Assimilation Of Altimeter Data Research Articles

Method of assimilation of altimeter data and its test in the tropical North Atlantic

Method of assimilation of altimeter data and its test in the tropical North Atlantic

Adjoint assimilation of altimetric, surface drifter, and hydrographic data in a quasi‐geostrophic model of the Azores Current

The flow characteristics in the region of the Azores Current are investigated by assimilating TOPEX/POSEIDON and ERS 1 altimeter data into the multilevel Harvard quasigeostrophic (QG) model with open boundaries (Miller et al., 1983) using an adjoint variational scheme (Moore, 1991). The study site lies in the path of the Azores Current, where a branch retroflects to the south in the vicinity of the Madeira Rise. The region was the site of an intensive field program in 1993, SEMAPHORE. We had two main aims in this adjoint assimilation project. The first was to see whether the adjoint method could be applied locally to optimize an initial guess field, derived from the continous assimilation of altimetry data using optimal interpolation (OI). The second aim was to assimilate a variety of different data sets and evaluate their importance in constraining our QG model. The adjoint assimilation of surface data was effective in optimizing the initial conditions from OI. After 20 iterations the cost function was generally reduced by 50–80%, depending on the chosen data constraints. The primary adjustment process was via the barotropic mode. Altimetry proved to be a good constraint on the variable flow field, in particular, for constraining the barotropic field. The excellent data quality of the TOPEX/POSEIDON (T/P) altimeter data provided smooth and reliable forcing; but for our mesoscale study in a region of long decorrelation times O(30 days), the spatial coverage from the combined T/P and ERS 1 data sets was more important for constraining the solution and providing stable flow at all levels. Surface drifters provided an excellent constraint on both the barotropic and baroclinic model fields. More importantly, the drifters provided a reliable measure of the mean field. Hydrographic data were also applied as a constraint; in general, hydrography provided a weak but effective constraint on the vertical Rossby modes in the model. Finally, forecasts run over a 2‐month period indicate that the initial conditions optimized by the 20‐day adjoint assimilation provide more stable, longer‐term forecasts.

Assimilation of satellite altimeter data into an open ocean model

Geosat sea surface height data are assimilated into an eddy‐resolving quasi‐geostrophic open ocean model using the adjoint technique. The method adjusts the initial conditions for all layers and is successful on the timescale of a few weeks. Time‐varying values for the open boundaries are prescribed by a much larger quasi‐geostrophic model of the Antarctic Circumpolar Current (ACC). Both models have the same resolution of approximately 20×20 km (1/3°×1/6°), have three layers, and include realistic bottom topography and coastlines. The open model box is embedded in the African sector of the ACC. For continuous assimilation of satellite data into the larger model the nudging technique is applied. These results are used for the adjoint optimization procedure as boundary conditions and as a first guess for the initial condition. For the open model box the difference between model and satellite sea surface height that remains after the nudging experiment amounts to a 19‐cm root‐mean‐square error (rmse). By assimilation into the regional model this value can be reduced to a 6‐cm rmse for an assimilation period of 20 days. Several experiments which attempt to improve the convergence of the iterative optimization method are reported. Scaling and regularization by smoothing have to be applied carefully. Especially during the first 10 iterations, the convergence can be improved considerably by low‐pass filtering of the cost function gradient. The result of a perturbation experiment shows that for longer assimilation periods the influence of the boundary values becomes dominant and they should be determined inversely by data assimilation into the open ocean model.

Assimilation of Altimeter Data into a Quasigeostrophic Model of the Gulf Stream System Part II: Assimilation Results

Abstract The improvement in the climatological behavior of a numerical model as a consequence of the assimilation of surface data is investigated. The model used for this study is a quasigeostrophic (QG) model of the Gulf Stream region. The data that have been assimilated are maps of sea surface height that have been obtained as the superposition of sea surface height variability deduced from the Geosat altimeter measurements and a mean field constructed from historical hydrographic data. The method used for assimilating the data is the nudging technique. Nudging has been implemented in such a way as to achieve a high degree of convergence of the surface model fields toward the observations. Comparisons of the assimilation results with available in situ observations show a significant improvement in the degree of realism of the climatological model behavior, with respect to the model in which no data are assimilated. The remaining discrepancies in the model mean circulation seem to be mainly associated wi...

Assimilation of Altimeter Data into a Quasigeostrophic Model of the Gulf Stream System Part I: Dynamical Considerations

Abstract The dynamical consequences of constraining a numerical model with sea surface height data have been investigated. The model used for this study is a quasigeostrophic model of the Gulf Stream region. The data that have been assimilated are maps of sea surface height obtained as the superposition of sea surface height variability deduced from the Geosat altimeter measurements and a mean field constructed from historical hydrographic data. The method used for assimilating the data is the nudging technique. Nudging has been implemented in such a way as to achieve a high degree of convergence of the surface model fields toward the observations. The assimilations of the surface data is thus equivalent to the prescription of a surface pressure boundary condition. The authors analyzed the mechanisms of the model adjustment and the characteristics of the resultant equilibrium state when the surface data are assimilated. Since the surface data are the superposition of a mean component and an eddy component...

Assimilation of simulated altimeter data in a two-layer linear Rossby wave model using the variational method

The variational assimilation method has been examined for ability of reconstructing mesoscale features in altimeter data using a simple dynamic model. A one-dimensional, two-layer Rossby wave model in a cross-track channel has been chosen. The “simulated” data are constructed from a theoretical solution, which is composed of any combination of two normal vertical (barotropic and baroclinic) modes. The data are collected along tracks and with repeat periods similar to those of the Geosat altimeter. The phase space of control variables is composed of initial and boundary conditions. A cost function is defined to measure differences between the simulated data and the model solution. Regularization (smoothing) terms are also included in the cost function in the form of secon-order spatial and time derivatives of the solution. In this paper, two potential problems existing in the altimeter data assimilation are addressed: one is low cross-track resolution, and the other is vertical projection of the data measured at the sea surface. A succesful metho is developed for reconstructing Rossby waves with wavelengths as short as twice the track intervals for any combination of two vertical modes. A key component to efficient assimilation is a preparation step prior to the actual variational assimilation: a uniform ratio of pressure amplitudes in the two layers is included as an optimization parameter. Starting with the first guess from the preparation step, the variational method is carried out based on adjoint equations without such constraint. Separation of the control variables into the two subsets of the initial and the boundary conditions is found useful. Characteristics of the Hessian matrix are related to the performance of this technique. The method developed for the linear system implies steps to be included in data assimilation for nonlinear meanders and eddies in a major current system as well.

Assimilation of altimeter data into a quasigeostrophic ocean model using optimal interpolation and eofs

Abstract An ensemble of fraternal-twin experiments is used to assess the utility of optimal interpolation and model-based vertical empirical orthogonal functions (eofs) of streamfunction variability to assimilate satellite altimeter data into ocean models. Simulated altimeter data are assimilated into a basin-wide 3-layer quasi-geostrophic model with a horizontal grid spacing of 15 km. The effects of bottom topography are included and the model is forced by a wind stress curl distribution which is constant in time. The simulated data are extracted, along altimeter tracks with spatial and temporal characteristics of Geosat, from a reference model ocean with a slightly different climatology from that generated by the model used for assimilation. The use of vertical eofs determined from the model-generated streamfunction variability is shown to be effective in aiding the model's dynamical extrapolation of the surface information throughout the rest of the water column. After a single repeat cycle (17 days), the analysis errors are reduced markedly from the initial level, by 52% in the surface layer, 41% in the second layer and 11% in the bottom layer. The largest differences between the assimilation analysis and the reference ocean are found in the nonlinear regime of the mid-latitude jet in all layers. After 100 days of assimilation, the error in the upper two layers has been reduced by over 50% and that in the bottom layer by 38%. The essence of the method is that the eofs capture the statistics of the dynamical balances in the model and ensure that this balance is not inappropriately disturbed during the assimilation process. This statistical balance includes any potential vorticity homogeneity which may be associated with the eddy stirring by mid-latitude surface jets.

The sea surface pressure formulation of rigid lid models. Implications for altimetric data assimilation studies

The sea surface pressure formulation of the rigid lid primitive equation oceanic problem is reviewed and clarified. The geostrophic limit for the sea surface pressure equation is then considered and a new diagnostic relationship is found that relates the surface pressure to the barotropic and baroclinic components of the subsurface flow field. We demonstrate that a direct insertion in the model equations of sea surface information, such as that provided by satellite altimetry, does not produce changes in the subsurface dynamics due to the divergenceless nature of the barotropic flow field. The geostrophic limit of the sea surface pressure field computed from a standard general circulation model of the world ocean is presented and the barotropic/baroclinic components of the absolute dynamic topography of the global general circulation are discussed.

Comments on “the sea surface pressure formulation of rigid lid models. Implications for altimetric data assimilation studies” by N. Pinardi, A. Rosati and R. Pacanowski

Comments on “the sea surface pressure formulation of rigid lid models. Implications for altimetric data assimilation studies” by N. Pinardi, A. Rosati and R. Pacanowski

Assimilation of ERS-1 altimeter data in a tidal model of the continental shelf

In this paper ERS-1 altimeter data are used for the calibration of a tidal model of the European Continental Shelf. The data assimilation procedure to estimate the uncertain parameters in the model is based on the adjoint method. Here the estimation problem is formulated as a large scale optimization problem, that is solved with a gradient based optimization method. The gradient is determined efficiently by using the solution of the adjoint problem. The data assimilation procedure is applied by using the ERS-1 data with and without other measurement information from water level stations.

Seamount topography by least-squares inversion of altimetric geoid heights and shipborne profiles of bathymetry and/or gravity anomalies

SUMMARY Altimetric measurements of sea-surface heights have long been used to locate uncharted marine features. Newly available data sets (Geosat-GM, ERS-1 and Topex/Poseidon), now allow the topography of submarine volcanoes to be accurately restored and a method based upon inverse modelling is presented in this paper. This method allows a joint assimilation of altimetric data and shipborne profiles of bathymetry and/or gravity anomalies when available. Furthermore, uncertainties can be computed together with the topography, which constitutes a noticeable improvement since these parameters are still missing in the usual bathymetric data bases. Tests are performed using simulated data, in order to characterize the errors which may or must occur with the modelled topography of a previously unknown volcano, in particular the errors due to data uncertainties and the lack of precise knowledge of sensitive parameters that are used in the modelling. These errors can be maintained under the 100 m level (rms) for altimetric data with a measurement noise of 5 cm (rms) or less, which is achieved for Geosat, ERS-1 and Topex/Poseidon altimeters. An example of inversion of Seasat data is presented in the case of a seamount in French Polynesia.

The assimilation of altimetric data into the barotropic mode of a rigid lid ocean model

Some aspects of the computation of the barotropic mode of a primitive equations, rigid lid, global ocean model are reviewed in the perspective of data assimilation. It is shown that “blending” the surface pressure predicted by the model and that obtained from altimetry is impracticable, since mass conservation must be enforced. By relaxing the momentum equation, however, it is possible to take the measurements into account. A variational method, requiring that the perturbation to the momentum equation be minimal, is suggested. A Poisson equation is obtained for the streamfunction—allowing evaluation of the transport—for which a noniterative solution method is derived. The difference between the curl of the assimilated transport and that ensuing from the model unconstrained by data is investigated, pointing to the role of the bathymetry and that of the “implicitness” in the evaluation of the Coriolis acceleration. Finally, implications for altimetric data assimilation of the elliptic nature of the rigid lid equations are presented, and a comparison with the hyperbolic equations of a free surface model is outlined.

Assimilation of Altimeter Data in a Two-Layer Primitive Equation Model of the Gulf Stream

A two-layer finite depth, primitive equation model of the Gulf Stream region is used to study the effect of updating the model with simulated altimeter data as observations. In this study both a complete field of sea surface height (SSH) and SSH sampled along satellite tracks are used as “observations.” A simulated 17-day repeat orbit corresponding to the Geosat-ERM and a 10-day repeat period corresponding to Topex/Poseidon are used. Satellite observations give only information about the sea surface height but previous studies have shown that it is important to transfer the surface information to the lower layer as fast as possible in order for the model to have a realistic evolution. A statistical inference technique is therefore used to update the lower-layer pressure field. The velocity fields in both layers are updated using a geostrophic correction calculated from the change in the pressure fields. It is shown that updating the velocities is important for the assimilation to be successful. When complete fields of sea surface heights from an identical twin experiment are assimilated the rms error between the model solution and the “true” ocean for the upper-layer pressure field is reduced to less than 5% after six weeks of assimilation. Assimilation of simulated observations along Geosat-ERM and Topex/Poseidon tracks show that both satellites give similar levels of rms. error at the end of the assimilation period. The effect of assimilating altimeter data from two satellites is also discussed. The results show there can be a reduction in the rms error of up to 40% with the addition of a second satellite with appropriate orbital characteristics.

Sequential updating assimilation of simulated Geosat altimeter data from mesoscale features into a two-layer quasi-geostrophic model

A sequential updating method for assimilating Geosat altimeter data into an eddyresolving, quasi-geostrophic model is examined using simulated data of mesoscale features taken from a control run solution. The upper-layer streamfunction in the model is updated by the “altimeter” data on satellite tracks (at ∼110 km intervals) at times of observations (with 17-day cycles). To evaluate data suitability, the correlation between the data and the assimilation solution just before update is compared with the correlation between the two data with a 1-cycle separation: i.e., predictability is compared with persistence. The assimilation method is tested on mesoscale features such as linear Rossby waves, unstable mesoscale meanders in a jet and dipole eddies over realistic deep ocean topography. The assimilation method is successful for reconstructing the mesoscale features that evolve gradually or extend over more than one track. Assimilation is degraded by quick evolution of smaller scale features; i.e the unstable meanders that have short wavelengths and are not well captured by the altimeter with the low cross-track resolution, and the mesoscale features, whose lower layer component receives effects of bottom topography in the data but is underestimated due to inefficient downward transfer of the surface data in the assimilation.

A comparison of two methods for the assimilation of altimeter data into a shallow-water model

Abstract Two simple methods for assimilating ocean surface pressure data into a three-layer adiabatic primitive equation model are tested and compared using a twin-experiment approach. One of the methods is based on a recently presented direct insertion scheme using a quasi-geostrophic model. The method explicitly avoids making changes to the potential vorticity fields in the lower layers and the vertical current structure after assimilation is determined by this criterion. The modification required for a primitive equation framework are discussed and a comparison is made with the more familiar ‘nudging’ method of assimilation. We use root mean square errors to quantify the response of the model to the assimilation of a single complete surface pressure field and also to repeated or intermittent data available every 20 days for a period of 1 year. The two methods are almost equally effective in the short term, although the direct insertion scheme appears to be more effective in the longer term. It is suggested that this is because insertion maintains a geostrophic balance at all levels and thus avoids the generation of internal gravity waves which are needed when nudging is used to restore geostrophic currents in the deeper layers.

Toward altimetric data assimilation in a tropical Atlantic model

We present three types of experiments of sequential assimilation of altimetric data in a linear tropical Atlantic model, using an analysis technique we developed. In the first experiments we assimilate dynamic height fields calculated from simulated data in order to study the impacts of assimilations. In the second ones we assimilate dynamic height anomaly fields calculated from simulated data, and finally, in the third ones we use sea level anomaly fields obtained from altimetric data from Geosat. Perturbations are observed in the eastern part of the basin, due to Kelvin waves artificially generated in the west equatorial basin. However, we show that altimetric data may be very useful for improving model simulations in the tropics when appropriate assimilation techniques are used.

Parallel ocean general circulation modeling

We have developed a global ocean model on the massively parallel CM-2 Connection Machine based on the Bryan-Cox-Semtner ocean general circulation model. This paper discusses the Connection Machine implementation of the model and its performance, as well as major changes that were made in the algorithms and numerical methods. In order to increase the parallel efficiency, we reformulated the barotropic equations to solve for the surface-pressure field rather than the volume-transport streamfunction. In addition to being more efficient, this method has several other advantages over the streamfunction formulation: (1) any number of islands can be included in the computational grid at no extra cost; (2) the model can handle steep gradients in the bottom topography; and (3) the free-surface height is a prognostic variable, which simplifies assimilation of altimetric data into the model. We have also developed a new parallelizable preconditioning method for the solution of the elliptic equation for the surface-pressure field. In order to use a standard conjugate gradient algorithm, an approximate-factorization technique is used to split off the Coriolis terms and obtain a symmetric operator. We then apply a symmetric local approximate-inverse operator as a preconditioning matrix, which is very effective in accelerating convergence to a solution.

Assimilation of altimeter data in a global third‐generation wave model

We investigated the effect of the assimilation of altimeter satellite data in the third‐generation ocean wave model WAM. We used a sequential method, where analyzed significant wave height fields are created by optimum interpolation, and the analyzed values are then used to construct the analyzed wave spectrum. The method provides also an estimate of the surface stress showing the possibility of using the analysis of the wave spectrum to derive an analyzed surface stress field. In a first set of numerical experiments, the data, provided by the Seasat altimeter, have been assimilated in the WAM model for 1½ days. The comparison between model results and satellite data during the continuation of the run shows a positive and persistent impact of the assimilation. In a second set of numerical experiments, Geosat altimeter data were assimilated for 10 days and the resulting analysis was compared with buoy data. Although the assimilation improves the model results, it is not capable of compensating the differences between model and buoys. Some failures are clearly derived from the absence in the satellite data of the high‐wave events that were reported by the buoys. Other failures may be the consequence of an excessive swell attenuation in the WAM model, which compromises the effect of a previous correction. In fact, the comparison of WAM model results with altimeter data suggests that there is a tendency of the model to overevaluate initially the wind sea, and successively to overestimate the decay of the wave energy, when the waves leave the area of the storm.

Continuous assimilation in an open domain of the northeast Atlantic: 1. Methodology and application to AthenA‐88

A scheme to perform suboptimal intermittent assimilation in an open‐ocean, quasi‐geostrophic model is presented and applied to the assimilation of altimeter data in a domain of the northeast Atlantic west of Ireland. Both simulated and real altimeter data are used. Three‐dimensional (3‐D) synoptic observation fields and error variance fields are derived from synoptic dynamic topography anomaly estimates, using an empirical orthogonal mode (EOF) vertical extension technique. The 3‐D estimates are suboptimally combined with the time‐dependent part of the forecast fields. Suboptimality here means that the combination does not directly use spatial correlations of errors; however, the 3‐D synoptic observation estimates used in the combination do contain error spatial statistical information. The resulting fields added to a mean model climatology are used as initial conditions for the model, which is integrated until a new dynamic topography anomaly estimate is available. Locally, the combination is optimal in the sense that it is based on the observational error variances, which reflect the measurement noise and the space‐time distribution of data, and that it minimizes the error variance of the results. A similar combination scheme using past as well as future observations is used to update the boundaries during model integration. Simulated surface topography anomaly maps, typical of the northeast Atlantic, are assimilated every 20 days for a 300‐day period, with different noise characteristics typical of altimetric sampling errors, in a three‐level version of the model. The assimilation fields, especially the vorticity at deeper levels, converge toward the reference fields. The convergence is obtained after O(100 days), regardless of the observational noise levels tested (up to −2 dB). Using a relevant observational error model seems to matter, in particular, the error level should not be underestimated. In the case of an 80‐day gap in the data inflow, the model predictability limits the reliability of the forecast beyond 20 days. The scheme makes the model converge back as soon as new observations are entered, with limited convergence loss due to the gap. The case of an unknown model climatology is also studied. Finally, the scheme is applied to Geosat altimeter data, added to the Robinson, Bauer and Schroeder annual climatology, in the AthenA‐88 cruise area, which is contained in the model box. The assimilation results compare favorably with hydrographical data from the cruise and help the synoptic interpretation of the observed phenomena.

Nudging satellite altimeter data into quasi‐geostrophic ocean models

This paper discusses the efficiency of several variants of the nudging technique (derived from the technique of the same name developed by meteorologists) for assimilating altimeter data into numerical ocean models based on quasi‐geostrophic formulation. Assimilation experiments are performed with data simulated in the nominal sampling conditions of the Topex‐Poseidon satellite mission. Under experimental conditions it is found that nudging on the altimetric sea level is as efficient as nudging on the vorticity (second derivative in space of the dynamic topography), the technique used thus far in studies of this type. The use of altimetric residuals only, instead of the total altimetric sea level signal, is also explored. The critical importance of having an adequate reference mean sea level is largely confirmed. Finally, the possibility of nudging only the signal of sea level tendency (i.e., the successive time differences of the sea level height) is examined. Apart from the barotropic mode, results are not very successful compared with those obtained by assimilating the residuals.