Ocean altimeter assimilation with observational- and model-bias correction

Transcription

1 QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Published online in Wiley InterScience ( Ocean altimeter assimilation with observational- and model-bias correction D. J. Lea, a,b * J.-P. Drecourt, a K. Haines a and M. J. Martin b a ESSC, Reading University, UK b Met Office, Exeter, UK ABSTRACT: We implement a combined on-line model- and observation-bias correction scheme in the UK Met Office Forecasting Ocean Assimilation Model (F) Unified Model ocean data assimilation system. The observation bias is designed to estimate the error in the mean dynamic topography that is used for altimeter data assimilation. In future, this mean dynamic topography and its errors may be derived from the Gravity field and steady-state Ocean Circulation Explorer (GOCE) mission geoid data. The mean dynamic topography field is added to the altimeter data supplied as sea-level anomalies, giving the absolute dynamic topography. The model-bias scheme separately estimates the remaining bias in the model s sea surface height field. The final unbiased estimate of the absolute dynamic topography is assimilated into the F model by adjusting the subsurface density field using the Cooper and Haines scheme. Various diagnostics, including the observation minus background statistics, show that both model- and observation-bias correction schemes improve the assimilation results. Combining the schemes provides better results than either used alone. Copyright c Royal Meteorological Society and Crown KEY WORDS Bayesian; F; mean dynamic topography Received 26 October 27; Revised 9 May 28; Accepted 14 August Introduction The application of bias analysis methods in atmospheric and particularly in oceanographic data assimilation is still relatively rare. The work of Dee and Da Silva (1998) brought a sound theoretical basis to this field for atmospheric science applications with a focus on the case of forecast bias alone. In this case, the evolving model is subject to drift and the aim is to estimate the bias as the time-mean error in the short-term model forecasts. The recent review paper of Dee (25) gives an excellent background to the application of modelbias approaches. Applications to observation bias have also been necessary, particularly for the assimilation of satellite observations in weather forecasting (Julian, 1991; Eyre, 1992). A general theory in which both observations and model are subject to bias makes the problem of distinguishing the two effects very difficult indeed. This is because we only see the observation and model differences, and therefore only the combined effect of the model and observation bias. We therefore require additional information in order to separate the model and observation bias. This includes spatial information given by the respective error covariances and temporal *Correspondence to: D. J. Lea, Met Office, FitzRoy Road, Exeter, EX1 3PB, UK. daniel.lea@metoffice.gov.uk The contributions of D.J. Lea and M.J. Martin were written in the course of their employment at the Met Office, UK and are published with the permission of the Controller of HMSO and the Queen s Printer for Scotland. information, which can be introduced with differing models for the time evolution of the bias. We propose, and demonstrate, that despite these difficulties, splitting the model and observation bias is a better solution than only assuming observation or model bias alone. Ocean altimeter data have provided routine sea-level or dynamic topography information over the global oceans for 15+ years now. However, altimetry can only measure the time-varying dynamic topography signal because the absolute dynamic topography (ADT) is the sea level with respect to the gravity geopotential surface (the geoid), and this geoid is still not well enough known. Indeed, the European Space Agency (ESA) are launching a new satellite, the Gravity Field and Steady-state Ocean Circulation Explorer (GOCE; European Space Agency, 1999) in 28, to measure the geoid more accurately. When assimilating satellite altimeter data into ocean circulation models, it is necessary to add a mean dynamic topography (MDT) to the altimeter measurements of the dynamic topography anomalies. This MDT can have an error that is considerably larger than that of the altimeter data, although it is constant in time. Thus, the ADT observation (MDT + altimeter anomaly), at any particular location, is subject to considerable observation bias. Two recent papers (Dobricic, 25; Drecourt, Haines and Martin, 26) have attempted to deal with this problem by estimating the MDT observation bias during altimeter data assimilation; both papers make the assumption that the model forecast bias can be neglected. This makes the analysis easier and is the common starting

2 1762 D. J. LEA ET AL. point for many studies of observation bias in atmospheric assimilation. However, ocean models probably have more substantial biases than many operational atmospheric models. Therefore, this really is a problem in which both observation and model biases should be accounted for together. In this paper, we use Bayesian methods, as in Lorenc (1986), to study the general theory of bias estimation, resolvable into a three-dimensional variational (3D-Var) problem in which both observation and model forecast bias are estimated at analysis time. This approach is helpful in clarifying the assumptions and simplifications that need to be made for real applications. In section 2, we present the Bayesian probability approach and derive the 3D-Var cost function for the general bias problem. This 3D-Var problem is solved by deriving the solution in closed form, which can be applied in a sequential data assimilation formulation. In section 3, we make approximations to this estimation problem in a number of different cases. In particular, by accepting that error covariances are often not well known, it is possible to derive simplifying steps to reduce the implementation effort and computational time. In section 4, we apply this method to altimeter assimilation within the Met Office operational Forecasting Ocean Assimilation Model (F) system. In section 5, we examine the results of four hindcast assimilation experiments including observation-bias correction, model-bias correction, both together or neither bias correction method. Section 6 contains a discussion and a brief summary. 2. Bias estimation theory Those more interested in the application of the method can proceed directly to the cost function equation (15) Assumptions The derivation of the bias assimilation using Bayesian theory relies on a set of assumptions. The central assumptions, as in Lorenc (1986), are that the problem is linear and all the probability distributions are multinormal. This ensures that the maximum a posteriori estimate found by Bayesian theory is also the minimum variance estimate, as well as the least-squares estimate. For simplicity of notation, the probability distribution of a vector x (of dimension n, the number of model grid points), with mean µ and covariance matrix is written as N x (µ, ) α exp [ (x µ) 1 (x µ) T], (1) where α is a normalization coefficient. The following three assumptions define the interaction between the model and the observation, in order to be able to separate observation from model bias. 1. The observations y o (of dimension m) are independent of the model bias c (of dimension n). This defines the model bias as an intrinsic property of the model instead of being dependent on the observations. In the case of altimeter assimilation, y o is the ADT (SLA + reference MDT). 2. The model state x is independent of the observation bias b (expressed on the model grid here, so of dimension n). This is the counterpart of the previous assumption for the state. 3. There is a priori knowledge b o on the observation bias, with an error covariance matrix T. Furthermore, this knowledge depends only on the bias itself. It is therefore possible to write 2.2. Bayesian derivation p ( b o b ) = N b o (b, T). (2) The analysis consists of finding the maximum likelihood estimate of the state, the state forecast bias and the observation bias, given the observations and any prior knowledge of the bias. This is equivalent to calculating the probability distribution p(x, b, c y o, b o ). This can be decomposed using the Bayes theorem to give p ( x, b, c y o, b o) p ( y o, b o x, b, c ) p(x, b, c). (3) Given assumption 3, we can decompose the first part of the right-hand side as p ( y o, b o x, b, c ) = p ( y o x, b, c ) p ( b o x, b, c ). (4) Given assumption 1, we can write p ( y o x, b, c ) = p ( y o x, b ), (5) and assumption 3 leads to p ( b o x, b, c ) = p ( b o b ). (6) The unbiased observations, y o Hb, and the unbiased model state x should have the same mean, equal to the mean of the true state of the system x t (when mapped to the observation locations). Note that the observation bias b is specified on the model grid and the observation operator H maps x and x t, which are on the model grid to the observation locations. We define R y o as the observation error covariance matrix with regard to the truth, and R x as the model state representativeness error covariance matrix with regard to the truth. Then the measurement error probability distribution is p ( y o x t, b ) ( = N y o Hx t ) + Hb, R y o, (7) and the representation error probability distribution is p ( x x t ) = N x ( x t, R x ). (8) As these two probability distributions are related to the true state of the system, they cannot be known. It is therefore necessary to define the observation uncertainty

3 ALTIMETER BIAS ASSIMILATION 1763 with regard to the state as a combination of measurement and representation errors. Using the observation operator, H, tomapr x to observation space, we set R = R y o + HR x H T, assuming H is linear or linearizable. This gives p ( y o x, b ) = N y o (H(x + b), R). (9) Using probability theory, we can write that p(x, b, c) = p(x, b c) p(c) = p(x b, c) p p(c). (1) Using assumption 2, it is possible to write that p(x b, c) = p(x c), and therefore p(x, b, c) = p(x c) p p(c). (11) Both b and c are assumed to be randomly distributed around their forecast values, with error covariances O and P, respectively: p = N b ( b f, O ) ; (12) p(c) = N c ( c f, P ). (13) The state is randomly distributed around the biascorrected model forecast x f, with error covariance B: p(x c) = N x ( x f c, B ). (14) Gathering the different probability distributions, it is possible to rewrite (3) as a product of normal distributions. Taking the natural logarithm of this equation and solving for the maximum a posteriori estimate of the state is equivalent to finding the minimum of the following cost function: 2J (x, b, c) = ( y o H(b + x) ) T R 1 ( y o H(b + x) ) + ( x x f + c ) T B 1 ( x x f + c ) + ( b o b ) T T 1 ( b o b ) + ( b b f ) T O 1 ( b b f ) + ( c c f ) T P 1 ( c c f ). (15) Some comments are worth making regarding the first three terms on the right of this cost function. The desired solution is an unbiased model state and thus there is no c in the first cost function term. The model forecast state x f, unlike the model state x, is assumed to be biased and hence the presence of the c correction in the second term of the cost function. The a priori observation bias b o is zero in the experiments presented later, so the third term always limits the size of b with the a priori error covariance T Sequential formulation The sequential data assimilation equations are found by setting the partial derivatives of J with respect to x, b and c, to zero, giving H T R 1 { y o H(x a + b a ) } + B 1 ( x a x f + c a) =, (16) H T R 1 {y o H(x a + b a )} T 1 (b o b a ) + O 1 ( b a b f ) =, (17) B 1 ( x a x f + c a) + P 1 ( c a c f ) =. (18) Here, x f, b f and c f are previously forecast values for the model state, observation bias and model bias, respectively, at the analysis time. After rearranging the terms, with respect to the analysis values, x a, b a and c a, we obtain three general equations for the 3D-Var problem with bias. Equation (16) gives x a = ( x f c a) + K {( y o Hb a) H ( x f c a)}, (19) with the Kalman gain given by K = BH T ( HBH T + R ) 1. (2) Expressing the analysis equation in this form clearly shows the role of the model forecast bias c a in correcting the model forecast, producing an unbiased analysed model state x a. Both the observation- and model-bias analyses can be written in closed form as a function of the forecast values. The observation-bias equation is b a = { Lb o + (I L) b f } + F [ y o H { Lb o + (I L) b f } H ( x f c f )], (21) with the observation-bias gain F = LTH T { H(B + P + LT)H T + R } 1, (22) and the a priori knowledge gain L = O (T + O) 1. (23) The model-bias equation is c a = c f G [ y o H { Lb o + (I L) b f } H ( x f c f )], (24) with the model-bias gain given by G = PH T { H(B + P + LT)H T + R } 1. (25)

4 1764 D. J. LEA ET AL. The a priori knowledge of the observation bias adjusts the observation forecast bias prior to the analysis step. That is, Lb o + (I L) b f appears instead of b f in the above equations because this is the best prior estimate of the observation bias. This quantity has an error covariance of LT. Using these equations for the analysed biases in (19) gives x a = ( x f c f ) [ + K 1 y o H { Lb o + (I L) b f } H ( x f c f )], (26) as an alternative to (19), where the new gain is K 1 = (B + P) H T { H(B + P + LT)H T + R } 1. (27) Comparing the two gains, K and K 1, in (2) and (27) we can see that the presence of P and LT inside the braces is due to the forecast bias terms in the misfit of (26). The presence of P in the first term in parentheses in (27) comes from the use of the model forecast bias rather than the analysed bias in the first term correcting x f. The advantage of this formulation is that all of the analysis equations (21), (24) and (26) are based on the same misfit term in square brackets, although with different gain matrices, so only one misfit term must be calculated. Dee (25) and Balmaseda et al. (27) assume a slowly evolving bias (c a = c f in (19)) to achieve this simplification but (26) shows that this is unnecessary. In the following sections, we consider different approximations to this method. 3. Approximations and simplifications 3.1. No initial observation-bias knowledge Having no observation-bias knowledge prior to the start of the estimation procedure is equivalent to having infinite uncertainty on b o (i.e. T ) with a suitable norm. So, L and LT O, which mean the bias analysis equations (21) and (24) become b a = b f + F {( y o Hb f ) H ( x f c f )}, (28) c a = c f G {( y o Hb f ) H ( x f c f )}, (29) with gains (22) and (25) now F = OH T { H(B + P + O)H T + R } 1, (3) G = PH T { H(B + P + O)H T + R } 1. (31) Both b and c now only have a constraint on the forecast error covariance, so this case is straightforward and easily understood in terms of the errors on the various terms. The disadvantage of this approximation is that there is no explicit control on how far the observation bias may evolve from an initial MDT estimate given a long sequence of analyses Model bias or observation bias alone To estimate model bias alone, we remove the observationbias variables by setting b, T and O to zero in (26) and (24), and in the gain matrices (27) and (25). The resulting equations are identical to the formulation proposed by Chepurin, Carton and Dee (25), Dee and Da Silva (1998) and Dee and Todling (2), but the current derivation stems from Bayesian estimation instead of the traditional estimation theory of Friedland (1969). Similarly, to estimate observation bias alone, we remove the model-bias variables by setting c and P to zero in (26) and (21), and in the gain matrices (27) and (22). The problem with attempting to estimate only one type of bias is that the result can be contaminated by the other unestimated bias type. For example, when estimating observation bias alone, some of the resulting bias will be due to model bias if it exists. Joint estimation does not guarantee a clean separation of bias types but it does allow the projection of errors according to some wellchosen criteria Modelling the covariances The key difficulty with using the bias estimator described above is in knowing the different error covariances. We may have estimates of the observation a priori error covariances T and R. We note that error covariances on the analysed model state and the observation and forecast biases can be calculated, assuming that the original error covariances B, O and P are known. However, to use these in a sequential assimilation scheme we need to forecast the evolution of the error covariance information. The only practical, but still expensive, method for large problems is to use an ensemble method for the model error covariances, or to make some greatly simplifying assumptions. For the biases and their error covariances, we now look at what simplifying assumptions are realistic. In the case of the observation forecast bias, we assume that it is constant in time (removing any need to forecast its evolution), and consistent with the planned use in altimetry assimilation in section 4. In addition, we model its error as a fraction of the error of the a priori observations. Therefore with <γ b < 1, and thus O = γ b T, (32) L = γ b 1 + γ b I, (33) is a constant. Or, for small γ b, L γ b I and the term LT γ b T. A number of works on model bias have used a similar approximation, giving it the same error covariance as the random model error; P = γ c B, as in Dee and Da Silva (1998), Bell, Martin and Nichols (24) and Balmaseda et al. (27). Here instead we assume a spatially

5 ALTIMETER BIAS ASSIMILATION 1765 uniform and large-scale model-bias error covariance (see section 4). In order to evolve c between analysis times, we assume persistence (as for b), but with a time decay c f i+1 = βca i, (34) where i is the analysis time in days. With no further model error analysis, the bias would decay exponentially with a time-scale 1/(ln β). In the experiments here, β =.98895, which gives a decay time-scale of 9 days. The aim of the decay is to allow the model forecast bias to represent seasonal time-scale variations. In the following section, we look at how this combination of observation and model forecast bias can be applied to the case of assimilation of ocean altimeter observations. 4. Application to altimeter assimilation in the Met Office F system This study uses the F system (see Martin, Hines and Bell, 27 for a recent description of F). The experiments all use the limited area 1/3 model of the Atlantic and Arctic oceans, forced by six-hourly winds and fluxes from the UK Met Office global meteorological forecasting model. In addition to altimeter data, the system assimilates in situ temperature and salinity data, and sea-surface-temperature (SST) data. To assimilate altimeter data, we must first combine the altimeter sea-level anomaly data with a MDT. The MDT used (Figure 1) is derived from a previous model integration combined with data from Singh and Kelly (1997) over the Gulf stream, and data from the GOCINA project (Knudsen et al., 27) for the region between Scotland and Greenland (45 W 15 E, 53 N 72 N). The sea-surface-height (SSH) data are then assimilated into the model using the Cooper and Haines (1996) scheme, which involves adjusting the subsurface density field in order to produce 9N 6N 3N 3S 9W 45W 45E Figure 1. The MDT (cm) before observation-bias adjustment, used in all the experiments here. Contour levels are shown at intervals of 1 cm, with solid and dashed lines for positive and negative values, respectively. the required SSH increment. The standard F assimilation has a model-bias system used for in situ data, the pressure correction scheme of Bell et al. (24), in the equatorial regions, but to avoid confusion we have not used this scheme here. The MDT error standard deviation used to specify the square root of the diagonal of the observation-bias covariance T is shown in Figure 2. The observationbias errors are based on the Rio et al. (25) MDT error increased by 5 in order to more closely match errors from the spread between several MDT products calculated in Bingham and Haines (26) and Vossepoel (27). In future, the GOCE geoid will give an improved estimate of T as well as an improved initial MDT. The observation forecast bias error covariance O = γ b T, where γ b =.1. Both of these observation-bias error covariances (O and T) are given a correlation length-scale of 4 km (using a SOAR function), consistent with the small error scales expected from Knudsen and Tscherning (27). 9N 6N 3N 3S 9W 45W 45E Figure 2. The MDT error standard deviation (cm) used to specify the square root of the diagonal of the observation-bias covariance T. This is the Rio et al. (25) MDT error multiplied by 5 to take account of errors other than the gridding error. The observation-bias forecast error O = γ b T,whereγ b =.1. Contour levels are shown at intervals of 2.5 cm. In the F system, the background error covariance B varies spatially with a standard deviation shown in Figure 3. The error is high in the Gulf Stream, because of the strong eddy activity, and lower elsewhere. The values are estimated using the Hollingsworth and Lönnberg (1986) method (see Martin et al., 27 for details). The correlation length-scale for B is given as 6 km. In the original F assimilation scheme, B is also given a second larger scale, but here we reserve that scale explicitly for model forecast bias. For the model-bias analysis, we assume the amplitude of P is spatially uniform with a value of cm 2 day 2, and with a large correlation scale of 4 km. Larger values of the variance have been tested and found not to change the estimated model bias significantly, and these can result in instability in the value of c.

6 1766 D. J. LEA ET AL. 9N 6N 3N 3S 9W 45W 45E Figure 3. The background error mesoscale standard deviation (cm) used in B and P. Contour levels are shown at intervals of 5 cm. Note that the model-bias units are cm. The model-bias values shown in the results that follow are more correctly described as normalized model-bias values, where the model bias is divided by the time period (in days) between analyses and expressed in units of cm day 1. The sealevel model bias is applied to the model along with available altimeter observations, at every time step using the Cooper and Haines (1996) scheme. The scales in the increments to the model state, the observational- and model-bias fields are directly controlled by the correlation scales in the appropriate gain matrices, and thus by the correlation scales of the error covariances. In (27), the correlation length-scales of B and P (6 and 4 km) set the correlation length-scale of the increment to the model state x. For the observation bias, the increment scale is 4 km, set by LT in (22), and for the model bias it is 4 km, set by P in (25). The observation error covariance matrix R is also estimated using the Hollingsworth and Lönnberg (1986) method. The basic assumption is that the errors in the (bias-corrected) observations are uncorrelated in space and time. This is less than ideal for altimeter assimilation, as there are sources of observational error that are correlated along-track (e.g. orbit error or atmospheric corrections). However, it is beyond the scope of this paper to improve these observational error covariances and we leave this as a topic for future work. 5. Results We now examine the results of four assimilation experiments, each integrated over a five-year period (7 January 21 to 31 December 25). These are the following: STD, a standard altimeter assimilation experiment with no model or observation bias; experiment, which estimates the observation bias b only; experiment, which estimates the model bias c only; experiment, which estimates both observation and model bias. Results are assessed by looking at the remaining uncorrected bias in the assimilation output (see sections 5.1 and 5.2). This is found by calculating the observation minus background values (y o H(b f + x f c f )), also known as innovations, which compare the bias-corrected model forecast before assimilation with the observations. The bias fields estimated by the assimilation experiments are examined in sections 5.3 and 5.4. The impact on subsurface fields is also assessed, in section Time-mean innovations Table I shows domain- and time-average innovations excluding the first year of the five-year model integration to eliminate spin-up effects. We have excluded regions where altimeter data are not assimilated, which are where the depth is less than 1 m and the temperature stratification is low (less than 5 C top to bottom difference). The stratification varies seasonally, so to produce a fixed area for diagnostics we select the area based on Levitus et al. (1998) temperatures from March. Table I. Domain average. The mean innovation (cm) averaged over space and time, and the r.m.s. and standard deviation (SD) averaged over the domain, for the four assimilation experiments. These are the standard altimeter assimilation (STD), observation-bias assimilation (), model-bias assimilation () and a combined observation- and model-bias assimilation (). Mean innovation r.m.s. SD STD a a The overall best result in this region. The main result is that both experiment and experiment improve the mean innovation and the standard deviation of the innovations. Experiment has a lower standard deviation than experiment. Combining the model- and observation-bias schemes in experiment produces the lowest mean bias and standard deviation of all. Experiment has a 1% lower standard deviation than experiment STD. Tables II IV show the results averaged over some standard regions in the model. Experiment has smaller mean and standard deviation innovations compared to STD in all cases. In the North-west Atlantic region (Table II), STD has a mean bias of 2 cm, which is substantially reduced in and. The mean innovation is not reduced further in because of the a priori bias constraint (T). The mean is lower in experiment, partly because of the a priori observation-bias constraint, but also because of the different spatial covariance scales of the two bias terms. In the North-west Atlantic, the innovation standard deviations are relatively high, compared to other regions, because of the strong eddy activity around the Gulf Stream. These should be reduced in a higher-resolution model capable of simulating the evolution of Gulf Stream eddies.

7 ALTIMETER BIAS ASSIMILATION 1767 Table II. As Table I, but for the North-west Atlantic region (1 W 3 W, 15 N 6 N). Mean innovation r.m.s. SD STD a a The overall best result in this region. Table III. As Table I, but for the North-east Atlantic region (3 W 1 E, 15 N 6 N). Mean innovation r.m.s. SD STD a a The overall best result in this region. Table IV. As Table I, but for the Tropics region (1 W 3 E, 15 S 15 N). Mean innovation r.m.s. SD STD a a The overall best result in this region. The North-east Atlantic (Table III) results are generally in accord with the whole domain results, with the bias-correcting experiments reducing the mean and standard deviation of the innovations compared to experiment STD. In the Tropics (Table IV), the mean innovations of STD are high (see the next section). Again, both the bias-correction experiments are able to reduce the mean and the standard deviation of the innovations. We can look at the spatial structure of the mean innovations in more detail by binning them into 1 1 boxes (Figure 4). In STD, there are large positive mean innovations (up to 2 cm) in the subtropical gyre, and negative innovations in the Gulf Stream and particularly to the north of the Gulf Stream. These values are significantly reduced in both and, and even more so in. Overall, and have the lowest mean innovations, presumably because the observation-bias scheme is tuned to correct the mean, whereas the model bias is intended to correct the seasonal time variations. Experiments and show a significant reduction of the negative innovations north of the Gulf Stream while at the same time the positive innovations in the subtropical gyre are reduced Time variability in the innovations The model-bias term is tuned to correct some of the seasonal biases in the model, and thereby some of the time variability of the innovations. The observation-bias correction can also improve these temporal variations, as using a better MDT should improve the behaviour of the system in general. The domain average of the altimeter observations has an annual cycle with an amplitude of 6 cm (not shown) because of seasonal steric expansion. The model is volume (not mass) conserving, with a rigid-lid pressure representing SSH, and therefore does not simulate domainaverage changes in SSH (e.g. Gordon et al., 2). However, the domain-average SSH can be calculated from the domain-average SSH increments, allowing the model to track the seasonal changes in the observed domainaverage SSH. The resulting domain mean model SSH plot is shown in Figure 5. Breaking down this result into the standard areas (as for Tables II IV) shows the latitudinally varying steric effect (Figures 5b d) with a strong seasonal cycle in the North Atlantic and little seasonal variation in the Tropics. Figure 6 shows a time series of the mean innovations averaged over the whole domain and over three standard regions. There is a clear adjustment period of around three months at the beginning of the integrations in January 21. The domain mean innovations show little variation for all the experiments (indicating the success of the steric SSH correction described above). What bias remains is further reduced by the model- and observationbias correction schemes. The standard deviation of the innovations (Figure 7) also shows small time variability. It clearly shows that, and consistently reduce the innovation standard deviations compared to STD. Also of note is the standard deviation reductions for all the experiments during 24 and lasting to the end of the integrations. This is consistent with the large increase in Argo data density over this period and shows that good high-density in situ data lead to a closer fit to the altimeter data variability. The standard deviation of the innovations in the Northwest Atlantic (Figure 7c) is higher than elsewhere and is only slightly reduced by bias correction. This is likely related to strong eddy activity here, which the 1/3 model cannot fully capture. The North-east Atlantic and the Tropics, in contrast, are significantly improved by bias correction. The standard deviation and mean of STD both peak in 23, coinciding with an increase in the observed mean SSH in 23, which is mirrored by the model SSH in Figure 5(c). Comparing the average SSH, from experiment STD, in (Figure 8) reveals a coherent pattern similar to the dipole pattern of decadal variability in tropical Atlantic SST seen in Chang, Ji and Li (1997). From 22 to 25, the SSH decreases in the eastern equatorial Atlantic and increases to the west and north, resulting in the average increase in the SSH seen over this period. The most likely explanation for the increase in the innovations in experiment STD is that the assimilation experiment initially fails to capture this change in the Tropics. The reduced mean and standard deviation of the innovations (see Figures 6 and 7) indicate that the bias correction experiments aid the model in tracking these changes more accurately.

8 1768 D. J. LEA ET AL. STD (cm) 45N 2 1 (c) (d) 45N 1 2 9W 45W 9W 45W Figure 4. Four-year (22 25) time-average mean innovations (cm) in 1 1 bins for experiments STD,, (c) and (d). Contour levels are shown at intervals of 2.5 cm with solid and dashed lines for positive and negative values, respectively. SSH (cm) Whole domain STD Tropics (c) 25 NW Atlantic (d) NE Atlantic 2 5 SSH (cm) Figure 5. Time series of area-averaged model SSH (cm), with a 15-day time smoothing, in the whole domain, the Tropics (1 W 3 W, 15 S 15 N), (c) the North-west Atlantic (1 W 3 W, 15 N 6 N) and (d) the North-east Atlantic (3 W 1 E, 15 N 6 N) for experiments STD,, and Time-mean bias fields We now examine the observation-bias b and model-bias c fields that are estimated in the assimilation experiments. Ignoring 21 to remove the spin-up period, the fouryear mean (22 25) of the observation-bias field, b, is shown in Figure 9 for experiments and. Although dominated by small scales consistent with the error covariances, the observation bias, particularly in, does develop some large-scale components; in particular, it is negative on the shelf north of the Gulf Stream and positive in the subpolar gyre and in the subtropical gyre. Where the b field is negative, for example north of the Gulf Stream, the implied MDT is higher so the observation bias is acting to reduce the MDT gradient across the Gulf Stream. The model four-year mean bias fields c (Figure 1) are dominated by large scales, again consistent with the error covariances, with patterns similar to the reverse

9 ALTIMETER BIAS ASSIMILATION 1769 Mean (cm) 4 2 Whole domain STD Tropics (c) 6 NW Atlantic (d) 6 NE Atlantic 4 4 Mean (cm) Figure 6. Monthly time-series plots of the average innovations in the whole domain, the Tropics (1 W 3 W, 15 S 15 N), (c) the North-west Atlantic (1 W 3 W, 15 N 6 N) and (d) the North-east Atlantic (3 W 1 E, 15 N 6 N) for experiments STD,, and. S.d. (cm) Whole domain STD Tropics (c) 15 NW Atlantic (d) 1 NE Atlantic 14 S.d. (cm) Figure 7. Monthly time-series plots of the standard deviation of innovations in the whole domain, the Tropics (1 W 3 W, 15 S 15 N), (c) the North-west Atlantic (1 W 3 W, 15 N 6 N) and (d) the North-east Atlantic (3 W 1 E, 15 N 6 N) for experiments STD,, and. of the larger-scale observation bias, with a positive value north of the Gulf Stream and negative values in the subpolar and subtropical gyres. These biases both act in the same sense to reduce the mean value of the innovations, and it is the respective covariance matrices and bias models that determine the split between b and c, although neither model is particularly tuned to a large-scale time-mean bias. Using experiment, we can examine how the two bias methods work together. Despite the use of a time-varying representation of model bias in order to select seasonal errors, and the use of very different error covariance scales, the two bias fields still compete to some extent in representing the large-scale mean errors. For example, the mean observation bias in the subtropical gyre is reduced in experiment compared

10 177 D. J. LEA ET AL. 1N 1S 45W Figure 8. Model mean SSH in 25 minus the mean SSH in 22 in the Tropics area (for experiment STD). Contour levels are shown at intervals of 2 cm with solid and dashed lines for positive and negative values, respectively. to experiment. The mean model bias is reduced in experiment as some of the mean innovations are accounted for by the observation bias. This fits our initial assumption that most of the time-mean bias is a result of the MDT error. The contribution of the model bias in the time mean is in all cases much less than its impact in reducing the time-varying seasonal cycle component. Figure 11 shows the four-year mean SSH of the model in experiment STD and the changes in the mean SSH for each assimilation experiment. The main point to note is that experiments and with model bias change the model mean state more than with observation bias alone, because the model bias is used directly to alter the model state. The observation-bias scheme assumes any mean bias is a result of the observations and is therefore not applied to the model. Comparing Figures 11(c) and 1, we see that a positive model bias c of O(1) cm day 1 leads to a decrease in the model mean SSH of O(1) cm. Note also the small-scale changes in the model mean SSH in experiment. Despite the model bias c being confined to large scales, this feeds back on smaller-scale circulation patterns of the model. The change in the mean SSH of (compared to STD) is a combination of and. The magnitude of the SSH change is slightly reduced in experiment compared to experiment, consistent with the reduction in the mean c in Figure Time variability in the bias fields The average observation-bias values in the various standard regions are shown in Figure 12. The equivalent fields 9N 6N 3N 3S 9W 45W 45E 9W 45W 45E Figure 9. Four-year mean of the observation-bias field b (cm) for the experiments that estimate this field: ;. 9N 6N 3N 3S 9W 45W 45E 9W 45W 45E Figure 1. Four-year mean of the model-bias field c (cm day 1 ) for the experiments that estimate this field: ;.

11 ALTIMETER BIAS ASSIMILATION N STD 45N (c) 9N (d) N 9W 9W Figure 11. Four-year mean SSH field (cm) for the STD experiment. Anomalies with respect to STD for experiments, (c) and (d). Contour levels are shown at intervals of 25 cm in, and 2 cm in (d), with solid and dashed lines for positive and negative values, respectively. for the model bias are shown in Figure 13. The model and observation fields both spin up in a few months, and both exhibit ongoing slower time variability after the initial adjustment period. The temporal variations in the observation bias are relatively small compared to the local mean values (see Figure 9). In the North-west Atlantic and the Tropics, a mean offset in the observation bias is removed when model bias is estimated (Figures 12b and c). We expect the observation bias would make corrections to the time mean and the model bias to the time variations; however, these plots represent larger scales, which emphasizes the effect of the model bias that operates on these scales. Looking at the small-scale time variability of the observation bias with a Hovmüller diagram (Figure 14) shows that the time variability is significantly reduced when we include model bias in, compared to. The time-mean bias is mostly retained except at around 25 N and around 15 S. The seasonal time-scale variability in the model bias is clearly seen in the Hovmüller diagram (Figure 15). Both the means, and some variability, are attenuated in compared to. However, overall in experiment, the two bias fields are addressing distinct aspects of the total bias, the observation bias generally accounting for the mean, and the model bias for the variability Verification against temperature and salinity Assimilation of biased altimeter data into an ocean model is likely to have a detrimental impact on the temperature and salinity fields. All the experiments so far discussed also have independent temperature and salinity profile assimilation through a separate scheme. We can therefore assess the impact of the different altimeter assimilation schemes on the background error misfits to the temperature and salinity profiles prior to their subsequent assimilation. To provide a baseline, we use a new control experiment (CTL) where no altimeter data are assimilated, only temperature and salinity data. The results are summarized in Table V. This shows that the lowest r.m.s. errors for temperature and salinity are from experiment CTL at 1.14 C and.265 psu. Altimeter assimilation with or without bias correction tends to increase the r.m.s. errors, but by relatively small amounts. Experiment STD has the highest temperature and salinity r.m.s. errors at 1.2 C and.29 psu. This suggests that assimilation of altimeter data with an incorrect MDT slightly degrades the temperature and salinity structure of the ocean. Both model- and observation-bias corrections then slightly reduce these r.m.s. errors compared to STD. The verification was also performed on each of the separate regions (results not shown). Generally, the results are similar to those in the whole domain, with STD being slightly worse than CTL and the bias experiments giving an improvement. The North-east Atlantic is notable as the altimeter assimilation experiments here have 4 9% lower r.m.s. compared to CTL. There are several reasons why the bias-correction experiments generally have larger errors than CTL. The main reason is that we are not using independent data for the comparison; other data types (e.g. currents) are likely to show better results with altimeter assimilation.

12 1772 D. J. LEA ET AL. Obs bias (cm) Whole domain Tropics (c) 1. NW Atlantic (d).6 NE Atlantic Obs bias (cm) Figure 12. Time series of the observation bias, b (cm) averaged over the whole domain, the Tropics (1 W 3 W, 15 S 15 N), (c) the North-west Atlantic (1 W 3 W, 15 N 6 N) and (d) the North-east Atlantic (3 W 1 E, 15 N 6 N) for experiments and Whole domain Tropics (c) Mod bias (cm day 1 ) NW Atlantic (d) NE Atlantic Figure 13. Time series of the model bias c (cm day 1 ) averaged over the whole domain, the Tropics (1 W 3 W, 15 S 15 N), (c) the North-west Atlantic (1 W 3 W, 15 N 6 N) and (d) the North-east Atlantic (3 W 1 E, 15 N 6 N) for experiments and. Other reasons are that we may not have accounted for all the altimeter bias or because of the limitations of the altimeter assimilation method. 6. Discussion We have developed a theory for the simultaneous application of observation and model bias in sequential data assimilation. In order to separate out the model and observation biases from the primary information of model data differences, we have used further information about the spatial and temporal behaviour of the biases. We have demonstrated the method in an application to altimeter assimilation in an operational oceanography model at the Met Office. In this case, we can make sensible assumptions about the model and observational biases, which help to separate them. This only works to the extent that the patterns of model- and observation-bias are sufficiently distinguishable and can be adequately represented by the respective covariance specifications. The observation bias is assumed to be time-invariant because it represents uncertainty in the time MDT, and has quite

13 ALTIMETER BIAS ASSIMILATION 1773 Latitude Figure 14. Hovmüller diagram of observation-bias field b (cm) at 3 W for the relevant experiments and. Contour levels are shown at intervals of.5 cm with solid and dashed lines for positive and negative values, respectively. 5 5 Latitude Figure 15. Hovmüller diagram of model-bias field c (cm day 1 )at3 W for the relevant experiments and. Contour levels are shown at intervals of.2 cm with solid and dashed lines for positive and negative values, respectively. small covariance scales. The model bias is assumed to be time-varying (particularly on a seasonal cycle) and has much larger covariance length-scales. The most important evidence of success is the improvement in the innovation (observation minus background) statistics when either model- or observation-bias corrections, or both, are applied. Experiments, and (both biases) all reduce the mean innovations (or residual biases) in SSH compared to experiment STD (see section 5.1). Experiments and are also very effective at reducing the innovation standard deviation. This indicates that assimilation with bias correction is able to track the evolution of the time-evolving SSH more accurately. In experiment, the observation bias does largely take up the time-invariant bias, and the model bias captures the time-varying bias (see sections 5.3 and 5.4). However, without knowing the true MDT we cannot know directly whether the separation is successful. Results (not shown) without the a priori observation-bias constraint (T ) allow larger scales to build up in the observation bias, causing the MDT to tend to the model mean SSH (calculated from an assimilation of temperature and salinity data alone). Table V. The mean and r.m.s. temperature ( C) and salinity (psu) errors averaged over the whole domain at all depths and times for the assimilation experiments CTL (no altimeter assimilation), STD,, and. Mean temperature r.m.s. temperature Mean salinity r.m.s. salinity error ( C) error ( C) error (psu) error (psu) CTL a STD a The experiment with the overall lowest r.m.s. error.

14 1774 D. J. LEA ET AL. Future work is needed, in particular using a wellcharacterized error estimate on the observation bias, which should be achieved with the GOCE satellite geoid data. Model error covariances also need to be better characterized. At the Met Office, a new modelling system (using NEMO) is being implemented and part of this work will involve calculating new and improved model background error covariances, which can be better attributed to B and P within this assimilation scheme. The results for the model bias on large spatial scales may also require using different assimilation methods for applying the innovations to subsurface fields. Further work is also required to combine the altimeter bias correction with the pressure correction scheme (Bell et al., 24) used for the in situ data assimilation, which corrects the model pressure biases at the Equator in the current operational system. 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