Glacial isostatic adjustment is largely governed by the rheological properties of the Earth's mantle. Large mass redistributions in the ocean–cryosphere system and the subsequent response of the viscoelastic Earth have led to dramatic sea level changes in the past. This process is ongoing, and in order to understand and predict current and future sea level changes, the knowledge of mantle properties such as viscosity is essential. In this study, we present a method to obtain estimates of mantle viscosities by the assimilation of relative sea level rates of change into a viscoelastic model of the lithosphere and mantle. We set up a particle filter with probabilistic resampling. In an identical twin experiment, we show that mantle viscosities can be recovered in a glacial isostatic adjustment model of a simple three-layer Earth structure consisting of an elastic lithosphere and two mantle layers of different viscosity. We investigate the ensemble behaviour on different parameters in the following three set-ups: (1) global observations data set since last glacial maximum with different ensemble initialisations and observation uncertainties, (2) regional observations from Fennoscandia or Laurentide/Greenland only, and (3) limiting the observation period to 10 ka until the present. We show that the recovery is successful in all cases if the target parameter values are properly sampled by the initial ensemble probability distribution. This even includes cases in which the target viscosity values are located far in the tail of the initial ensemble probability distribution. Experiments show that the method is successful if enough near-field observations are available. This makes it work best for a period after substantial deglaciation until the present when the number of sea level indicators is relatively high.

Glacial isostatic adjustment (GIA) describes the continual response of the Earth to mass redistribution between continental glaciers, ice sheets, and the ocean during glacial cycles
(e.g.

Understanding GIA processes is essential for the quantification of past and recent sea level changes. In particular, the rheology of the Earth's mantle plays a significant role in surface deformation in the near-field of ice sheets

There have been a large number of studies that use different
techniques and data to infer the viscosity structure of the Earth's
mantle.

In this study, we present a method that allows us to draw conclusions about mantle viscosity values by assimilating relative sea level (RSL) observations into a viscoelastic Earth model. In an identical twin experiment, we assimilate RSL rates computed from a reference model. We apply a particle filter and study how the mean parameter state of the model ensemble converges to the target parameter state. First, we demonstrate the applicability of the method and then show two special cases that are relevant for the assimilation of real RSL observations.

The paper is organised as follows. In Sect.

In this study, we use the VIscoelastic model of the Lithosphere and
MAntle

We consider a 1D Earth structure, i.e. the mantle viscosity varies
only with depth, and it is set to respective constant values of

Depth structure of VILMA model in this study with number of spectral finite elements (SFEs) in the vertical direction per region and viscosity of the reference model.

As forcing, the surface mass load of the last glacial cycle in the
parameterisation of the ICE-5G reconstruction

Data assimilation provides a way to combine dynamic models with
observations

A well-known method for solving non-linear filtering problems with
non-Gaussian error statistics is the extended Kalman filter by

In our study, we used the particle filter (see Sect.

The particle filter is an ensemble-based data assimilation technique.
It follows the Monte Carlo view that any probability distribution can
be represented by a discrete sample from that distribution

The model update in our particle filter is based directly on Bayes' theorem (for an introduction, see, for example,

We use a particle filter with importance resampling and perturbation. Its principle is illustrated in Fig.

The particle filter principle. In the forecast phase, the dynamic model ensemble is propagated in time until observations become available. Then, the ensemble members are assigned a weight factor based on the residuals between the observations and the according values computed from the model states. Based on the weight, the ensemble is resampled. Members with low weights are disregarded, members with high weights are copied. The ensemble size stays constant. Finally, the model states of the ensemble members are perturbed, and the next model integration cycle starts with the updated ensemble.

Weighting the particles reduces the ensemble variance. If the variance
becomes very small, there is the risk of filter degeneracy

Figure

Resampling principle. After drawing a random number

For implementing the particle filter into the VILMA model, the parallel
data assimilation framework (PDAF) by

The experiments we present are conducted as sandbox experiments. We
use an identical twin set-up for the reference model run and the
assimilation simulation. Each ensemble is initialised from the model
state of the reference run

In Fig.

Set-up of the identical twin experiment. The ensemble of particles is initialised from the model state of the target run

As a starting point for the assimilation we chose

Parameters of the test cases investigated, with the standard deviation of RSL observations (

Geological sea level index points (SLIPs) allow us to reconstruct the
relative sea level (RSL) during the glacial cycle. RSL is defined as
the change of the water height,

To circumvent this problem, we consider the RSL rate, i.e. its time
derivative, which depends much less on the initial state. Despite the
RSL rate being a non-standard type of observation and knowing that
this quantity has to be derived from a series of SLIPs resulting in
increased uncertainties, we consider it as a tractable procedure. The
uncertainty of RSL rates has the following relationship to RSL
uncertainties:

The spatial distribution of collected SLIPs is rather heterogeneous, mainly spreading along the coasts of the continents, at islands, and concentrating to regions of large ice–water changes since the LGM. In order to run realistic scenarios, the synthetic observations were limited to locations where such data are available.

The grid point closest to each SLIP site was chosen for the
representation of the synthetic data. In that way, 1807 observation
points were obtained. They are unevenly distributed and located mostly
along the coasts of regions where large sea level changes have
occurred in the past or are still ongoing, e.g. Laurentide and
Fennoscandia (Fig.

The locations of synthetic observations are based on locations of real
observations in older compilations by Fleming (

Locations of SLIPs, which are used to generate synthetic observations. The observation locations were sub-divided into the following four regions: Laurentide and Greenland (green), Fennoscandia (blue), far field (purple), and other (red).

This study consists of three set-ups. The first set-up investigates the influence of observation uncertainty on the assimilation. In the second set-up, the observations were restricted to certain regions in order to test the performance of our approach when observations are not available globally. In the third set-up, the time interval with available observations was restricted to after 10 ka until the present day.

The first set-up is split into two scenarios based on the Cases A–D
listed in Table

In scenario one (Cases A–C), the initialisation is equal in all three
cases, i.e. perturbation with noise drawn from a normal distribution

In scenario two (Cases B and D), the influence of the initial offset (i.e. the mean of the initial perturbation; see Sect.

In Case E the RSL observation uncertainties vary with time. They are
set to realistic values of sea level indicator uncertainties that, on
average, agree with values found in the literature (e.g.

For the assimilation in set-up 2, four sets of observations were compiled. In the first set, all observations were used. This gives the best possible spatial and temporal coverage. In the three following scenarios, observations were restricted to (1) Laurentide and Greenland (Case LG), (2) Fennoscandia and northern Europe (Case FS), or (3) the far field. This was done to investigate under which conditions in the 1D model set-up regional observations can be used to obtain correct global viscosity values. When considering real SLIPs, observations might be available only in certain parts of the world, and it is important that our approach is proven successful under those conditions.

Looking at the temporal distribution of real SLIP observations, it becomes clear that most of them date from after the last glaciation. This is due to the fact that regions showing the largest post-glacial uplift were covered by ice during the glaciation period and only after deglaciation SLIPs could form. Therefore, in set-up 3 we tested our approach for the case of observations being available only after 10 ka. The parameters of the test in those cases correspond to Cases B10 and C10 with observation uncertainties of 0.25 and 0.5 m, respectively (see Table

In general, large ensemble sizes (especially for high-dimensional problems) are necessary to properly sample the model PDF. Due to the low dimensionality of the problem described here (only two distinct viscosity values), an ensemble size of 50 proved to be sufficient in the presented experiments.

In set-up 1, we studied the convergence of the weighted mean to the
target values of the reference model when all available observations
from the time interval 25.5 ka until the present day are considered.
Figure

Measures of misfit variation for the ensemble of set-up 1 (Cases A–C) in grey. Shown are the RMS values of the difference between the sea level rate of change observations and the model predictions of each ensemble member. The black lines show the total ice volume according to the external ice model. There are spikes following large changes in total ice mass. They appear due to the large amount of fresh water that changes the RSL independently of the viscosity model.

Figure

Variations in the viscosity in the course of the assimilation in the lower mantle

In scenario two (Cases B and D), we verify ensemble convergence for the case of the target viscosity value being in the tail of the initial ensemble PDF. That means we added an offset to the initial viscosities such that the target viscosities are far from the initial ensemble mean. Figure

Measures of misfit variation for the ensemble of scenario two in set-up 1 (Cases B and D) in grey. Shown are the RMS values of the difference between the observations and the model predictions. The black lines show the total ice volume according to the external ice model.

Figure

Variations in the viscosity in the course of the assimilation in the lower mantle

The final experiment in set-up 1 was designed to investigate the algorithm's behaviour under more realistic uncertainty conditions. In the first part of the model runs up to 17.5 ka, the assumed observations uncertainty is very high, and subsequently, the algorithm convergence does not start yet. When more precise observations become available, the algorithm starts to converge. The variation in RMS errors of the RSL rates is shown in Fig.

Measures of misfit variation for the ensemble of scenario three in set-up 1 (Case E) in grey. Shown are the RMS values of the difference between the observations and the model predictions. The black line shows the total ice volume according to the external ice model.

Figure

Variations in the viscosity in the course of the assimilation in the lower mantle

The standard deviation (SD) of the ensemble represents the uncertainty of the parameter estimation. Figure

Variation in the ensemble standard deviation of

Variation in the ensemble standard deviation of

Figure

For Case E, the picture is different. The effective ensemble size is very large at the beginning and slowly decreases during the assimilation run. At assimilation steps when the observation uncertainty is reduced significantly compared to the previous value, there is a more pronounced drop in the effective ensemble size. The lowest values are at 20 % of the nominal ensemble size after changes in observation uncertainty, but it rises to 50 % at the end of the assimilation.

Variation in the effective ensemble size over time. The effective ensemble size is low in the first assimilation step if the assumed observation uncertainty is too low for the initial ensemble spread. The ratio between the observation uncertainty and ensemble variance is low for Case A and larger for Cases B and C. It is largest for Case E with very high observation uncertainties. After resampling and perturbation in the first assimilation step, the effective ensemble size almost equal to the nominal ensemble size. For Cases A–D, there is a reduction in the effective ensemble size after large meltwater pulses at 14.5 and 9.5 ka, respectively. For Case E the effective ensemble size reduces when the observation uncertainty is reduced significantly from one assimilation step to the next.

In the first regional test (Case LG), we restricted the observations to
those located in the area of the Laurentide ice sheet and Greenland
(see Fig.

Figure

Measures of misfit variation in the ensemble of two regional observation sets, namely North America and Greenland

Figure

Variations in the viscosity in the course of the assimilation
in the lower mantle

Variation in the ensemble standard deviation of

In set-up 3, only observations taken after 10 ka were used in the assimilation. This corresponds to times after the last deglaciation. With this set-up, we demonstrate that the algorithm can reach convergence in a short period of time that is very relevant for real observations, as most SLIPs originate from the period since the early Holocene.

Figure

Measures of misfit variation for the ensemble of two sets of observations dating from after 10 ka. The statistical parameters of Cases B10 and C10 are equal to those of Cases B and C in Table

Figure

Variations in the viscosity in the course of the assimilation in the lower mantle

Figure

Variation in the ensemble standard deviation of

The results of set-up 1 show that the weighted ensemble mean converges to the target values during the assimilation. The final misfit of RSL and the ensemble variance scales with the assumed observation uncertainty. This is expected since, with increasing observation uncertainty, the correction of the dynamic models in the assimilation step is reduced. In the particle filter (PF) we used, ensemble members with low likelihood are resampled to model states with high likelihood. Larger observation uncertainties reduce the separability of models based on that measure. As a consequence, fewer models are resampled to better model states, the convergence slows down, and the final ensemble shows a larger variability.

Although, in general, the convergence of the ensemble is very good, there are some peaks in the RMS error of RSL rates at about 13.5 and 10 ka that appear suddenly and slow down the convergence. These peaks coincide with larger changes in ice volume (meltwater pulses) with a delay of 1 to 2 kyr. They can be explained by the large amount of meltwater flowing into the oceans and resulting in an RSL signal that dominates the sea level change at those times. However, that signal is independent of the viscosity model. As a consequence, the ensemble members cannot be effectively evaluated based on the current RSL rate misfit. Models that are relatively far from the ground truth are able to stay in the ensemble and produce large misfits in the first evaluation step after the meltwater pulse. In the following time steps with reduced melt rate, the mantle rheology again dominates the RSL rate, and the RMS errors are reduced to previous levels. The general level of misfit, and the peaks after sudden changes in ice volume, scale with the observation uncertainty assumed in that specific test case. This result shows that the interplay between meltwater and the Earth's response hinders the inference of structural parameters during this phase as the barystatic sea level change dominates, which is independent of the Earth's structure.

The convergence of the viscosities to the target values is very good. Generally, the convergence decreases slightly from Case A to Case C. The larger observation uncertainties allow particles to survive which are farther from the target model. The viscosity in the lower mantle show slower convergence than in the upper mantle. This is a general problem in GIA modelling (e.g.

The convergence of the ensemble mean viscosities to the target values of the reference runs in the presented cases show that, with our approach, we are able to recover mantle viscosities within a reasonable uncertainty range. This is even the case if the initial ensemble's PDF is far from the target values, i.e. the target value is in one of the tails of the initial PDF. A requirement is, however, that the sampling density of the ensemble near the target value is still high enough, such that the filter does not degenerate. Furthermore, the convergence is strongly influenced by the assumed observation uncertainties. Large uncertainties on the one hand slow down the convergence and lead to larger final variance within the ensemble. On the other hand, they reduce the chance of degeneration since particles with larger deviations from the target values are assigned higher likelihoods if observation uncertainties are higher.

Test Case E was run with considerably higher assumed observation uncertainties. The aim was to check the algorithm's success under near-realistic circumstances. That involves high uncertainties for older observations, and step-wise uncertainty decrease as observations become more recent. Under those conditions, the ensemble converges only after some time, when the uncertainty has become lower than 7 m. Clearly, the initial observation uncertainty results in an ensemble variance that is larger than the variance of the initial ensemble. Only after observations with lower uncertainty are available does the ensemble converge towards the target values. Prior to that, members relatively far from the ground truth are also assigned likelihood values that allow them to survive, and the ensemble mean does not yet converge towards the target value.

The results of set-up 2 show that we are able to recover target viscosities with only a subset of available observations. The uncertainty of the final estimations seems to depend only a little on the size of the observation region. The three-layer model with two mantle layers is simple enough to recover the viscosities with only a little more than 200 observation locations. However, there are differences in the lower mantle viscosity estimation. The larger variance in lower mantle viscosity from Fennoscandia when compared to Laurentide can be explained by the smaller size of the Fennoscandian ice
sheet. Accordingly, the GIA of Fennoscandia is less sensitive to the lower mantle viscosity structure (e.g.

In set-up 3, we show that it is also possible to estimate the target viscosities when only observations from a short time interval, i.e. from 10 ka until the present day, are available. The reasoning behind those test cases is that the 10 ka marks the end of the last deglaciation, and most real observations date from after that. The RMS misfit of the RSL rates obtained in these tests drop quickly after the onset of the assimilation. At 8 ka, the melting of Laurentide and Fennoscandian ice shields is finished, and from there on, ice mass changes can be seen only in Greenland and Antarctica. Therefore, in the time interval investigated here, the dominating ongoing process is the post-glacial rebound. All model realisations were able to reproduce (relatively slow) GIA processes taking place, and there is little variance in the ensemble. Such behaviour was already observed in set-ups 1 and 2, where the variance also drops significantly after 10 ka.

The results of this experiment show that our algorithm can quickly converge to the target values of the reference model under quiet conditions. With the term quiet, we mean that there are no large changes in the global ice mass within a short period of time, and therefore, the models have enough time to react viscoelastically to the new mass load. In that case, the RSL development is strictly a function of viscosity describing an exponential decrease (the relaxation process). On the other hand, if larger ice mass changes are present, the algorithm also converges, but it takes longer, and several assimilation steps are needed until the models adapt to the new mass load.

Ice models are a source of large uncertainties in GIA modelling. Usually, no uncertainties are provided for global ice models. However, ice histories from different approaches, e.g. ICE-5G by

Development of ensemble with erroneous ice load. The ice load was increased globally by 5 %. Red lines denotes the ensemble means and black lines the target values.

The focus of GIA-related sea level research lies on reconstructions of
the deglaciation since the last glacial maximum

Number of observations in ice-free locations over time. The grey line shows the development of number of observations. The horizontal black line is the number of far-field observations. Far-field observations do not contribute to constraining mantle viscosities.

In order to compute RSL rates from the RSL observations, several data points are necessary at one location. From the available sites which are commonly used to reconstruct the temporal evolution of the sea level from the late Pleistocene or Holocene to the present day, about 20 % to 30 % contain more than 10 samples. This, of course, further reduces the number of observations. However, from set-up 2, we have also seen that, with a limited number of observation sites (e.g. 209 in case of Fennoscandia), it is possible to recover the mantle viscosity target values.

In this study, we present a somewhat idealised set-up that is used for the development of the approach. There are several factors and ambiguities that play a role that could either not be addressed in the scope of this study or are beyond the current ability of the approach. One significant parameter in GIA is lithospheric thickness. First, it strongly varies laterally while we only assume a 1D Earth model. Second, there is a trade-off between the assumed lithospheric thickness and the obtained mantle viscosity values

In order to estimate the results of the PF approach, we made a simple
comparison to the so-called classical approach. In the latter, an
ensemble of models with fixed viscosity values is propagated in time,
and the resulting sea level rates of change are computed. At the end
of the model run period, RMS errors of the predicted sea level rates of
change are calculated. Here, the RSL rate misfits are compared to the
misfits of rates obtained with the PF approach. We constructed an
ensemble of 50 models that was used as the starting ensemble for both
methods. The ensemble is shown in Fig.

The RMS errors of the ensemble in the classical approach ranged from a
maximum of 4.36 m ka

The computation times of each approach are in the same order. It took
61 min for classical approach and 64 min for the PF approach. The relative overhead of the PF is smaller in this case than in the scenarios described in Sect.

If one member of the initial guess ensemble in the classical approach is very close to the true values, the resulting RMS error can be very small and outperform the PF approach. However, the PF approach yields a result that is as close to the true values as the observation uncertainty permits.

We have shown that our algorithm is able to recover a synthetic mantle viscosity structure through assimilation of palaeo sea level rates of change. This is the case even if the target viscosities are located in a tail of the initial distribution, thereby reducing the need for a very accurate initial guess. Furthermore, in contrast to the classical approach, the viscosity values obtained as the final result need not be part of the initial ensemble. This has the potential to lead to viscosity values that are closer to the truth than any member of the initial ensemble. This is even more important when the number of mantle layers is increased, and due to the larger number of possible combinations, it becomes more difficult to cover the whole space of parameter combinations in the classical approach.

Another advantage of our approach is the intermediate results that are
obtained in every assimilation step. The behaviour of RMS errors with
time points to specific events that are difficult to handle play an
important role in the modelling process. Those are, for example, the meltwater pulses. Also, the possibility to constrain mantle parameters with the observations at a given point in time can be investigated. This
depends on the number of observations available at a certain time span
and on their distribution. With only observations from the far field,
for example, we were not able to recover the GIA processes and the governing
mantle viscosities. This is due to the eustatic sea level change
controlling the signal in those regions

Some of the assumptions made about observation uncertainties need
closer inspection. Real SLIP uncertainties are usually in the range of
0.5 to 1 m for stratigraphic data and as precise as 0.1 to
0.2 m ka

In addition to SLIPs, which are defined as the band limits of palaeo sea level, other sea level data only indicate an upper or lower bound of palaeo sea level (e.g.

When applying the approach to real observations, it will be most promising if observations from after substantial deglaciation are used. This ensures a sufficient number of data points to constrain mantle viscosities, and we have shown to be successful in that set-up.

The ensemble size was limited to 50 members, mainly due to computational costs. For a proof of concept with synthetic data, this ensemble size is sufficient. In order to recover real viscosities, and as target values are unknown, a larger ensemble sampling a wider range of viscosity values might be necessary. On the other hand, when starting with large observation uncertainties that are gradually reduced as observations become more recent, a wide range of parameters can be sampled without the danger of filter degeneracy.

The effective ensemble size which is a measure for the robustness of the ensemble is very close to the nominal ensemble size in most of the cases. This indicates the filter is far from degeneration. Only in Case E, where realistic observation uncertainties were assumed, is the effective ensemble size reduced to about 50 % towards the end of the assimilation. Stronger drops in effective ensemble size appear at times when the observation uncertainty is reduced from one assimilation step to the next. However, the ensemble recovers from that reduction within a few assimilation steps. A smoother transition from high to low observation uncertainties could help to reduce this effect.

The computational costs are higher than for a pure forward model run with the same ensemble size. This is due to the overhead of the particle filter. The amount of overhead depends on the ensemble size, since some parts of the filter are performed in serial mode. With our ensemble size, the overhead was about 30 % to 40 %. The exact values depend on the allocation of computed nodes in the cluster.

Another crucial point is that we compute the rates of sea level change from two observations at a given location. The linear rate is attributed to the younger boundary of the time interval, while it really is a mean value for the whole interval. This introduces errors in the time of the observation and magnitude of sea level change. This is not a problem in our twin experiment, since observations and model predictions are treated in the same way. However, if real SLIP data are used, then there are additional dating errors for the sea level estimates. They have an impact on the rate of change uncertainty, and their consequential influence on the uncertainty of the viscosity estimates is a point that future investigations need to pay attention to. The motivation for using sea level rates of change was the fact that, in the applied assimilation method over time, we cannot iteratively determine the initial topography. With rates, we were able to overcome that problem; however, they have other disadvantages. For example, the number of locations with repeated observations (where rates can be computed) is much smaller than the total number of observation locations (about 30 % for the considered data). This limits the quality of the viscosity estimation. But other observation types are possible, e.g. differences to a defined base level. This has the advantages of both the absolute sea level values and rates.

Although RSL rates are a non-standard type of observation, there are
studies that model RSL rates during the Holocene, e.g.

In this first approach, we cover a rather simple 1D Earth structure with two mantle layers of constant viscosity, and we did not consider uncertainty in lithosphere thickness. While this is helpful for the algorithm development, more realistic scenarios involve radial viscosity profiles and even 3D viscosity variations. For a profound impact on the viscosity parameter estimation and regional sea level changes, this is an issue that will be addressed in future work.

Available SLIPs are sparse in the distant past and become more
numerous as the recent time is approached. But the situation improves due
to various groups working on constraining palaeo sea level rise under
PAGES like the HOLocene relative SEA level

Nowadays, even more sea level and ice mass data based on GPS, tide
gauges, or measurements of mass redistribution with satellite missions
such as, GRACE (Gravity Recovery and Climate Experiment) and GRACE-FO (Follow-On), are available. Uplift rates from GPS
measurements are widely used to invert mantle properties (e.g.

In the following, we list the basic equations for the response of a
self-gravitating Maxwell viscoelastic, incompressible sphere to surface mass load. The complete set of equations and the spectral–finite element approach can be found in

Consider a viscoelastic sphere

The fundamental properties of

At an internal discontinuity

Initial and boundary conditions are prescribed on the external surface
of

To summarise, “…the initial boundary value problem for the determination of the displacement field

In the following, we list the fundamental equations governing sea level
behaviour. The equations are taken from

Sea level (SL) is defined globally, and the difference between the radial
position of the geoid,

The synthetic data set used in this study has been submitted to a data repository at Helmholtz Centre Potsdam GFZ. It
is available at

RS took responsibility for the methodology, investigation, formal analysis, visualisation, and preparation of the initial draft. JSW was responsible for the conceptualisation, methodology, project administration, and project supervision. VK contributed the code for the VILMA model and supported the methodology. MB provided the auxiliary software and supported the investigation. MT was involved in the project administration and funding acquisition. All authors reviewed and edited the paper.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The numerical simulations for this study were performed at the German Climate Computing Centre (DKRZ). Some figures were produced using the generic mapping tool (GMT) by

This research has been supported by the Helmholtz Association (Advanced Earth System Modelling Capacity – ESM) and the Bundesministerium für Forschung und Technologie (grant nos. 01LP1502E, 01LP1503A, and 01LP1918A).The article processing charges for this open-access publication were covered by the Helmholtz Centre Potsdam – GFZ German Research Centre for Geosciences.

This paper was edited by Olivier Talagrand and reviewed by Peter Jan van Leeuwen, Dan Crisan, and two anonymous referees.