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CHAPTER 5 Data Assimilation by Models ICHIRO FUKUMORI Jet Propulsion Laboratory California Institute of Technology Pasadena CA 91109 1. INTRODUCTION Data assimilation is a procedure that combines observa- tions with models. The combination aims to better estimate and describe the state of a dynamic system, the ocean in the context of this book. The present article provides an overview of data assimilation with an emphasis on applica- tions to analyzing satellite altimeter data. Various issues are discussed and examples are described, but presentation of results from the non-altimetric literature will be limited for reasons of space and scope of this book. The problem of data assimilation belongs to the wider field of estimation and control theories. Estimates of the dy- namic system are improved by correcting model errors with the observations on the one hand and synthesizing observa- tions by the models on the other. Much of the original math- ematical theory of data assimilation was developed in the context of ballistics applications. In earth science, data as- similation was first applied in numerical weather forecast- ing. Data assimilation is an emerging area in oceanography, stimulated by recent improvements in computational and modeling capabilities and the increase in the amount of available oceanographic observations. The continuing in- crease in computational capabilities have made numerical ocean modeling a commonplace. A number of new ocean general circulation models have been constructed with dif- ferent grid structures and numerical algorithms, and incorpo- rating various innovations in modeling ocean physics (e.g., Gent and McWilliams, 1990; Holloway, 1992; Large et al., 1994). The fidelity of ocean modeling has advanced to a stage where models are utilized beyond idealized process studies and are now employed to simulate and study the actual circulation of the ocean. For instance, model results are operationally produced to analyze the state of the ocean (e.g., Leetmaa and Ji, 1989), and modeling the global ocean circulation at eddy resolution is nearing a reality (e.g., Fu and Smith, 1996). Recent oceanographic experiments, such as the World Ocean Circulation Experiment (WOCE) and the Tropical Ocean and Global Atmosphere Program (TOGA), have gen- erated unprecedented amounts of in situ observations. More- over, satellite observations, in particular satellite altimetry such as TOPEX/POSEIDON, have provided continuous syn- optic measurements of the dynamic state of the global ocean. Such extensive observations, for the first time, provide a suf- ficient basis to describe the coherent state of the ocean and to stringently test and further improve ocean models. However, although comprehensive, the available in situ measurements and those in the foreseeable future are and will remain sparse in space and time compared with the energy-containing scales of ocean circulation. An effective means of synthesizing such observations then becomes es- sential in utilizing the maximum information content of such observing systems. Although global in coverage, the na- ture of satellite altimetry also requires innovative approaches to effectively analyze its measurements. For instance, even though sea level is a dynamic variable that reflects circula- tion at depth, the vertical dependency of the circulation is not immediately obvious from sea-level measurements alone. The nadir-pointing property of altimeters also limits sam- pling in the direction across satellite ground tracks, making analyses of meso-scale features problematic, especially with a single satellite. Furthermore, the complex space-time sam- pling pattern of satellites caused by orbital dynamics makes analyses of even large horizontal scales nontrivial, especially Satelhte Altimetry and Earth Sciences 237 Copyright 9 2001 by Academic Press All rights of reproduction in any form reserved 23 8 SATELLITE ALTIMETRY AND EARTH SCIENCES for analyzing high-frequency variability such as tides and wind-forced barotropic motions. Data assimilation provides a systematic means to untan- gle such degeneracy and complexity, and to compensate for the incompleteness and inaccuracies of individual observing systems in describing the state of the ocean as a whole. The process is effected by the models' theoretical relationship among variables. Data information is interpolated and ex- trapolated by model equations in space, time, and into other variables including those that are not directly measured. In the process, the information is further combined with other data, which further improves the description of the oceanic state. In essence, assimilation is a dynamic extrapolation as well as a synthesis and averaging process. In terms of volume, data generated by a satellite altime- ter far exceeds any other observing system. Partly for this reason, satellite altimetry is currently the most common data type explored in studies of ocean data assimilation. (Other reasons include, for example, the near real-time data avail- ability and the nontrivial nature of altimetric measurements in relation to ocean circulation described above.) This chap- ter introduces the subject matter by describing the issues, particularly those that are often overlooked or ignored. By so doing, the discussion aims to provide the reader with a perspective on the present status of altimetric assimilation and on what it promises to accomplish. An emphasis is placed on describing what exactly data assimilation solves. In particular, assimilation improves the oceanic state consistent with both models and observations. This also means, for instance, that data assimilation does not and cannot correct every model error, and the results are not altogether more accurate than what the raw data mea- sure. This is because, from a pragmatic standpoint, mod- els are always incomplete owing to unresolved scales and physics, which in effect are inconsistent with models. Over- fitting models to data beyond the model's capability can lead to inaccurate estimates. These issues will be clarified in the subsequent discussion. We begin in Section 2 by reviewing some examples of data assimilation, which illustrate its merits and motivations. Reflecting the infancy of the subject, many published studies are of relatively simple demonstration exercises. However, the examples describe the diversity and potential of data as- similation's applications. The underlying mathematical problem of assimilation is identified and described in Section 3. Many of the issues, such as how best to perform assimilation, what it achieves, and how it differs from improving numerical models and/or data analyses per se, are best understood by first recognizing the fundamental problem of combining data and models. Many of the early studies on ocean data assimilation cen- ter on methodologies, whose complexities and theoretical nature have often muddied the topic. A series of different assimilation methods are heuristically reviewed in Section 4 with references to specific applications. Mathematical de- tails are minimized for brevity and the emphasis is placed in- stead on describing the nature of the approaches. In essence, most methods are equivalent to each other so long as the as- sumptions are the same. A summary and recommendation of methods is also presented at the end of Section 4. Practical Issues of Assimilation are discussed in Sec- tion 5. Identification of what the model-data combination resolves is clarified, in particular, how assimilation differs from model improvement per se. Other topics include prior error specifications, observability, and treatment of the time- mean sea level. We end this chapter in Section 6 with con- cluding remarks and a discussion on future directions and prospects of altimetric data assimilation. The present pace of advancement in assimilation is rapid. For other reviews of recent studies in ocean data assimila- tion, the reader is referred to articles by Ghil and Malanotte- Rizzoli (1991), Anderson et al. (1996), and by Robinson et al. (1998). The books by Anderson and Willebrand (1989) and Malanotte-Rizzoli (1996) contain a range of articles from theories and applications to reviews of specific prob- lems. A number of assimilation studies have also been col- lected in special issues of Dynamics of Atmospheres and Oceans (1989, vol 13, No 3-4), Journal of Marine Systems (1995, vol 6, No 1-2), Journal of the Meteorological Society of Japan (1997, vol 75, No 1B), and Journal of Atmospheric and Oceanic Technology (1997, vol 14, No 6). Several pa- pers focusing on altimetric assimilation are also collected in a special issue of Oceanologica Acta (1992, vol 5). 2. EXAMPLES AND MERITS OF DATA ASSIMILATION This section reviews some of the applications of data as- similation with an emphasis on analyzing satellite altimetry observations. The examples here are restricted because of limitation of space, but are chosen to illustrate the diversity of applications to date and to point to further possibilities in the future. One of the central merits of data assimilation is its ex- traction of oceanographic signals from incomplete and noisy observations. Most oceanographic measurements, including altimetry, are characterized by their sparseness in space and time compared to the inherent scales of ocean variability; this translates into noisy and gappy measurements. Figure 1 (see color insert) illustrates an example of the noise-removal aspect of altimetric assimilation. Sea-level anomalies mea- sured by TOPEX (left) and its model equivalent estimates (center and right) are compared as a function of space and time (Fukumori, 1995). The altimetric measurements (left panel) are characterized by noisy estimates caused by mea- surement errors and gaps in the sampling, whereas the as- similated estimate (center) is more complete, interpolating 5. DATA ASSIMILATION BY MODELS 23 9 FIGURE 2 A time sequence of sea-level anomaly maps based on Geosat data; (Left) model assimilation, (Right) statistical interpolation of the altimetric data. Contour interval is 2 cm. Shaded (unshaded) regions indicate negative (positive) values. The model is a 7-layer quasi-geostrophic (QG) model of the California Current, into which the altimetric data are assimilated by nudging. (Adapted from White et al. (1990a), Fig. 13, p. 3142.) over the data dropouts and removing the short-scale tempo- ral and spatial variabilities measured by the altimeter. In the process, the assimilation corrects inaccuracies in model sim- ulation (right panel), elucidating the stronger seasonal cycle and westward propagating signals of sea-level variability. The issue of dynamically interpolating sea level informa- tion is particularly critical in studying meso-scale dynam- ics, as satellites cannot adequately measure eddies because the satellite's ground-track spacing is typically wider than the size of the eddy features. Figure 2 compares a time se- quence of dynamically (i.e., assimilation; left column) and statistically (right column) interpolated synoptic maps of sea level by White et al. (1990a). The statistical interpolation is based solely on spatial distances between the analysis point and the data point (e.g., Bretherton et al., 1976), whereas the dynamical interpolation is based on assimilation with an ocean model. While the statistically interpolated maps tend to have maxima and minima associated with meso-scale eddies along the satellite ground-tracks, the assimilated es- timates do not, allowing the eddies to propagate without significant distortion of amplitude, even between satellite ground tracks. An altimeter's resolving power of meso-scale variability can also significantly improve variabilities simu- lated by models. For instance, Figure 3 shows distribution of sea-surface height variability by Oschlies and Willebrand (1996), comparing measurements of Geosat (middle) and an eddy-resolving primitive equation model. The bottom and top panels show model results with and without assimilation, respectively. The altimetric assimilation corrects the spatial distribution of variability, especially north of 30~ reducing the model's variability in the Irminger Sea but enhancing it in the North Atlantic Current and the Azores Current. The virtue of data assimilation in dynamically interpo- lating and extrapolating data information extends beyond the variables that are observed to properties not directly measured. Such an estimate is possible owing to the dy- namic relationship among different model properties. For in- stance, Figure 4 shows estimates of subsurface temperature (left) and velocity (right) anomalies of an altimetric assimi- lation (gray curve) compared against independent (i.e., non- assimilated) in situ measurements (solid curve) (Fukumori et al., 1999). In spite of the assimilated data being limited to sea-level measurements, the assimilated estimate (gray) is found to resolve the amplitude and timing of many of the subsurface temperature and velocity "events" better than the model simulation (dashed curve). The skill of the model re- sults are also consistent with formal uncertainty estimates (dashed and solid gray bars) that reflect inaccuracies in data and model. Such error estimates are by-products of assimi- lation that, in effect, quantify what has been resolved by the model (see Section 5.3 for further discussion). Although uncertainties in our present knowledge of the marine geoid (cf., Chapter 10) limit the direct use of alti- metric sea-level measurements to mostly that of temporal variabilities, the nonlinear nature of ocean circulation allows estimates of the mean circulation to be made from measure- 240 SATELLITE ALTIMETRY AND EARTH SCIENCES FIGURE 3 Root-mean-square variability of sea surface height; (a) model without assimilation, (b) Geosat data, (c) model with assimilation. Contour interval is 5 cm. The model is based on the Community Modeling Effort (CME; Bryan and Holland, 1989). Assimilation is based on optimal interpolation. (Adapted from Oschlies and Willebrand (1996), Fig. 7, p. 14184.) 5. DATA ASSIMILATION BY MODELS 241 FIGURE 4 Comparison of model estimates and in situ data; (A) temperature anomaly at 200 m 8~ 180~ (B) zonal velocity anomaly at 120 m 0~ 110~ The different curves are data (black), model simulation (gray dashed), and model estimate by TOPEX/POSEIDON assimilation (gray solid). Bars denote formal uncertainty estimates of the model. The model is based on the GFDL Modular Ocean Model, and the assimilation scheme is an approximate Kalman filter and smoother. This model and assimilation are further discussed in Sections 5.1.2, 5.1.4, and 5.2. (Adapted from Fukumori et al. (1999), Plates 4 and 5.) 240 220 200 180 160 140 120 ' ' ' ~ I , 'r , . I ' " ' '" ' ' I ' ' ' ' - . - . D20 NOASS ASS2 I00 ~ I ~ . L, ~ _., , ~ l -20 -15 -10 -5 latitude FIGURE 5 Time-mean thermocline depth (in m) along 95~ 20~ isotherm depth (plain), model simulation (dashed), and model with assimilating Geosat data (chain-dashed). The model is a non- linear 1.5-layer reduced gravity model of the Indian Ocean. Geosat data are assimilated over 1-year (November 1986 to October 1987) employing the adjoint method. The 20~ isotherm is deduced from an XBT analyses (Smith, 1995). (Adapted from Greiner and Perigaud (1996), Fig. 10, p. 1744.) ments of variabilities alone. Figure 5 compares such an esti- mate by Greiner and Perigaud (1996) of the time-mean depth of the thermocline in the Indian Ocean, based solely on as- similation of temporal variabilities of sea level measured by Geosat. The thermocline depth of the altimetric assimila- tion (chain-dash) is found to be significantly deeper between 10~ and 18~ than without assimilation (dash) and is in closer agreement with in situ observations based on XBT measurements (solid). Data assimilation's ability to estimate unmeasured prop- erties provides a powerful tool and framework to analyze data and to combine information systematically from mul- tiple observing systems simultaneously, making better esti- mates that are otherwise difficult to obtain from measure- ments alone. Stammer et al. (1997) have begun the process of synthesizing a wide suite of observations with a gen- eral circulation model, so as to improve estimates of the complete state of the global ocean. Figure 6 illustrates im- 242 SATELLITE ALTIMETRY AND EARTH SCIENCES FIGURE 6 Mean meridional heat transport (in 1015 W) estimate of a constrained (solid) and unconstrained (dashed lines) model of the At- lantic, the Pacific, and the Indian Oceans, respectively. The model (Mar- shall et al., 1997) is constrained using the adjoint method by assimilating TOPEX/POSEIDON data in addition to a hydrographic climatology and a geoid model. Bars on the solid lines show root-mean-square variability over individual 10-day periods. Open circles and bars show similar estimates and their uncertainties of Macdonald and Wunsch (1996). (Adapted from Stam- mer et al. (1997), Fig. 13, p. 28.) provements made in the time-mean meridional heat transport estimate from assimilating altimetric measurements from TOPEX/POSEIDON, along with a geoid estimate and a hy- drographic climatology. For instance, in the North Atlantic, the observations require a larger northward heat transport (solid curve) than an unconstrained model (dashed curve) that is in better agreement with independent estimates (cir- cles). Differences in heat flux with and without assimilation are equally significant in other basins. One of the legacies of TOPEX/POSEIDON is its im- provement in our understanding of ocean tides. Refer to Chapter 6 for a comprehensive discussion on tidal research using satellite altimetry. In the context of this chapter, a sig- nificant development in the last few years is the emergence of altimetric assimilation as an integral part of developing accurate tidal models. The two models chosen for reprocess- ing TOPEX/POSEIDON data are both based on combining observations and models (Shum et al., 1997). In particu- lar, Le Provost et al. (1998) give an example of the benefit of assimilation, in which the data assimilated tidal solution (FES95.2) is shown to be more accurate than the pure hy- drodynamic model (FES94.1) or the empirical tidal estimate (CSR2.0) used in the assimilation. That is, assimilated esti- mates are more accurate than analyses based either on data or model alone. FIGURE 7 Hindcasts of Nifio3 index of sea surface temperature (SST) anomaly with (a) and without (b) assimilation. The gray and solid curves are observed and modeled SSTs, respectively. The model is a simple coupled ocean-atmosphere model, and the assimilation is of altimetry, winds, and sea surface temperatures, conducted by the adjoint method. (Adapted from Lee et al. (2000), Fig. 10.) Data assimilation also provides a means to improve pre- diction of a dynamic system's future evolution, by provid- ing optimal initial conditions and other model parameters from which forecasts are issued. In fact, such applications of data assimilation are the central focus in ballistics ap- plications and in numerical weather forecasting. In recent years, forecasting has also become an important application of data assimilation in oceanography. For example, oceano- graphic forecasts in the tropical Pacific are routinely pro- duced by the National Center for Environmental Prediction (NCEP) (Behringer et al., 1998; Ji et al., 1998), with par- ticular applications to forecasting the E1 Nifio-Southern Os- cillation (ENSO). Of late, altimetric observations have also been utilized in the NCEP system (Ji et al., 2000). Lee et al. (2000) have explored the impact of assimilating al- timetry data into a simple coupled ocean-atmosphere model of the tropical Pacific. For example, Figure 7 shows improve- ments in their model's skill in predicting the so-called Nifio3 sea-surface temperature anomaly as a result of assimilating TOPEX/POSEIDON altimeter data. The model predictions (solid curves) are in better agreement with the observed in- dex (gray curve) in the assimilated estimate (left panel) than without data constraints (right panel). Apart from sea level, satellite altimetry also measures significant wave height (SWH), which is another oceano- graphic variable of interest. In particular, the European Cen- tre for Medium-Range Weather Forecasting (ECMWF) has been assimilating altimetric wave height (ERS 1) in produc- ing global operational wave forecasts (Janssen et al., 1997). Figure 8 shows an example of the impact of assimilating al- timetric SWH in improving predictions made by this wave model up to 5-days into the future (Lionello et al., 1995). 5. DATA ASSIMILATION BY MODELS 243 0,10 ~ 50 i~G ~ [G • 40 E 0,05 o ,,, ~ 30 L 'iiii.'il " -0,05 10 -0,I0 ~ 0 A 24 48 72 96 120 A 24 48 72 96 120 Forecast period in hours Forecast period in hours FIGURE 8 Bias and scatter index of significant wave height (SWH) analysis (denoted A on the abscissa) and various forecasts. Comparisons are between model and altimeter. Full (dotted) bars denote the reference experiment without (with) assimilating ERS-I significant wave height data. The scatter index measures the lack of correlation between model and data. The model is the third generation wave model WAM. Assimilation is performed by optimal interpolation. (Adapted from Lionello et al. (1995), Fig. 12, p. 105.) The figure shows the assimilation (dotted bars) resulting in a smaller bias (left panel) and higher correlation (i.e., smaller scatter) (right panel) with respect to actual wave- height measurements than those without assimilation (full bars). Further discussions on wave forecasting can be found in Chapter 7. In addition to the state of the ocean, data assimilation also provides a framework to estimate and improve model parameters, external forcing, and open boundary conditions. For instance, Smedstad and O'Brien (1991) estimated the phase speed in a reduced-gravity model of the tropical Pa- cific Ocean using sea-level measurements from tide gauges. Fu et al. (1993) and Stammer et al. (1997) estimated uncer- tainties in winds, in addition to the model state, from assim- ilating altimetry data. (The latter study also estimated errors in atmospheric heat fluxes.) Lee and Marotzke (1998) esti- mated open boundary conditions of an Indian Ocean model. Data assimilation in effect fits models to observations. Then, the extent to which models can or cannot be fit to data gives a quantitative measure of the model's consistency with measurements, thus providing a formal means of hy- pothesis testing that can also help identify specific deficien- cies of models. For example, Bennett et al. (1998) identified inconsistencies between moored temperature measurements and a coupled ocean-atmosphere model of the tropical Pa- cific Ocean, resulting from the model's lack of momentum advection. Marotzke and Wunsch (1993) found inconsisten- cies between a time-invariant general circulation model and a climatological hydrography, indicating the inherent nonlin- earity of ocean circulation. Alternatively, excessive model- data discrepancies found by data assimilation can also point to inaccuracies in observations. Examples of such analysis at present can be best found in meteorological applications (e.g., Hollingsworth, 1989). Lastly, data assimilation has also been employed in eval- uating merits of different observing systems by analyz- ing model results with and without assimilating particu- lar observations. For instance, Carton et al. (1996) found TOPEX/POSEIDON altimeter data having larger impact in resolving intra-seasonal variability of the tropical Pacific Ocean than data from a mooring array or a network of expendable bathythermographs (XBTs). Verron (1990) and Verron et al. (1996) conducted a series of numerical experi- ments (observing system simulation experiments, OSSEs, or twin experiments) to evaluate different scenarios of single- and dual-altimetric satellites. OSSEs and twin experiments are numerical experiments in which a set of pseudo obser- vations are extracted from a particular numerical simula- tion and are assimilated into another (e.g., with different ini- tial conditions and/or forcing, etc.) to examine the degree to which the former results can be reconstructed. The rela- tive skill of the estimate among different observing scenarios provides a measure of the observation's effectiveness. From such an analysis, Verron et al. (1996) conclude that a 10- 20 day repeat period is satisfactory for the spatial sampling of mid-latitude meso-scale eddies but that any further gain would come from increased temporal, rather than spatial, sampling provided by a second satellite that is offset in time. Twin experiments are also employed in testing and evaluat- ing different data assimilation methods (Section 4). 3. DATA ASSIMILATION AS AN INVERSE PROBLEM Recognizing the mathematical problem of data assimila- tion is essential in understanding what assimilation could achieve, where the difficulties exist, and where the issues arise from. For example, there are theoretical and practi- cal difficulties involved in solving the problem, and various assumptions and approximations are necessarily made, of- tentimes implicitly. A clear understanding of the problem is 244 SATELLITE ALTIMETRY AND EARTH SCIENCES critical in interpreting the results of assimilation as well as in identifying sources of inconsistencies. Mathematically, as will be shown, data assimilation is simply an inverse problem, such as, ,A(x) ~ y (1) in which the unknowns, vector x, are estimated by inverting some functional ,,4 relating the unknowns on the left-hand- side to the knowns, y, on the right-hand-side9 Equation (1) is understood to hold only approximately (thus ~ instead of =), as there are uncertainties on both sides of the equation9 Throughout this chapter, bold lowercase letters will denote column vectors. The unknowns x in the context of assimilation, are inde- pendent variables of the model that may include the state of the model, such as temperature, salinity, and velocity over the entire model domain, and various model parameters as well as unknown external forcing and boundary conditions9 The knowns, y, include all observations as well as known elements of the forcing and boundary conditions. The func- tional .,4 describes the relationships between the knowns and unknowns, and includes the model equations that dic- tate the temporal evolution of the model state. All variables and functions will be assumed discretized in space and time as is the case in most practical numerical model implemen- tations. The data assimilation problem can be identified in the form of Eq. (1) by explicitly noting the available relation- ships. Observations of the ocean at some particular instant (subscript i), yi, can be related to the state of the model (in- cluding all uncertain model parameters), xi, by some func- tional 7-r "~'~i (Xi) ~ Yi. (2) (The functional '~'~i is also dependent on i because the par- ticular set of observations may change with time i.) In case of a direct measurement of one of the model unknowns, 7"ti is simply a functional that returns the corresponding element of xi. For instance, if Yi were a scalar measurement of the j th element of xi, 7~i would be a row vector with zeroes except for its jth element being one: "]'~i (0 0, 1, 0, , 0). (3) Functional 7-r would be nontrivial for diagnostic quantities of the model state, such as sea level in a primitive equation model with a rigid-lid approximation (e.g., Pinardi et al., 1995). However, even for such situations, a model equiva- lent of the observation can be expressed by some functional 7"r as in Eq. (2), be it explicit or implicit. In addition to the observation equations (Eq. [2]), the model algorithm provides a constraint on the temporal evo- lution of the model state, that could be brought to bear upon the problem of determining the unknown model states x: Xi + 1 "~ -~'i (Xi). (4) Equation (4) includes the initialization constraint, x0 Xfirst guess" (5) Function ,~'i is, in practice, a discretization of the continu- ous equations of the ocean physics and embodies the model algorithm of integrating the model state in time from one ob- served instant i to another i + 1. The function generally de- pends on the state at i as well as any external forcing and/or boundary condition. (For multi-stage algorithms that involve multiple time-steps in the integration, such as the leap-frog or Adams-Bashforth schemes, the state at i could be defined as concatenated states at corresponding multiple time-steps.) Combining observation Eq. (2) and model evolution Eq. (4), the assimilation problem as a whole can be written as, i "~i (Xi) Yi Xi+l '.~'i (Xi) 0 (6) By solving the data and model equations simultaneously, as- similation seeks a solution (model state) that is consistent with both data and model equation. Eq. (6) defines the assimilation problem and can be rec- ognized as a problem of the form Eq. (1), where the states in Eq. (6) at different time steps ( xT ' xr+l )7" define the unknown x on the left-hand side of Eq. (1). Typically, the number of unknowns far exceed the number of independent equations and the problem is ill-posed. Thus, data assimi- lation is mathematically equivalent to other inverse prob- lems such as the classic box model geostrophic inversion (Wunsch, 1977) and the beta spiral (Stommel and Schott, 1977). However, what distinguishes assimilation problems from other oceanic inverse problems is the temporal evolu- tion and the sophistication of the models involved. Instead of simple constraints such as geostrophy and mass conserva- tion, data assimilation employs more general physical prin- ciples applied at much higher resolution and spatial extent. The intervariable relationship provided by the model equa- tions solved together with the observation equations allows data information to affect the model solution in space and time, both with respect to times that formally lie in the future and past of the observed instance, as well as among different properties. From a practical standpoint, the distinguishing property of data assimilation is its enormous dimensionality. Typical ocean models contain on the order of several million inde- pendent variables at any particular instant. For example, a global model with 1 ~ horizontal resolution and 20 vertical 5. DATA ASSIMILATION BY MODELS 245 levels is a fairly coarse model by present standards, yet it would have 1.3 million grid points (360 x 180 x 20) over the globe. With four independent variables per grid node (the two components of horizontal velocity, temperature, and salinity), such as in a primitive equation model with the rigid-lid approximation, the number of unknowns would equal 5 million globally or approximately 3 million when counting points only within the ocean. The amount of data is also large for an altimeter. For TOPEX/POSEIDON, the Geophysical Data Record pro- vides a datum every second, which over its 10-day repeat cy- cle amount to approximately 500,000 points over the ocean, which is an order of magnitude larger than the number of horizontal grid points of the 1 ~ model considered above. In light of the redundancy the data would provide for such a coarse model, the altimeter could be thought of as providing sea level measurements at the rate of one measurement at ev- ery grid point per repeat cycle. Then, assuming for simplic- ity that all observations within a repeat cycle are coincident in time, each observation equation of form Eq. (2) would have approximately 50,000 equations, and there would be 180 such sets (time-levels or different i's) over a course of a 5-year mission amounting to 9 million individual observa- tion equations. The number of time-levels involved in the observation equations would require at least as many for the model equations in Eq. (6), amounting to 540 million (180 x 3 million) individual model equations. The size of such a problem precludes any direct approach in solving Eq. (6), such as deriving the inverse of the opera- tor on the left-hand side even if it existed. In practice, there is generally no solution that exactly satisfies Eq. (6), because of inaccuracies of models and uncertainties in observations. In- stead, an approximate solution is sought that solves the equa- tions as "close" as possible in some suitably defined manner. Several ingenious inverse methods are known and/or have been developed, and are briefly reviewed in the section be- low. 4. ASSIMILATION METHODOLOGIES Because of the problem's large computational task, de- vising methods of assimilation has been one of the central issues in data assimilation. Many assimilation methods have been put forth and explored, and they are heuristically re- viewed in this section. The aim of this discussion is to elu- cidate the nature of different methods and thereby allow the reader familiarity with how the problems are approached. Rigorous descriptions of the methods are deferred to refer- ences herein. Assimilation problems are in practice ill-posed, in the sense that no unique solution satisfies the problem Eq. (6). Consequently, many assimilation methodologies are based on "classic" inverse methods. Therefore, for reference, we will begin the discussion with a simple review of the na- ture of inverse methods. Different assimilation methodolo- gies are then individually described, preceded by a brief overview so as to place the approaches into a broad per- spective. A Summary and Recommendation is given in Sec- tion 4.11. 4.1. Inverse Methods Comprehensive mathematical expositions of oceano- graphic inverse problems and inverse methods can be found, for example, in the textbooks of Bennett (1992) and Wunsch (1996). Here we will briefly review their nature for refer- ence. Inverse methods are mathematical techniques that solve ill-posed problems that do not have solutions in the strict mathematical sense. The methods seek solutions that ap- proximately satisfy constraints, such as Eq. (6), under suitable "optimality" criteria. These criteria include, vari- ous least-squares, maximum likelihood, and minimum-error variance (Bayesian estimates). Differences among the crite- ria lie in what are explicitly assumed. Least-squares methods seek solutions that minimize the weighted sum of differences between the left- and right-hand sides of an inverse problem (Eq. [1 ]): ,5" = (y - .A(x)) r W-1 (y _ .A(x)) (7) where W is a matrix defining weights. Least-squares methods do not have explicit statistical or probabilistic assumptions. In comparison, the maximum likelihood estimate seeks a solution that maximizes the a posteriori probability of the right-hand side of Eq. (6) by invoking particular probability distribution functions for y. The minimum variance estimate solves for solutions x with minimum a posteriori error variance by assuming the error covariance of the solution's prior expectation as well as that of the right-hand side. Although seemingly different, the methods lead to iden- tical results so long as the assumptions are the same (see for example Introduction to Chapter 4 of Gelb [1974] and Sec- tion 3.6 of Wunsch [ 1996]). In particular, a lack of an explicit assumption can be recognized as being equivalent to a par- ticular implicit assumption. For instance, a maximum likeli- hood estimate with no prior assumptions about the solution is equivalent to assuming an infinite prior error covariance for a minimum variance estimate. For such an estimate, any solution is acceptable as long as it maximizes the a posteriori probability of the right-hand side (Eq. [6]). Based on the equivalence among "optimal methods," Eq. (7) can be regarded as a practical definition of what various inverse methods solve (and therefore assimilation). Furthermore, the equivalence provides a statistical basis for prescribing weights used in Eq. (7). In particular, W can be 246 SATELLITE ALTIMETRY AND EARTH SCIENCES identified as the error covariance among individual equations of the inverse problem Eq. (6). When the weights of each separate relation are uncorre- lated in time, Eq. (7) may be expanded as, ,.q,- M T (Yi "~i (Xi)) ]~i=o(Yi 7-~i(Xi)) R~ -1 qt_ ]~M0(xi+I __ ff~'i(Xi))TQ-~l(xi+l __ .~'i (Xi)) (8) where R and Q denote weighting matrices of data and model equations, respectively, and M is the total number of obser- vations of form Eq. (2). Most assimilation problems are for- mulated as in Eq. (8), i.e., uncertainties are implicitly as- sumed to be uncorrelated in time. The statistical basis of optimal inverse methods allows explicit a posteriori uncertainty estimates to be derived. Such estimates quantify what has been resolved and is an inte- gral part of an inverse solution. The errors identify what is accurately determined and what remains indeterminate, and thereby provide a basis for interpreting the solution and a means to ascertain necessary improvements in models and observing systems. 4.2. Overview of Assimilation Methods Many of the so-called "advanced" assimilation methods originate in estimation and control theories (e.g., Bryson and Ho, 1975; Gelb, 1974), which in turn are based on "clas- sic" inverse methods. These include the adjoint, represen- ter, Kalman filter and related smoothers, and Green's func- tion methods. These techniques are characterized by their explicit assumptions under which the inverse problem of Eq. (6) is consistently solved. The assumptions include, for example, the weights W used in the problem identification (Eq. [7]) and specific criteria in choosing particular "opti- mal" solutions, such as least-squares, minimum error vari- ance, and maximum likelihood. As with "classic" inverse methods, these assimilation schemes are equivalent to each other and result in the same solution as long as the assump- tions are the same. Using specific weights allows for explic- itly accounting for uncertainties in models and data, as well as evaluation of a posteriori errors. However, because of sig- nificant algorithmic and computational requirements in im- plementing these optimal methods, many studies have ex- plored developing and testing alternate, simpler approaches of combining model and data. The simpler approaches include optimal interpolation, "3D-var," "direct insertion," "feature models," and "nudg- ing." Many of these approaches originate in atmospheric weather forecasting and are largely motivated in making practical forecasts by sequentially modifying model fields with observations. The methods are characterized by various ad hoc assumptions (e.g., vertical extrapolation of altimeter data) to effect the simplification, but the results are at times obscured by the nature of the choices made without a clear understanding of the dynamical and statistical implications. Although the methods aim to adjust model fields towards ob- servations, it is not entirely clear how the solution relates to the problem identified by Eq. (6). Many of the simpler ap- proaches do not account for uncertainties, potentially allow- ing the models to be forced towards noise, and data that are formally in the future are generally not used in the estimate except locally to yield a temporally smooth result. However, in spite of these shortcomings, these methods are still widely employed because of their simplicity, and, therefore, warrant examination. 4.3. Adjoint Method Iterative gradient descent methods provide an effec- tive means of solving minimization problems of form Eq. (7), and a particularly powerful method of obtaining such gradients is the so-called adjoint method. The adjoint method transforms the unconstrained minimization problem of Eq. (7) into a constrained one, which allows the gradi- ent of the "cost function" (Eq. [7]), 03"/0x, to be evaluated by the model's adjoint (i.e., the conjugate transpose [Her- mitian] of the model derivative with respect to the model state variables [Jacobian]). Namely, without loss of general- ity, uncertainties of the model equations (Eq. [4]) are treated as part of the unknowns and moved to the left-hand side of Eq. (6). The resulting model equations are then satisfied identically by the solution that also explicitly includes er- rors of the model as part of the unknowns. As a standard method for solving constrained optimization problems, La- grange multipliers are introduced to formally transform the constrained problem back to an unconstrained one. The La- grange multipliers are solutions to the model adjoint, and in turn give the gradient information of ,3" with respect to the unknowns. The computational efficiency of solving the adjoint equations is what makes the adjoint method partic- ularly useful. Detailed derivation of the adjoint method can be found, for example, in Thacker and Long (1988). Methods that directly solve the minimization problem (7) are sometimes called variational methods or 4D-var (four- dimensional variational method). Namely, four-dimensional for minimization over space and time and variational be- cause of the theory based on functional variations. However, strictly speaking, this reference is a misnomer. For example, Kalman filtering/smoothing is also a solution to the four- dimensional optimization problem, and to the extent that as- similation problems are always rendered discrete, the adjoint method is no longer variational but is algebraic. Many applications of the adjoint are of the so-called "strong constraint" variety (Sasaki, 1970), in which model equations are assumed to hold exactly without errors making initial and boundary conditions the only model unknowns. As a consequence, many such studies are of short dura- tion because of finite errors in f" in Eq. (4) (e.g., Greiner [...]... and where A r 2 days (Munk and Cartwright, 1966) Ulmn and Vls n are the orthoweights This formalism has been used in several of the models, which will be introduced later For example, Desai and Wahr (1995) computed their orthoweights using 161 tidal components in the diurnal band and 116 in the semidiurnal band 270 SATELLITE ALTIMETRY AND EARTH SCIENCES 3 STATUS BEFORE HIGH-PRECISION SATELLITE ALTIMETRY. .. buoy and altimeter data, Weather Forecasting, 12, 763-784 264 SATELLITE ALTIMETRYAND EARTH SCIENCES Ji, M., D W Behringer, and A Leetmaa, (1998) An improved coupled model for ENSO prediction and implications for ocean initialization Part II: The coupled model Mon Weather Rev., 126, 1022-1034 Ji, M., R W Reynolds, and D W Behringer, (2000) Use of TOPEX/POSEIDON sea level data for ocean analyses and. .. adjoint may be found in Stammer et al (1997) and Lee and Marotzke (1998) (See also Griffith and Nichols, 1996.) Adjoint methods have been used to assimilate altimetry data into regional quasi-geostrophic models (Moore, 1991; Schr6ter et al., 1993; Vogeler and Schr6ter, 1995; Morrow and De Mey, 1995; Weaver and Anderson, 1997), shallow water models (Greiner and Perigaud, 1994, 1996; Cong et al., 1998),... minimization of Eq (8) For instance, Sheinbaum and Anderson (1990), in investigating assimilation of XBT data, used a smoothness constraint of the form, (VHX) 2 -q- (VH2X) 2 (27) 25 6 SATELLITE ALTIMETRY AND EARTH SCIENCES where V H is a horizontal gradient operator The gradient and Laplacian operators are linear operators and can be expressed by some matrix, G and L, respectively Then Eq (27) can be written... example, Fu and Fukumori (1996) examined effects of the differences in covariances of orbit and residual tidal errors in altimetry Orbit error is a slowly decaying function of time following the satellite ground track, and is characterized by a dominating period of once per satellite revolution around the globe Geographically, errors are positively correlated along satellite ground tracks, and weakly... (Accad and Pekeris, 1978; Parke and Hendershott, 1980) A strong improvement of the numerical tidal models resulted from the introduction of earth tides, ocean tide loading, and self-attraction (Hendershott, 1977; Zahel, 1977; Accad and Pekeris, 1978; Parke and Hendershott, 1980) Although these hydrodynamic numerical models brought very significant contributions to our understanding of the tidal regimes and. .. function" (Eq [8]) require careful selection Different assimilations often make different assumptions, and the adequacy and implication of their particular suppositions must properly be assessed These and other practical issues of assimilation are reviewed in the following section 25 2 SATELLITE ALTIMETRY AND EARTH SCIENCES 5 PRACTICAL ISSUES OF ASSIMILATION As described in the previous section, assimilation... large The assumed data locations are (A) 60~ 170~ and (B) 0~ 170~ Corresponding effects of the changes on sea level are small due to relatively large magnitudes of data error with respect to model error; changes are 0.02 and 0.03 cm at the respective data locations for (A) and (B) (Adapted from Fukumori et al (1999), Fig 4.) 25 8 SATELLITE ALTIMETRY AND EARTH SCIENCES without assimilation or the assimilated... While on one hand, sequential assimilation transfers surface information into the interior of the ocean, on the other hand, future observations also contain 260 SATELLITE ALTIMETRY AND EARTH SCIENCES information of the past state That is, the entire temporal evolution of the measured property, viz., indices i = 0 M in Eq (34), provides information in determining the model state and thus the observability... are often chosen more or less subjectively, and a systematic effort is required to better characterize and understand the a priori uncertainties and thereby the weights In particular, the significance of representation error is often under-appreciated Quantifying what models and observing systems respectively do and do not represent is arguably the most urgent and important issue in estimation In fact, . in Sections 5.1 .2, 5.1.4, and 5 .2. (Adapted from Fukumori et al. (1999), Plates 4 and 5.) 24 0 22 0 20 0 180 160 140 120 ' ' '. illustrates im- 24 2 SATELLITE ALTIMETRY AND EARTH SCIENCES FIGURE 6 Mean meridional heat transport (in 1015 W) estimate of a constrained (solid) and unconstrained