine grids (e.g. 1′ \(\times \)1′) and contain the averaged SSH of multiple mission collected over decades with spatial resolution of a few kilometers (Andersen et al 2015).

The geoid information is typically based on a spherical harmonic model generated from satellite observations, e.g. GRACE and GOCE. Consequently, it has a limited spatial resolution of about 100 km with an accuracy level of 1 cm to 3 cm (e.g. Brockmann et al 2021).

To derive a consistent MDT from the difference of a MSS and the geoid information (N)

$$\displaystyle \begin{aligned}{} \zeta &= \text{MSS} - N \end{aligned} $$
(1)

filtering is required to overcome the spatial inconsistencies. The approaches mostly differ in the chosen filtering strategy, the filter characteristics or the domain the filtering is applied in (e.g. Albertella et al 2008; Bingham et al 2008; Čunderlík et al 2013; Siegismund 2013; Gilardoni et al 2015; Knudsen et al 2021).

In this study, we utilize a parametric least-squares approach (cf. Becker et al 2014; Neyers 2017) which jointly estimates both, a refined geoid as well as a model of the MDT. Here, we analyse the feasibility of coestimating a spatio-temporal model component which is supposed to compensate the OV. This will reduce the reliance on prior models used for reduction and/or assumptions about implicit canceling. For this purpose, Sect. 2 summarizes the parametric approach and introduces the applied estimation strategy. Section 3 introduces the data, study region and the configuration details for the numerical experiments applied to study the extended model. The results obtained for both configurations, i.e. with and without the spatio-temporal extension, are presented and compared in Sect. 4. Finally, Sect. 5 summarizes the results and draws conclusions.

2 Parametric Modelling Approach

2.1 Parametric Model Functions

2.1.1 Modelling the Geoid

The Earth’s gravity field is typically modelled by global spherical harmonic basis functions. Although not optimal when working regionally, this representation results from the used gravity field information (cf. Sect. 3.2). The disturbing potential at an evaluation point with spherical coordinates (r, \(\lambda \), \(\varphi \)) then reads (e.g. Hofmann-Wellenhof and Moritz 2005)

$$\displaystyle \begin{aligned}{} \begin{aligned} T\!\left(r,\lambda,\varphi\right) = \frac{GM}{R}\sum_{l=0}^{l_{\max}}\left(\frac{R}{r}\right)^{l+1} \sum_{m=0}^{l}P_{lm}\left(\sin \varphi\right)\\ \left(c_{lm}\cos\!\left(m\lambda\right)+s_{lm}\sin\!\left(m\lambda\right)\right) - U\left(r,\varphi\right), \end{aligned} \end{aligned} $$
(2)

with the maximum degree \(l_{\max }\) of the expansion. GM and R are the gravitational constant of the Earth and the equatorial radius, and \(P_{lm}\!\left (\cdot \right )\) the fully normalized associated Legendre basis functions and \(U\left (\cdot \right )\) the normal potential. \(c_{lm}\) and \(s_{lm}\) are the Stokes coefficients. Using this representation, the geoid can be approximated in the spherical harmonic domain and represented as a function of ellipsoidal coordinates (\(h=0\), \(\lambda \), \(\phi \))

$$\displaystyle \begin{aligned}{} N(\lambda, \phi) &= T\!\left(r\left(0,\lambda,\phi\right),\,\lambda,\,\varphi\left(0,\lambda,\phi\right)\right) / \gamma(\phi), \end{aligned} $$
(3)

where \(\gamma (\cdot )\) is the normal gravity.

2.1.2 Modelling the MDT

Since there is no natural choice of suitable basis functions for the MDT, a finite element approach on a triangulation is chosen to approximate the unknown function. The MDT is modelled by the continuous function

$$\displaystyle \begin{aligned}{} \zeta(\lambda,\phi)=\sum_{k\in K}{a_{\text{MDT},k}}\,b_{k}(\lambda, \phi) \end{aligned} $$
(4)

where \(a_{\text{MDT},k}\) are K unknown scaling coefficients of the basis functions \(b_{k}(\lambda ,\phi )\) resulting from the chosen finite element space. In this study, the Argyris element (Argyris et al 1968) is selected to achieve a \(C^1\)-smooth definition of \(\zeta \) over the domain of interest \(\Omega \). Within each triangle, polynomials of degree 5 are spanned by 21 degrees of freedom. The spatial resolution of zeta is then controlled by the mesh resolution, i.e. triangle size.

2.1.3 Modelling the Ocean Variability

Similar to Borlinghaus et al (2023), a spatio-temporal finite element space is constructed by a tensor product of spatial and temporal basis functions. Whereas for the spatial domain the same finite element space as for the MDT is chosen (cf. (4)), the temporal model function is a linear combination of a trend and seasonal harmonics

$$\displaystyle \begin{aligned}{} \begin{array}{ll} a_{\text{OV},l}(t) = e_{\text{OV},1,l} (t-t_0) + \\ e_{\text{OV},2,l}\sin{\left(\omega (t - t_0) \right)} + e_{\text{OV},3,l}\cos{\left(\omega (t-t_0)\right)} \end{array} \end{aligned} $$
(5)

with reference epoch \(t_0\) and the fixed annual frequency \(\omega \).

Combining these functions with a tensor product as in Borlinghaus et al (2023) yields the spatio-temporal model function

$$\displaystyle \begin{aligned} OV(\lambda,\phi,t)&=\sum_{l\in L}{a_{\text{OV},l}}(t)b_{l}(\lambda,\phi){} \end{aligned} $$
(6)
$$\displaystyle \begin{aligned} &=\sum_{l\in L} e_{\text{OV},1,l} (t-t_0) b_{l}(\lambda,\phi)\notag\\ + &e_{\text{OV},2,l}\sin{\left(\omega (t - t_0) \right)} b_{l}(\lambda,\phi)\notag\\ + &e_{\text{OV},3,l}\cos{\left(\omega (t-t_0)\right)} b_{l}(\lambda,\phi){} \end{aligned} $$
(7)

which can be used to absorb long term signals of the ocean variability. The scaling coefficients \(e_{\text{OV},m,l}\) of the spatio-temporal basis functions are estimated in the least-squares adjustment from the altimetric SSH observations.

2.2 Combined Estimation Procedure

The unknown Stokes coefficients \(c_{lm}\) and \(s_{lm}\) for the geoid, \(a_{\text{MDT},k}\) for the MDT as well as optionally \(e_{\text{OV},m,l}\) for the ocean variability are estimated from a joint adjustment of the satellite-based geoid information and the altimetric sea surface height measurements. As the gravity field information is already available in form of normal equations of global satellite-only gravity field models in spherical harmonic domain, observation equations have to be set up for the altimetric SSH measurements only.

SSH Observation Equations for Scenario A

No ocean variability is estimated. Thus, it is assumed that the ocean variability cancels due to the implicit spatio-temporal averaging within the least-squares adjustment. Consequently the observation equation for the i-th SSH measurement \(l_i\) at location \(\lambda _i, \phi _i\) simply reads

$$\displaystyle \begin{aligned}{} l_i + v_i &= N(\lambda_i, \phi_i) + \zeta(\lambda_i, \phi_i) + o_j, \end{aligned} $$
(8)

Here, \(v_i\) are the residuals and \(o_j\) is a mission specific bias correction parameter which is estimated in addition. The bias of one selected reference mission is fixed to zero.

SSH Observation Equations for Scenario B The ocean variability is coestimated using the model function from (6). Especially for the geodetic missions it is not guaranteed that the ocean variability cancels, due to the missing, or at least very long repeat cycle. Thus, the observation equations, which now depend on the measurement epoch, read

$$\displaystyle \begin{aligned}{} \begin{aligned} l_i + v_i = &N(\lambda_i, \phi_i) + \zeta(\lambda_i, \phi_i)\\ &+ OV(\lambda_i, \phi_i, t_i) + o_j. \end{aligned} \end{aligned} $$
(9)

Parts of the ocean variability can now be absorbed by the deterministic function \(OV(\lambda _i, \phi _i, t_i)\), which can reduce aliasing signals in the MDT.

2.3 Smoothness Conditions

To support the separation and to make the resulting system of equations solvable, smoothness conditions are formulated for the geoid, the MDT and optionally the ocean variability. For this purpose, regularization matrices are constructed and applied in the adjustment process.

Regularization of the Spherical Harmonics

The Kaula rule is used to determine degree dependent weights of a diagonal regularization matrix for all spherical harmonic coefficients above degree 200, which cannot – at least not accurately – be determined from the satellite based geoid information assembled (indicated as SH medium and SH high deg in Fig. 1). Regularization towards zero is applied using a small weight of 10−4 just to make all spherical harmonic coefficients estimable.

Fig. 1
figure 1

Schematic view on the combination of normal equations for scenario A without, and B with coestimation of OV. (a) Structure of normal equations without estimating the ocean variability. (b) Structure of normal equations when coestimating the ocean variability

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Regularization of the MDT

The separation of the SSH into geoid and MDT only works for the long wavelengths, for which the geoid is accurately known from the satellite based geoid. To support the separation, the assumption that the MDT is smooth can be added. For that purpose, the least-squares objective function is extended by the minimization of the norm of the gradient of the MDT, i.e. \(\left \lVert \nabla \zeta \right \lVert _{L^2\left (\Omega \right )}\). As the Argyris element uses full polynomials of degree five, the condition can be expressed as a (non-diagonal) regularization matrix, which is derived by a numerical quadrature using the control points and weights from Taylor et al (2005). An empirical weight is used in the combination.

Regularization of OV

Similar to the MDT, a regularization matrix is derived for the function which represents the ocean variability. Consequently, the objective function is further extended by adding the norm of OV’s spatial gradient, \(\left \lVert \nabla OV \right \lVert _{L^2\left (\Omega _T\right )}\). This leads to a block-diagonal regularization matrix with three blocks, that are all identical to the MDT’s regularization block. Individual weights for the amplitudes and the trend are determined by variance component estimation, resulting in 5.7, 2.9 and 6,816,929.4 respectively.

2.4 Combined Solution

With the contributors, i.e. the satellite based geoid information, the SSH normal equations as derived from the observation equations in (8) or (9) and the regularization matrices, the least-squares normal equations can be assembled and solved for the unknown parameters.

For scenario A, without coestimation of the ocean variability, the schematic overview of the normal equations which are assembled and solved is provided in Fig. 1a. For scenario B, which coestimates the ocean variability, the structure of normal equations is provided in Fig. 1b. The large dimensional normal equations are assembled and solved in a massive parallel implementation on a high performance compute cluster.

3 Configuration for the Numerical Experiments

3.1 Study Region and Mesh

To study the proposed coestimation of the long-term ocean variability, a numerical real data experiment is conducted. The Agulhas region (10 ° E to 40 ° E and 42 ° S to 20 ° S, cf. Fig. 2) is chosen as a local study region as it contains both regions with smooth and strong geoid signal as well as of low and high ocean variability. The domain of the finite element space is limited by a polygon derived from these borders and the coastlines. Using the jigsaw-geo package (Engwirda 2017), a triangulation is generated (see Fig. 2) inside the polygon. The resulting mean length of the edges is about 175 km in the region of interest.

Fig. 2
figure 2

Overview of the Agulhas study region used in the numerical experiments: the expected signal from the CNES-CLS18 MDT and the used triangulation (gray)

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3.2 Used Data Sets

For the gravity field information, the unregularized normal equations from the GOCO06s satellite-only gravity field model (Kvas et al 2021) are used. They are assembled in the spherical harmonic domain for degrees 2 to 300 and can be directly included in the estimation (cf. Fig. 1).

Ten years of along-track SSH measurements are analysed for the period 01/2010 to 12/2019. The corrected L2P data as distributed by Aviso+ is selected and used for the ERM Jason-1, Jason-2 and Jason-3 and the GM Cryosat-2. Geodetic mission phases from the Jason missions are used in addition (see Table 1). In total more then 5 × 106 SSH observations are used. In the experiment, they are assumed to be uncorrelated with a mission specific variance which is derived by variance component estimation.

Table 1 Characteristics of the used altimetry missions
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3.3 Scenario Configuration

To show the effect of the coestimation of ocean variability on the target quantities, i.e. the MDT and the geoid, two scenarios are computed. In scenario A, geoid and the MDT are estimated (cf. Fig. 1a) using (8) as least-squares observation equations for altimetric SSH. Contrary, scenario B utilizes (9) to jointly estimate geoid, MDT and the spatio-temporal model for the ocean variability (Fig. 1b).

In both scenarios, the basic settings are the same to obtain comparable solutions: The geoid is estimated from spherical harmonic degree 2 to \(l_{\max } = 600\), i.e. 361,197 parameters. As only local SSH data are used, medium and high degree global spherical harmonics are regularized towards zero using the Kaula rule for the degree dependent weighting. In both scenarios, the MDT is estimated with the same finite element space, i.e. using the mesh shown in Fig. 2 and the Argyris element. This results in 1195 unknown parameters for the MDT. To support the separation, the MDT is regularized applying the smoothness condition (cf. Sect. 2.3). Scenario B additionally coestimates the ocean variability in form of a linear trend and annual harmonics (cf. 6) and thus estimates \(3\times \)1195 additional parameters for which additional smoothness conditions are applied.

Full altimetry normal equations are assembled which takes about 15 h to 20 h with 576 cores on the JUWELS supercomputer with a massive parallel implementation. The combination and solution of the normal equations (cf. Fig.1) for the unknown parameters takes additional 1 h to 2 h. Weights are derived by variance component estimation and some empirical tuning. The derived results for the geoid, the MDT and the ocean variability are presented and discussed in Sect. 4.

4 Results and Evaluation

4.1 Comparison of the MDT and Geoid Estimates

For both scenarios the parameters \(a_{\text{MDT},k}\) for the MDT and \(c_{lm}\)/\(s_{lm}\) for the Earth’s gravity field are estimated. From these parameters, the MDT as well as the geoid can be evaluated and compared among each other and to reference models.

Figure 3 shows both MDT solutions as a difference to the established CNES-CLS18 MDT (Mulet et al 2021), which is adopted to the reference epoch 01/01/2015, as well as the difference between both solutions. In terms of RMS the MDT estimated in scenario A shows a consistency of 5.1 cm compared to CNES-CLS18. It is dominated by a large systematic difference close to the coast of South Africa. The RMS in regions of low ocean variability is about 1 cm to 2 cm, i.e. 0.9 cm in the north-western part (orange box in Fig. 3a) and 1.6 cm in the north-eastern part (green box in Fig. 3a). It is significantly larger close to the main Agulhas current with 3.6 cm (red box in Fig. 3a) where a higher OV is expected. Despite the large systematic difference, the solutions shows a good agreement to CNES-CLS18. The RMS is in the same order of magnitude as the RMS between the CNES-CLS18 MDT and the alternative DTU22 MDT model (Knudsen et al 2022), which is about 3.1 cm in the region of interest. But, the strong coastal difference indicates a systematic error in the MDT derived in scenario A – the MDT estimate does not include the strong coastal gradient which clearly shows up in both CNES-CLS18 (cf. Fig. 2) and DTU22 MDT.

Fig. 3
figure 3

Differences of the two MDT solutions evaluated on the grid as provided by the CNES-CLS18 MDT. The coloured boxes indicate regions for which the statistics are provided in the discussion. (a) A \(-\) CNES-CLS18 MDT. (b) B \(-\) CNES-CLS18 MDT. (c) A \(-\) B MDT

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Figure 3b shows the same for the MDT estimated in scenario B (including the coestimation of OV). A comparison of Fig. 3a and b as well as the direct difference in Fig. 3c shows that both solutions are very similar and thus equivalent. RMS values with respect to CNES-CLS18 are the same, and the RMS of the difference is below 2 mm (and maximal/minimal differences are below ±1 cm in regions of strongest variations). At the level of the MDT and the comparison shown here, it can not be concluded that the estimated MDT benefits from the coestimation of OV.

Similarly, the estimated gravity field can be compared to existing higher resolution models. For this purpose, Fig. 4 shows both estimated spherical harmonic series evaluated to degree 600 in terms of geoid height differences to the XGM2019 (Zingerle et al 2019) model evaluated to degree 760. As a comparison, Fig. 4a shows the difference of the used GOCO06S (at degree 250) model to the XGM2019 model (degree 760). This difference is dominated by the additional higher frequency signal of XGM2019 and the RMS is about 20 cm. The difference for the estimated geoid from scenario A is shown in Fig. 4b. Obviously, the differences are significantly reduced, the RMS is 2.0 cm for the orange, 3.5 cm for the green and 5.3 cm for the red region. In the entire region, the RMS is 10.0 cm and again dominated by a large difference close to the coast of South Africa. This allows to draw two conclusions: Firstly, as the RMS is significantly reduced, the geoid of GOCO06S is significantly improved for the higher frequencies in the joint estimation. Thus, the estimated gravity field is successfully improved locally from the SSH measurements. Secondly, as now the large coastal difference shows up with an inverted sign compared to the MDT, it is confirmed that the separation failed in the coastal area, the missing strong gradient in the MDT entered the geoid.

Fig. 4
figure 4

Differences of the two gravity field solutions evaluated for geoid heights. The coloured boxes indicate regions for which the statistics are provided in the discussions. (a) GOCO06S(\(l_{ \max }=250\)) \(-\) XGM2019. (b) A \(-\) XGM2019. (c) B \(-\) XGM2019. (d) A \(-\) B

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Similar to the MDT results, the geoid determined in scenario B is equivalent to the geoid derived in scenario A, Fig. 4c shows hardly a difference. Figure 4d shows the differences for which the RMS is below 1.0 cm. Same conclusions as for the MDT solutions can be drawn: it cannot be demonstrated, that the solution which coestimates OV is superior compared to the solution without.

4.2 Estimates of Ocean Variability

As neither MDT nor the geoid improved, the estimates for the ocean variability are compared to gridded sea level anomaly products (daily DUACS Level 4 gridded SLA DT2018, Taburet et al (2019)) to validate the coestimated spatio-temporal signal. For this purpose, trends, amplitudes and phases are estimated independently for each cell of the DUACS grid (see Fig. 5, first row) using a least-squares regression.

Fig. 5
figure 5

Spatial maps for trends, seasonal amplitudes (cm) and phases (\(^{\circ }\)). First row: from regression of the DUACS gridded SLA maps, Second row: derived from the parameters estimated in scenario B. (a) Trends (cm/yr). (b) Amplitudes (cm). (c) Phases (\(^{\circ }\)). (d) Trends (cm/yr). (e) Amplitudes (cm). (f) Phases (\(^{\circ }\))

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Using (7), the same quantities can be derived for all the grid locations from the parameters estimated in scenario B. There, \(\sum _{l\in L} e_{\text{OV},1,l} b_{l}(\lambda ,\phi )\) reflects the trend estimated for location \((\lambda ,\phi )\). Similarly, amplitudes and phases of the seasonal harmonic can be derived from \(\sum _{l\in L} e_{\text{OV},2,l} b_{l}(\lambda ,\phi )\) and \(\sum _{l\in L} e_{\text{OV},3,l} b_{l}(\lambda ,\phi )\) in the domain of interest. These derived quantities are shown in Fig. 5, second row. A good agreement is visible, most of the dominant features which are visible in the maps derived from the gridded data are visible in the maps derived from the coestimated quantities as well. This confirms, that the coestimation works – under the assumption that the chosen model (5) sufficiently reflects the true OV.

To show the quality of this simple model, individual time series can be analysed. From the gridded SLA product, a time series for a single grid point at location \((\lambda _c,\phi _c)\) can be easily extracted. The time series for the coestimated model follows from (7) for the location \((\lambda _c,\phi _c)\) as a one-dimensional function in the time domain

$$\displaystyle \begin{aligned} f(t) := OV(\lambda_c,\phi_c,t) \quad \mathbb{R} \mapsto\mathbb{R}. \end{aligned} $$
(10)

Figure 6 shows both, the time series from the DUACS product as well as the function estimated in scenario B for a region of low as well as a region of high ocean variability. The first time series is evaluated in the north of the orange region of very low ocean variability (13.375 ° E, 20.875 ° S). It is visible in Fig. 6a, that the seasonal signal is dominant, but seasonal variations are below ±5 cm. The estimated model (blue curve) nicely captures this main feature and approximates the variability. On the contrary, the second time series shown in Fig. 6b is close to the Agulhas current, thus strong ocean variability is expected (red box, 17.325 ° E, 39.125 ° S). The DUACS SLA product shows strong high-frequency variations in the range of ±1 m, whereas the seasonal signal is hardly visible. Consequently, the estimated model is a poor approximation of the OV, the dominant signal is not captured by a model like (5).

Fig. 6
figure 6

Sea level anomaly time series from the DUACS product (orange) for two grid points compared to the function coestimated in scenario B (blue). (a) Location in region of low ocean variability. (b) Location in region of high ocean variability

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5 Summary and Conclusion

In this contribution, the parametric joint estimation of the MDT and geoid is extended for the coestimation of a spatio-temporal model component. This is supposed to model and compensate long-term ocean variability to avoid aliasing into the mean – i.e. static – geoid and MDT models.

For this purpose, a parametric approach is chosen in which both geoid and MDT are modelled by continuous functions. The geoid in terms of spherical harmonics and the MDT by finite element basis functions. Similar to Borlinghaus et al (2023), combining separable functions – finite element basis functions in the spatial and a trend and seasonal harmonic functions in the time domain – a spatio-temporal function is designed to model the OV. The parameters are jointly estimated from altimetric SSH measurements with (9) and (6) as flexible observation equations, and a satellite-based gravity field model, applying some smoothness conditions.

A numerical real-data experiment is performed to study the coestimation of the OV. For this purpose, MDT and geoid are estimated without (scenario A) and with this extension (scenario B). Comparing the results among each other and to reference models, no obvious improvements could be shown for MDT or geoid. It is concluded that

  • the analysed 10 year period is too short for stable trend estimate and strong regional signals cause leakage,

  • due to the mean mesh spacing of 175 km of the MDT model, the implicit spatio-temporal filtering effect of the finite elements is already quite strong, as the temporal sampling of SSH observations in each triangle is sufficient,

  • there is a quite homogeneous data sampling of geodetic and exact repeat altimetry missions in the study region (in the analysed period), which supports the implicit filtering of the least-squares approach.

Although it is shown that a basic model of (linear) trend and annual harmonic is not sufficient to model the OV close to strong current systems, it is demonstrated that the coestimation is working in general. Estimated trends, amplitudes and phases are similar compared to those derived from gridded sea level anomaly products.

Consequently, more advanced models in the temporal domain are required. E.g. Borlinghaus et al (2023) proposed the use of B-Splines when coestimating the OV while estimating the mean sea surface. But, this will significantly increase the parameter space, which is already large (more than 360,000 parameters), by additional hundreds of thousands parameters, which is not yet operational and subject to future work.