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- Author or Editor: Robert N. Miller x
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Abstract
As ocean models improve, assimilation of data with the help of models becomes increasingly important. The Kalman filter provides a method for assimilation of data that are arbitrarily distributed in time and space and have differing error characteristics. Its desirable features are optimality in the least squares sense for a broad class of systems, and recursiveness, i.e., the algorithm depends only upon statistical quantities that are updated with each successive observation. The observations themselves may then be discarded, and no actual history of the system under study need be retained.
The full Kalman filter, however, presents considerable demands on computing resources. There are few examples with solutions in closed from, relatively little is known about the case in which the system under study is governed by partial rather than ordinary differential equation, and the effects of nonlinearity are still incompletely understood.
In this study a first step is undertaken toward the formulation of a suitably simplified, computationally efficient form of the Kalman filter for estimation and prediction of ocean eddy fields. In this step, the full Kalman filter is applied to simplified systems designed to capture some of the properties of open ocean models, and computational results are analyzed and interpreted in terms of realistic models are datasets.
Abstract
As ocean models improve, assimilation of data with the help of models becomes increasingly important. The Kalman filter provides a method for assimilation of data that are arbitrarily distributed in time and space and have differing error characteristics. Its desirable features are optimality in the least squares sense for a broad class of systems, and recursiveness, i.e., the algorithm depends only upon statistical quantities that are updated with each successive observation. The observations themselves may then be discarded, and no actual history of the system under study need be retained.
The full Kalman filter, however, presents considerable demands on computing resources. There are few examples with solutions in closed from, relatively little is known about the case in which the system under study is governed by partial rather than ordinary differential equation, and the effects of nonlinearity are still incompletely understood.
In this study a first step is undertaken toward the formulation of a suitably simplified, computationally efficient form of the Kalman filter for estimation and prediction of ocean eddy fields. In this step, the full Kalman filter is applied to simplified systems designed to capture some of the properties of open ocean models, and computational results are analyzed and interpreted in terms of realistic models are datasets.
Abstract
The Kalman filter is implemented and tested for a simple model of sea level anomalies in the tropical Pacific, using tide gauge data from six selected island stations to update the model. The Kalman filter requires detailed statistical assumptions about the errors in the model and the data. In this study, it is assumed that the model errors are dominated by the errors in the wind stress analysis. The error model is a simple covariance function with parameters fit from the observed differences between the tide gauge data and the model output. The fitted parameters are consistent with independent estimates of the errors in the wind stress analysis. The calibrated error model is used in a Kalman filtering scheme to generate monthly sea level height anomaly maps for the tropical Pacific. The filtered maps, i.e., those which result from data assimilation, exhibit fine structure that is absent from the unfiltered model output, even in regions removed from the data insertion points. Error estimates, an important byproduct of the scheme, suggest that the filter reduces the error in the equatorial wave guide by about 1 cm. The few independent verification points available are consistent with this estimate. Given that only six data points participate in the data assimilation, the results are encouraging, but it is obvious that model errors cannot be substantially reduced without more data.
Abstract
The Kalman filter is implemented and tested for a simple model of sea level anomalies in the tropical Pacific, using tide gauge data from six selected island stations to update the model. The Kalman filter requires detailed statistical assumptions about the errors in the model and the data. In this study, it is assumed that the model errors are dominated by the errors in the wind stress analysis. The error model is a simple covariance function with parameters fit from the observed differences between the tide gauge data and the model output. The fitted parameters are consistent with independent estimates of the errors in the wind stress analysis. The calibrated error model is used in a Kalman filtering scheme to generate monthly sea level height anomaly maps for the tropical Pacific. The filtered maps, i.e., those which result from data assimilation, exhibit fine structure that is absent from the unfiltered model output, even in regions removed from the data insertion points. Error estimates, an important byproduct of the scheme, suggest that the filter reduces the error in the equatorial wave guide by about 1 cm. The few independent verification points available are consistent with this estimate. Given that only six data points participate in the data assimilation, the results are encouraging, but it is obvious that model errors cannot be substantially reduced without more data.
Abstract
The Geosat altimeter sea level observations in the tropical Pacific Ocean are used to evaluate the Performance of a linear wind-driven equatorial wave model. The question posed is the extent to which such a model can describe the observed sea level variations. The Kalman filter and optimal smoother are used to obtain a solution that is an optimal fit to the observation in a weighted least-squares sense. The total mean variance of the Geosat sea level observation is 98.1 cm2, of which 36.6 cm2 is due to measurement errors, leaving 61.5 cm2 for the oceanographic signal to be explained. The model is found to account for about 68% of this signal Variance and the remainder is ascribed to the effects of physical mechanisms missing from the model. This result suggests that the Geosat data contains sufficient information for testing yet more sophisticated models. Utility of an approximate filter and smoother based on the asymptotic time limit of the estimation error covariance is also examined and compared with the estimates of the full time-evolving filter. The results are found to be statistically indistinguishable from each other, but the computational requirements are more than an order of magnitude less for the approximate filter/smoother. Corrections to the wind field that drives the model are also obtained by the smoother, but they are found only to be marginally improved when compared with in situ wind measurements. The substantial errors in the Geosat data and the simplicity of the present model prevents a reliable wind estimate from being made.
Abstract
The Geosat altimeter sea level observations in the tropical Pacific Ocean are used to evaluate the Performance of a linear wind-driven equatorial wave model. The question posed is the extent to which such a model can describe the observed sea level variations. The Kalman filter and optimal smoother are used to obtain a solution that is an optimal fit to the observation in a weighted least-squares sense. The total mean variance of the Geosat sea level observation is 98.1 cm2, of which 36.6 cm2 is due to measurement errors, leaving 61.5 cm2 for the oceanographic signal to be explained. The model is found to account for about 68% of this signal Variance and the remainder is ascribed to the effects of physical mechanisms missing from the model. This result suggests that the Geosat data contains sufficient information for testing yet more sophisticated models. Utility of an approximate filter and smoother based on the asymptotic time limit of the estimation error covariance is also examined and compared with the estimates of the full time-evolving filter. The results are found to be statistically indistinguishable from each other, but the computational requirements are more than an order of magnitude less for the approximate filter/smoother. Corrections to the wind field that drives the model are also obtained by the smoother, but they are found only to be marginally improved when compared with in situ wind measurements. The substantial errors in the Geosat data and the simplicity of the present model prevents a reliable wind estimate from being made.
Abstract
The latitudinal structure of annual equatorial Rossby waves in the tropical Pacific Ocean based on sea surface height (SSH) and thermocline depth observations is equatorially asymmetric, which differs from the structure of the linear waves of classical theory that are often presumed to dominate the variability. The nature of this asymmetry is such that the northern SSH maximum (along 5.5°N) is roughly 2 times that of the southern maximum (along 6.5°S). In addition, the observed westward phase speeds are roughly 0.5 times the predicted speed of 90 cm s−1 and are also asymmetric with the northern phase speeds, about 25% faster than the southern phase speeds. One hypothesized mechanism for the observed annual equatorial Rossby wave amplitude asymmetry is modification of the meridional structure by the asymmetric meridional shears associated with the equatorial current system. Another hypothesis is the asymmetry of the annually varying wind forcing, which is stronger north of the equator. A reduced-gravity, nonlinear, β-plane model with rectangular basin geometry forced by idealized Quick Scatterometer (QuikSCAT) wind stress is used to test these two mechanisms. The model with an asymmetric background mean current system perturbed with symmetric annually varying winds consistently produces asymmetric Rossby waves with a northern maximum (4.7°N) that is 1.6 times the southern maximum (5.2°S) and westward phase speeds of approximately 53 ± 13 cm s−1 along both latitudes. Simulations with a symmetric background mean current system perturbed by asymmetric annually varying winds fail to produce the observed Rossby wave structure unless the perturbation winds become strong enough for nonlinear interactions to produce asymmetry in the background mean current system. The observed latitudinal asymmetry of the phase speed is found to be critically dependent on the inclusion of realistic coastline boundaries.
Abstract
The latitudinal structure of annual equatorial Rossby waves in the tropical Pacific Ocean based on sea surface height (SSH) and thermocline depth observations is equatorially asymmetric, which differs from the structure of the linear waves of classical theory that are often presumed to dominate the variability. The nature of this asymmetry is such that the northern SSH maximum (along 5.5°N) is roughly 2 times that of the southern maximum (along 6.5°S). In addition, the observed westward phase speeds are roughly 0.5 times the predicted speed of 90 cm s−1 and are also asymmetric with the northern phase speeds, about 25% faster than the southern phase speeds. One hypothesized mechanism for the observed annual equatorial Rossby wave amplitude asymmetry is modification of the meridional structure by the asymmetric meridional shears associated with the equatorial current system. Another hypothesis is the asymmetry of the annually varying wind forcing, which is stronger north of the equator. A reduced-gravity, nonlinear, β-plane model with rectangular basin geometry forced by idealized Quick Scatterometer (QuikSCAT) wind stress is used to test these two mechanisms. The model with an asymmetric background mean current system perturbed with symmetric annually varying winds consistently produces asymmetric Rossby waves with a northern maximum (4.7°N) that is 1.6 times the southern maximum (5.2°S) and westward phase speeds of approximately 53 ± 13 cm s−1 along both latitudes. Simulations with a symmetric background mean current system perturbed by asymmetric annually varying winds fail to produce the observed Rossby wave structure unless the perturbation winds become strong enough for nonlinear interactions to produce asymmetry in the background mean current system. The observed latitudinal asymmetry of the phase speed is found to be critically dependent on the inclusion of realistic coastline boundaries.
Abstract
Kaiman filter theory and autoregressive time series are used to map sea level height anomalies in the tropical Pacific. Our Kalman filters are implemented with a linear state space model consisting of evolution equations for the amplitudes of baroclinic Kelvin and Rossby waves and data from the Pacific tide gauge network. In this study, three versions of the Kalman filter are evaluated through examination of the innovation sequences, that is, the time series of differences between the observations and the model predictions before updating. In a properly tuned Kalman filter, one expects the innovation sequence to be white (uncorrelated, with zero mean). A white innovation sequence can thus be taken as an indication that there is no further information to be extracted from the sequence of observations. This is the basis for the frequent use of whiteness, that is, lack of autocorrelation, in the innovation sequence as a performance diagnostic for the Kalman filter.
Our long-wave model embodies the conceptual basis of current understanding of the large-scale behavior of the tropical ocean. When the Kalman filter was used to assimilate sea level anomaly data, we found the resulting innovation sequence to be temporally correlated, that is, nonwhite and well fitted by an autoregressive process with a lag of one month. A simple modification of the way in which sea level height anomaly is represented in terms of the state vector for comparison to observation results in a slight reduction in the temporal correlation of the innovation sequences and closer fits of the model to the observations, but significant autoregressive structure remains in the innovation sequence. This autoregressive structure represents either a deficiency in the model or some source of inconsistency in the data.
When an explicit first-order autoregressive model of the innovation sequence is incorporated into the filter, the new innovation sequence is white. In an experiment with the modified filter in which some data were held back from the assimilation process, the sequences of residuals at the withheld stations were also white. To our knowledge, this has not been achieved before in an ocean data assimilation scheme with real data. Implications of our results for improved estimates of model error statistics and evaluation of adequacy of models are discussed in detail.
Abstract
Kaiman filter theory and autoregressive time series are used to map sea level height anomalies in the tropical Pacific. Our Kalman filters are implemented with a linear state space model consisting of evolution equations for the amplitudes of baroclinic Kelvin and Rossby waves and data from the Pacific tide gauge network. In this study, three versions of the Kalman filter are evaluated through examination of the innovation sequences, that is, the time series of differences between the observations and the model predictions before updating. In a properly tuned Kalman filter, one expects the innovation sequence to be white (uncorrelated, with zero mean). A white innovation sequence can thus be taken as an indication that there is no further information to be extracted from the sequence of observations. This is the basis for the frequent use of whiteness, that is, lack of autocorrelation, in the innovation sequence as a performance diagnostic for the Kalman filter.
Our long-wave model embodies the conceptual basis of current understanding of the large-scale behavior of the tropical ocean. When the Kalman filter was used to assimilate sea level anomaly data, we found the resulting innovation sequence to be temporally correlated, that is, nonwhite and well fitted by an autoregressive process with a lag of one month. A simple modification of the way in which sea level height anomaly is represented in terms of the state vector for comparison to observation results in a slight reduction in the temporal correlation of the innovation sequences and closer fits of the model to the observations, but significant autoregressive structure remains in the innovation sequence. This autoregressive structure represents either a deficiency in the model or some source of inconsistency in the data.
When an explicit first-order autoregressive model of the innovation sequence is incorporated into the filter, the new innovation sequence is white. In an experiment with the modified filter in which some data were held back from the assimilation process, the sequences of residuals at the withheld stations were also white. To our knowledge, this has not been achieved before in an ocean data assimilation scheme with real data. Implications of our results for improved estimates of model error statistics and evaluation of adequacy of models are discussed in detail.
Abstract
A linearized baroclinic, spectral-in-time tidal inverse model has been developed for assimilation of surface currents from coast-based high-frequency (HF) radars. Representer functions obtained as a part of the generalized inverse solution show that for superinertial flows information from the surface velocity measurements propagates to depth along wave characteristics, allowing internal tidal flows to be mapped throughout the water column. Application of the inverse model to a 38 km × 57 km domain off the mid-Oregon coast, where data from two HF radar systems are available, provides a uniquely detailed picture of spatial and temporal variability of the M 2 internal tide in a coastal environment. Most baroclinic signal contained in the data comes from outside the computational domain, and so data assimilation (DA) is used to restore baroclinic currents at the open boundary (OB). Experiments with synthetic data demonstrate that the choice of the error covariance for the OB condition affects model performance. A covariance consistent with assumed dynamics is obtained by nesting, using representers computed in a larger domain. Harmonic analysis of currents from HF radars and an acoustic Doppler profiler (ADP) mooring off Oregon for May–July 1998 reveals substantial intermittence of the internal tide, both in amplitude and phase. Assimilation of the surface current measurements captures the temporal variability and improves the ADP/solution rms difference. Despite significant temporal variability, persistent features are found for the studied period; for instance, the dominant direction of baroclinic wave phase and energy propagation is always from the northwest. At the surface, baroclinic surface tidal currents (deviations from the depth-averaged current) can be 10 cm s–1, 2 times as large as the depth-averaged current. Barotropic-to-baroclinic energy conversion is generally weak within the model domain over the shelf but reaches 5 mW m–2 at times over the slopes of Stonewall Bank.
Abstract
A linearized baroclinic, spectral-in-time tidal inverse model has been developed for assimilation of surface currents from coast-based high-frequency (HF) radars. Representer functions obtained as a part of the generalized inverse solution show that for superinertial flows information from the surface velocity measurements propagates to depth along wave characteristics, allowing internal tidal flows to be mapped throughout the water column. Application of the inverse model to a 38 km × 57 km domain off the mid-Oregon coast, where data from two HF radar systems are available, provides a uniquely detailed picture of spatial and temporal variability of the M 2 internal tide in a coastal environment. Most baroclinic signal contained in the data comes from outside the computational domain, and so data assimilation (DA) is used to restore baroclinic currents at the open boundary (OB). Experiments with synthetic data demonstrate that the choice of the error covariance for the OB condition affects model performance. A covariance consistent with assumed dynamics is obtained by nesting, using representers computed in a larger domain. Harmonic analysis of currents from HF radars and an acoustic Doppler profiler (ADP) mooring off Oregon for May–July 1998 reveals substantial intermittence of the internal tide, both in amplitude and phase. Assimilation of the surface current measurements captures the temporal variability and improves the ADP/solution rms difference. Despite significant temporal variability, persistent features are found for the studied period; for instance, the dominant direction of baroclinic wave phase and energy propagation is always from the northwest. At the surface, baroclinic surface tidal currents (deviations from the depth-averaged current) can be 10 cm s–1, 2 times as large as the depth-averaged current. Barotropic-to-baroclinic energy conversion is generally weak within the model domain over the shelf but reaches 5 mW m–2 at times over the slopes of Stonewall Bank.
Abstract
A linearized baroclinic, spectral-in-time tidal inverse model has been developed for assimilation of surface currents from coast-based high-frequency (HF) radars. Representer functions obtained as a part of the generalized inverse solution show that for superinertial flows information from the surface velocity measurements propagates to depth along wave characteristics, allowing internal tidal flows to be mapped throughout the water column. Application of the inverse model to a 38 km × 57 km domain off the mid-Oregon coast, where data from two HF radar systems are available, provides a uniquely detailed picture of spatial and temporal variability of the M 2 internal tide in a coastal environment. Most baroclinic signal contained in the data comes from outside the computational domain, and so data assimilation (DA) is used to restore baroclinic currents at the open boundary (OB). Experiments with synthetic data demonstrate that the choice of the error covariance for the OB condition affects model performance. A covariance consistent with assumed dynamics is obtained by nesting, using representers computed in a larger domain. Harmonic analysis of currents from HF radars and an acoustic Doppler profiler (ADP) mooring off Oregon for May–July 1998 reveals substantial intermittence of the internal tide, both in amplitude and phase. Assimilation of the surface current measurements captures the temporal variability and improves the ADP/solution rms difference. Despite significant temporal variability, persistent features are found for the studied period; for instance, the dominant direction of baroclinic wave phase and energy propagation is always from the northwest. At the surface, baroclinic surface tidal currents (deviations from the depth-averaged current) can be 10 cm s–1, 2 times as large as the depth-averaged current. Barotropic-to-baroclinic energy conversion is generally weak within the model domain over the shelf but reaches 5 mW m–2 at times over the slopes of Stonewall Bank.
Abstract
A linearized baroclinic, spectral-in-time tidal inverse model has been developed for assimilation of surface currents from coast-based high-frequency (HF) radars. Representer functions obtained as a part of the generalized inverse solution show that for superinertial flows information from the surface velocity measurements propagates to depth along wave characteristics, allowing internal tidal flows to be mapped throughout the water column. Application of the inverse model to a 38 km × 57 km domain off the mid-Oregon coast, where data from two HF radar systems are available, provides a uniquely detailed picture of spatial and temporal variability of the M 2 internal tide in a coastal environment. Most baroclinic signal contained in the data comes from outside the computational domain, and so data assimilation (DA) is used to restore baroclinic currents at the open boundary (OB). Experiments with synthetic data demonstrate that the choice of the error covariance for the OB condition affects model performance. A covariance consistent with assumed dynamics is obtained by nesting, using representers computed in a larger domain. Harmonic analysis of currents from HF radars and an acoustic Doppler profiler (ADP) mooring off Oregon for May–July 1998 reveals substantial intermittence of the internal tide, both in amplitude and phase. Assimilation of the surface current measurements captures the temporal variability and improves the ADP/solution rms difference. Despite significant temporal variability, persistent features are found for the studied period; for instance, the dominant direction of baroclinic wave phase and energy propagation is always from the northwest. At the surface, baroclinic surface tidal currents (deviations from the depth-averaged current) can be 10 cm s–1, 2 times as large as the depth-averaged current. Barotropic-to-baroclinic energy conversion is generally weak within the model domain over the shelf but reaches 5 mW m–2 at times over the slopes of Stonewall Bank.
