Keywords

1 Introduction

To estimate a continuous model in the spatial and temporal domain of both, the mean sea surface (MSS) and sea level anomalies (SLA) from altimetric sea surface height (SSH) measurements is the key idea of the proposed work. The sea surface can be represented as the sum of the long time mean (i.e. MSS) and its temporal variability (i.e. SLA). Both products have various areas of application such as the computation of mean dynamic topography (MDT) models and the derived long-term stable ocean currents (Knudsen et al. 2011; Becker et al. 2014; Mulet et al. 2021). Sea level anomalies can be furthermore used to detect mesoscale eddies (Bolmer et al. 2022).

Common MSS estimation approaches use multi-step procedures in which the temporal variability is eliminated from the SSH observations in a first step (e.g. Pujol et al. 2018; Andersen and Knudsen 2009; Jin et al. 2016). Whereas temporal averaging of all cycles is done for the exact repeat missions (ERM) to obtain a mean profile along the reference track, prior information like gridded SLA products (e.g. Taburet et al. 2019) are used to reduce the ocean variability from SSH observations obtained in Geodetic Mission phases (GM). Alternatively, binned SSH observations can be approximated by a temporal model per grid cell, to estimate a cell dependent temporal correction. Afterwards, collocation-like interpolation and gridding techniques are used to combine the corrected data and to estimate MSS (e.g. Pujol et al. 2018; Andersen and Knudsen 2009; Jin et al. 2016) models on pre-defined over-sampled grids.

In contrast to this, the approach used here is a generic one-step approach based on finite elements as basis functions to describe the spatial signal of both, the MSS and the SLA (Borlinghaus et al. 2023). In this context, the SLA model is mainly used to absorb the temporal ocean variability. The finite elements are set up on triangulations which have initially a constant edge length over the entire study area to start with a homogeneous spatial resolution. To describe the temporal domain, B-Splines with a constant node spacing are used as finite basis function to obtain a high flexibility. But, these initial models show – especially in high variable areas – large least-squares residuals. In this study the spatial distribution of the residuals are used as an indicator of an insufficient parameterization and thus to refine the triangulations.

The manuscript is organized as follows, in Sect. 2 the theoretical background to construct the \(C^1\)-smooth finite element space (FES) is summarized. Based on this space the least-squares observation equations are set up to estimate the spatio-temporal model. The used altimetric satellite data and the analyzed estimation scenarios are explained in Sect. 3. In Sect. 4 a reference scenario configuration is introduced which serves as a basis for the refinements. The scenario with the refined FES for the static model component is analyzed in Sect. 5. In Sect. 6 the impact of the refinement of the FES for the temporal model component is evaluated. Finally, a summary, some conclusions and an outlook are provided in Sect. 7.

2 Summary of the MSS Estimation Approach

The basis for this study is the finite element based spatio-temporal estimation approach for the MSS and the SLA as proposed in Borlinghaus et al. (2023). Here the key idea is shortly summarized (cf. Borlinghaus et al. 2023). The geophysically corrected SSH is represented as the sum of the long time mean (MSS) and the sea level anomalies (SLA)

$$\displaystyle \begin{aligned} f_{\text{SSH}}(\lambda,\phi,\Delta t) = g_{\text{MSS}}(\lambda,\phi) + f_{\text{SLA}}(\lambda,\phi,\Delta t) {} \end{aligned} $$
(1)

where both \( g_{\text{MSS}}: \mathbb {R}^2 \to \mathbb {R} \) and \( f_{\text{SLA}}: \mathbb {R}^3 \to \mathbb {R} \) are continuous functions and the time is represented as \( \Delta t := t-t_0 \) with the reference epoch \( t_0 \). To model the temporal behavior of the SLA, it is assumed that changes in time are continuous and separable from spatial variability. The spatial domain is modeled with finite elements as basis functions, which have only a local support. This allows to model complex signals by a continuous mathematical function which have no accessible closed expression that is directly derived from physical laws. Within this study finite elements defined on triangular meshes are chosen, as they can easily be adapted to regions with complex boundaries (e.g. coastal regions). There are different finite elements which guarantee a \( C^1 \)-smooth surface. Here the Argyris element (Argyris et al. 1968) is selected because it is the element with lowest degrees of freedom which guarantees \( C^1 \)-continuity while spanning a complete polynomial space. In particular this is the local space of polynomials of degree 5 with 21 degrees of freedom including the function value, two first and three second derivatives for each of the three nodes, as well as the three normal derivatives in the centers of the edges.

The entire domain of interest is partitioned into a finite number of triangular sub-regions, each of which has its own locally defined basis functions and corresponding parameters. To construct the triangulations utilized in this study the software package jigsaw (Engwirda 2017) is used. It allows for an automatic generation of meshes given geometrical boundary constraints which define the local study area. The desired location specific size of the triangles can be configured via an input map which defines the target length of the edges in the region of interest. Based on that input a mesh of well-defined triangles is optimized by the software.

As described in Eq. (1), the SSH is modeled by two components, one for the static part and one for the temporal ocean variability. The static MSS signal is described by the function

$$\displaystyle \begin{aligned} g_{\text{MSS}}(\lambda,\phi) = \sum_{i\in I_{\text{MSS}}} a_{\text{MSS},i} b_{\text{MSS},i}(\lambda,\phi) \end{aligned} $$
(2)

where \( i\in I_{\text{MSS}} \) describes the indexing of all \( I_{\text{MSS}} \) piece-wise defined basis functions \( b_{\text{MSS},i}(\lambda ,\phi ) \) and \( a_{\text{MSS},i} \) the corresponding scaling coefficients/parameters.

To model the spatio-temporal SLA signal

$$\displaystyle \begin{aligned} f_{\text{SLA}}(\lambda,\phi,\Delta t) = \sum_{i \in I_{\text{SLA}}} \sum_{j \in J} e_{\text{SLA},i,j} b_{\text{SLA},i}(\lambda,\phi) B_j^3(\Delta t) {} \end{aligned} $$
(3)

is used, which is build by tensor product of the spatial and temporal basis functions. Here \( e_{\text{SLA},i,j} \) are the unknown spatio-temporal parameters and \( b_{\text{SLA},i}(\lambda ,\phi ) \) again the spatial finite elements.

In detail, uniform B-Splines of degree 3 (cf. De Boor 2001; Fahrmeir et al. 2021) with a constant node spacing of \( \Delta \kappa {\thickapprox } {6}^{d}\) of the temporal nodes \( \kappa \) are used in this study to obtain a high temporal resolution. \( B_j^3(\Delta t) \) describes the \( j \)-th B-Spline basis function \( (j \in \kappa ) \). As the chosen B-Spline function corresponds to a low-pass filter with cut-off frequency of \(\nu _c \approx \frac {1}{2 \Delta \kappa } = {1/12}{1{/}d}\) (Sünkel 1985), the model can represent signals down to a 12 d period. Given a single ERM of the Jason family with a temporal repeat of \(\delta t = {10}^{d}\), the Nyquist frequency \(\nu _N = \frac {1}{2\delta t} = {1/20}{1/ d}\) follows, thus 20 d periods are resolvable. Given a combination of simultaneous operating ERM missions (e.g. Jasons and SARAL, HY-2A, and Sentinel-3) it turned out that in the joint spatio-temporal analysis the 6 d node spacing in close to the highest possible temporal resolution which can be estimated stably.

In Borlinghaus et al. (2023) it is shown that for a stable estimation of both components, two different FES are required. A fine resolution for \(g_{\text{MSS}}\) to capture the high frequency static (geoid) signal, and a significantly coarser space for \(f_{\text{SLA}}\). Consequently, the resolvable spatial resolution is limited. As for both functions the Argyris element is selected, the spatial resolution of the functions completely depends on the mesh.

To estimate the unknown parameters \(a_{\text{MSS},i}\) and \(e_{\text{SLA},i,j}\) in a least-squares adjustment, Eq. (1) is used to setup the linear observation equations for all SSH observations as left hand side. Here, the SSH observations are assumed to be uncorrelated with a variance of \( \sigma _0^2 \), i.e the covariance matrix of all SSH observations is

(4)

although it is known that the noise standard deviation of SSHs is spatially not homogeneous.

To be complete, the estimation is stabilized applying a Tikhonov regularization (cf. Tikhonov et al. 1977) with manually adapted individual weights from variance component analysis (cf. Koch and Kusche 2002) for each parameter group (i.e. individual identity matrices for parameters of the same kind, i.e. SLA and MSS as well as for the function values, first and second derivatives which are the local parameters of the finite elements). This compensates spatial or temporal data gaps and inappropriate observation distribution close to the boundary of the region of interest. Furthermore, linear zero-mean constraints are applied to the SLA model to prevent leakage of static MSS signal into \(f_{\text{SLA}}\). Additionally, the temporal model is stabilized at its borders via forcing the second derivatives to zero (for further details see Borlinghaus et al. 2023).

3 Real Data Experiment

Focus of the presented study is the improvement of the triangulations for both the spatial MSS modeling and the spatial SLA modeling. Therefore, the approach summarized in Sect. 2 is applied in a real data experiment to obtain optimized triangulations. In Borlinghaus et al. (2023), it is assumed that both meshes are known a priori, they have been generated as homogeneous meshes, i.e. homogeneous edge lengths which have been chosen motivated by the spatial sampling of the satellites and computational resources.

Figure 1 shows the investigated region south of Africa where the methodology is tested. This region is selected because of its high spatial and temporal variability and to keep the computational effort handy. The high spatial variability results from high frequency geoid signal and is especially visible in the southwest of the region. Furthermore the Agulhas current is the dominant temporal feature in this region which is subject to large temporal variability.

Fig. 1
figure 1

Test region south of Africa with an estimated model of the mean sea surface

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To obtain a best possible spatial resolution for the MSS and a sufficient temporal resolution to compensate the SLA, observations of both exact repeat altimetry missions (ERM, e.g. Jason-1) and geodetic missions (GM, e.g. CryoSat-2) have to be jointly analyzed. Because of the high temporal resolution ERMs have a poor spatial resolution. For GMs, the opposite is true.

Thus, all available ERM and GM altimetry missions for which a L2P data product is available on AVISO+ for the study region and period (2010 to 2019, inclusive) are selected.Footnote 1 These are in total 3.2 × 106 observations collected by nine satellite missions. The reference epoch \(t_0\) is set to January 1st, 2015, which is in the mid of the study period.

This study configuration is used to estimate models of the MSS and the SLA utilizing the summarized estimation approach, while targeting a data adaptive refinement of the meshes of both FES. Table 1 summarizes the main configuration details of the three scenarios considered here.

Table 1 Description of the three estimation scenarios with the involved FES and the temporal node spacing \( \Delta \kappa \)
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4 Reference Scenario and Objectives of the Study

The initial scenario R serves as a reference. MSS and SLA are estimated in order to have a kind of internal baseline for comparisons of models estimated with refined triangulations. This reference scenario utilizes a FES with a homogeneous target edge length of 35 km in the entire domain for the MSS and 175 km for the SLA. Please note that it is not easy to provide a precise measure of the spatial resolution of the FES. But, given the definition of the ARGYRIS element, the local polynomials within a triangle are of degree five. Consequently, this corresponds to one dimensional polynomials of degree five along all slices as well. Given the six parameters of the polynomial, the polynomial has four degrees of freedom accounting for two constraints required to guarantee the \(C^1\)-smoothness at the borders of the triangles. Thus, we expect a spatial resolution in the order of edge length divided by four kilometer, which is confirmed by numerical experiments (mot shown here). For the MSS it is ≈9 km, and thus slightly above the along-track sampling of the 1 Hz SSH sampling (≈7.5 km).

In Borlinghaus et al. (2023) it is shown that the FES for the temporal model component requires a coarser resolution to avoid overparameterization. As it is mainly determined by the ERMs, the reference edge length is tailored to the ground track spacing of the ERMs which is in the order of 100 km to 315 km (e.g. Sentinel-3 and the Jason family). For the reference scenario, a edge length of 175 km was chosen. The temporal resolution given by the node spacing of the B-Splines (cf. Table 1) remains constant for all scenarios. It is chosen as \(\Delta \kappa {\thickapprox } \) 6 d, thus differences in the results obtained only relate to the refined triangulation.

The configuration of the reference scenario summarized above is used to estimate a MSS and the model for the SLAs. Based on the resulting least-squares residuals, this study addresses the research question, whether it is possible to improve the spatial meshes of both – MSS and SLA – based on this internal quality measure.

Figure 2 shows the empirical standard deviations of all residuals within a single triangle for the reference scenario R. Whereas Fig. 2a uses the fine MSS mesh for the computation of the standard deviations, Fig. 2b shows them computed for the coarser SLA mesh. The standard deviations in both figures are not homogeneous, they are in a range of 4 cm to 5 cm in the northwestern and southeastern part, but reach more than 8 cm in the central part where the Agulhas current is the dominant feature. Additionally, some small regions with higher values are visible in the southwestern part in Fig. 2a. The higher variances are an indicator for an insufficient parameterization, either in the spatial or in the temporal domain. Furthermore, larger difference at the eastern boundary become visible, which are attributed to numerical issues and boundary effects. The overall standard deviation of the residuals in the test region of \( R \) is 5.5 cm.

Fig. 2
figure 2

Standard deviation of the residuals per triangle of \( R \) of the FESs (\( \text{F}_{35} \) and \( \text{F}_{175} \)) for the static (a) and temporal (b) model component

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The goal of this study is to use the maps shown in Fig. 2 to refine both triangulations, for the MSS as well as the SLA to identify the unmodeled higher resolution signals. As the larger standard deviations can either result from (i) unmodeled spatial high resolution MSS signal, (ii) unmodeled spatial higher resolution SLA signal, or (iii) unmodeled high-resolution temporal SLA signal, these maps are a good proxy for mesh refinement. They are converted to a to jigsaw input map of target edge lengths, from which optimized meshes are generated. For regions of lower standard deviation, edge length at the upper limit are requested, whereas for regions of high standard deviations edge length of the lower limit are requested. In scenario S, this is studied for the mesh of the MSS, where the target edge length is allowed to vary between 17 km to 35 km depending on the standard deviation (\(\text{F}_{17,35}\)). Consequently the spatial resolution is doubled in regions of highest variance. Technically speaking, the spatial map of standard deviation (cf. Fig. 2a) is converted to a map of target edge lengths, mapping the standard deviation to a target edge length of the interval 17 km to 35 km. Based on this, a new mesh is optimized by jigsaw, trying to obtain the regionally requested edge lengths. The mesh for the SLA is not changed (\(\text{F}_{175}\), cf. Table 1).

This is modified in scenario T, which uses the homogeneous \(\text{F}_{35}\) for the MSS, but refines the mesh for the SLA component to 130 km to 200 km, again depending on the regional standard deviations shown in Fig. 2b. As the mesh of the SLA component dominates the number of unknown parameters and to avoid overparameterization, the lower bound of the interval for the target edge length is limited to 130 km. Here, the upper limit is chosen as 200 km, which allows even larger triangles in regions of low variance.

In the following two sections (Sects. 4 and 5), the results obtained with the newly generated meshes in the two different scenarios are analyzed and compared to the reference scenario R.

5 Refined MSS Component

The first refined scenario S combines the refined FES for the MSS with the original homogeneous \(\text{F}_{175}\) for the SLA. The homogeneous reference mesh is shown in Fig. 3a and the refined mesh optimized by jigsaw is shown in Fig. 3b with the area of the individual triangles color coded. Although the correlation of the mesh to the map of standard deviations is visible (cf. Fig. 2a), it is obvious that the mesh, due to the transition from coarse to fine, is blurry. As targeted, the major refinements are seen in the area of the Agulhas current with the highest standard deviation of the residuals, but also some refinements with a smaller extend can be seen in the southwest.

Fig. 3
figure 3

Approximate triangle sizes of the reference (\(\text{F}_{35}\), a) and refined FES (\(\text{F}_{17,35}\), b), the standard deviation of the residuals per triangle (c) and the change of the standard deviations per triangle compared to the reference scenario (d), the RMS per triangle of the differences to CNES_CLS15 MSS (e) and the differences of the RMS compared to the reference scenario (f)

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After estimating the MSS and SLA model with the refined mesh for the MSS, new residuals are computed. Figure 3c shows the standard deviation of the residuals per triangle of \(\text{F}_{17,35}\). Compared to Fig. 2a there is no difference in magnitude of the standard deviation visible and the Agulhas current is still the dominant feature. Compared to the reference scenario in Fig. 2a, no obvious difference is visible. Figure 3d shows the differences of the standard deviations evaluated on \(\text{F}_{35}\). Here, red colors indicate a reduced standard deviation of the refined scenario compared to the reference scenario. The highest improvements can be seen in the southwestern part, but additional improvements are visible in the northern part. But, in the center of the region where the main refinement happens, improvements are only minor. However, the overall standard deviation of the residuals could not be improved significantly (sub mm level).

To only rely on the residuals to judge the quality of the refinement is disadvantageous. As external comparison, the MSS model component is compared to the well established model CNES_CLS15 MSSFootnote 2 (Pujol et al. 2018). To do so the model is evaluated on the grid provided by the comparison model. Figure 3e shows the RMS of the difference between the estimated MSS and the CNES_CLS15 computed per triangle. In general, a good agreement of 1 cm to 4 cm is achieved. The differences have a more or less random structure over the complete study area with highest values in the region of the Agulhas current and in the southwestern region. To show again the effect of the mesh refinement, the differences of the RMS of differences to the CNES_CLS15 MSS are computed (see Fig. 3f). Again, red colors show a reduced RMS and thus an improvement, green colors correspond to larger RMS and thus a degradation.

The highest improvement are again visible in the southwest and northern part. This confirms, that the refined model captures an additional MSS signal. But, the central area shows higher RMS values of the difference of the refined model to the CNES_CLS15 MSS compared to the reference model from scenario R. This can either indicate an overparameterization or a lower filtering effect caused by the smaller triangles in this area. Due to the smaller triangles the refined model has 27,446 degrees of freedom, compared to 14,043 of the reference solution. This suggests the conclusion, that the increased standard deviation of the residuals in the central area cannot be attributed to unmodeled MSS signal. It has to be attributed to unmodeled spatio-temporal SLA signal, which is studied in scenario T which uses the reference mesh for the MSS (cf. Table 1).

6 Refined SLA Component

For the second model the refined FES (\( \text{F}_{130,200} \)) for temporal model component is used together with the original homogeneous mesh for the MSS (as for the reference solution). Figures 4a and b show the reference and refined mesh. As desired the main refinement appears in the high variable area of the Agulhas current, but especially in the northwestern and southeastern part regions with a coarser triangle sizes are visible. Because of more then 600 temporal B-Spline nodes, small changes in the degrees of freedom of the spatial FES have a large impact on the total number of unknown parameters. Therefore the lower and the upper limit for the target edge length are defined as 130 km to 200 km and optimized by jigsaw depending on the standard deviation to obtain \(\text{F}_{130,200}\). This leads to only small changes in the total number of parameters of the spatio-temporal SLA from 414,864 to 416,673.

Fig. 4
figure 4

Approximate triangle sizes of the reference (\( \text{F}_{175} \), a) and refined FES (\(\text{F}_{130,200} \), b), the standard deviation of the residuals per triangle (c) and the change of the standard deviations per triangle compared to the reference scenario (d), the mean RMS per triangle of the differences to DUACS SLA maps (e) and the differences of the mean RMS compared to the reference scenario (f)

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Figure 4c shows the standard deviation of the residuals of the model T. Again the main features in the area of the Agulhas current are visible. To highlight the difference to the reference solution, the differences of the standard deviations are computed and are shown in Fig. 4d. The differences of the standard deviations computed from the residuals of scenario R and the refined scenario T show the largest improvements as expected in the area of the refinement. But, some higher values are visible in regions with a coarser triangulation structure. The overall standard deviation of the residuals is slightly reduced by 1 mm, it is 5.4 cm.

For an external comparison, DUACS Level 4 gridded SLA DT2018 maps are used. The spatio-temporal SLA model (cf. Eq. 3) is evaluated on the grid and at the time stamps provided by the product. Then the differences between both time series are computed and the RMS is computed for each pixel of the grid. Afterwards, RMS values for the triangles of the different epochs are averaged (see Fig. 4e). Here the trend of the Agulhas current is not visible as a dominant feature but again larger mean RMS values are visible in regions with higher temporal variability.

To get an impression of the improvement obtained by the refinement the differences of the mean RMS values of the reference model and the refined model are computed (see Fig. 4f). Here the differences show a more or less random characteristic with no dominant features in the area of the FES refinement. Additionally the regional improvements visible in Fig. 4d are not visible in Fig. 4f. This shows that, due to the regional refinements, signal is modeled which has no impact on the differences to the DUACS SLA maps. Again, the overall mean RMS is slightly improved by 1 mm from 5.7 cm to 5.6 cm.

7 Summary, Discussion and Outlook

In this contribution, the refinement of the triangulation of the FES which are used to model the MSS and the SLA as a continuous mathematical function are studied. MSS and SLA are jointly estimated from along-track SSH observation in a least-squares adjustment. For the refinement, least-squares observation residuals from an initial reference solution with a homogeneous mesh are used to identify conspicuous spatial regions which are candidates for mesh refinement. The model functions used in this study are compound by two components, one is static which models the MSS and one spatio-temporal which approximates the temporal ocean variability (SLA). Both parts utilize finite element basis functions for spatial description, a high resolution FES for the MSS and a lower resolution FES for the SLA. The latter is composed to a spatio-temporal model using B-Splines basis functions for the temporal domain.

To test this strategy, a small region south of Africa is selected where a high temporal as well as high spatial variability is expected. The reference scenario \( R \) for MSS modeling uses a homogeneous FES with a target edge length of 35 km and 175 km for the SLA component. In this contribution, the refinement of both components is individually studied, to better access the effects on the derived MSS and SLA. The first scenario \( S \) refines the mesh for the MSS to a target edge length of 17 km to 35 km depending on the empirically derived standard deviation of the least-squares residuals within a triangle of the mesh. The second scenario \( T \) adjusts the FES for the temporal model component, to target edge lengths of 130 km to 200 km, again depending on the empirically derived standard deviations of the residuals. For both refined scenarios, MSS as well as SLA models are estimated and used to access the performance of the refinement.

The spatial pattern of estimated standard deviations per triangle are a good internal quality measure to indicate potential regions of an insufficient parameterization. It is shown that they can help locally to better adapt the FES in high-residual regions, e.g. the south-western part. In regions of high temporal variability, the mesh can be refined as well, leading to smaller least-squares residuals. But, when comparing the results to reference models for the MSS, degradations are visible as well, which indicates the risk of overparameterization. However, the impact of the FES refinement on both resulting scenarios is small, we conclude that the choice of the initial meshes based on the data sampling was reasonable. The regions of lowest standard deviation are in the order of 4 cm which is inline with the typical assumption of the accuracy of a few centimeter of a single SSH measurement. The largest problem for the FES refinement based on lest-squares residuals is to differentiate between larger standard deviations resulting from (i) unmodeled MSS signal, (ii) unmodeled spatial SLA signal or (iii) insufficient temporal resolution. This can lead to an iterative process to compute refined FES. In general three different design choices related to the approximation capabilities can be adjusted:

  • The fine spatial FES for the static model component (MSS).

  • The coarser spatial FES for the temporal model component (SLA).

  • The temporal resolution defined by the node spacing \( \Delta \kappa \) of the B-Spline basis functions.

The first two design criteria were shown to have a negligible impact on the overall results. For the FES of the MSS component, the positive impact of the refinement is only visible in some very local areas. Thus, the iterative refinement seems to be a useful technique to refine the FES locally to avoid an overparameterization in smoother areas.

But still, the highest potential to improve the estimated models seems to be an increased resolution of the temporal domain by reducing the node spacing of the B-Splines. However, this is on the one hand limited by the available data – i.e. the repeat cycle of the ERM (10 d) and the number of available satellite missions operating in parallel. On the other hand, this significantly increases the number of unknown parameters which results in computational challenges and might cause numerical problems. This requires more advanced and adopted regularization techniques.