Analysis sensitivity calculation in an Ensemble Kalman Filter

Transcription

1 Analysis sensitivity calculation in an Ensemble Kalman Filter Junjie Liu 1, Eugenia Kalnay 2, Takemasa Miyoshi 2, and Carla Cardinali 3 1 University of California, Berkeley, CA, USA 2 University of Maryland, College Park, MD, USA 3 ECMWF, Data Assimilation Section, Reading, UK 1

2 Abstract Analysis sensitivity indicates the sensitivity of an analysis to the observations, which is complementary to the sensitivity of the analysis to the background. In this paper, we discuss a method to calculate this quantity in an Ensemble Kalman Filter (EnKF). The calculation procedure and the geometrical interpretation, which shows that the analysis sensitivity is proportional to the analysis error and anti-correlated with the observation error, are experimentally verified with the Lorenz 40-variable model. With the analysis sensitivity, the cross-validation in its original formulation can be efficiently computed in EnKFs, and this property can be used in observational quality control. Idealized experiments based on a simplified-parameterization primitive equation global model show that the information content (the trace of the analysis sensitivity of any subset of observations) qualitatively agrees with the actual observation impact calculated from much more expensive data denial experiments, not only for the same type of dynamical variable, but also for different types of dynamical variables. 2

3 1. Introduction Observations are the central information introduced into numerical weather prediction systems through data assimilation, which is a process that combines observations with background forecasts based on their error statistics. With the increase of observation datasets assimilated in modern data assimilation systems, such as the assimilation of Advanced InfraRed Satellite (AIRS) and the Infrared Atmospheric Sounding Interferometer (IASI), it is important to examine questions such as how much information content a new observation dataset has, and what is the relative influence of the background forecasts and the observations on the analyses. Analysis sensitivity (also called self-sensitivity), a quantity introduced by Cardinali et al. (2004), can address these questions. The larger the analysis sensitivity is, the larger is the influence of the corresponding observation on the analysis, and the smaller is the influence of the corresponding background forecast. Since the analysis sensitivity is a function of the analysis error covariance, which is not explicitly calculated in variational data assimilation schemes, Cardinali et al. (2004) proposed an approximate method to calculate this quantity within a 4D-Var data assimilation framework. They showed that the trace of the analysis sensitivity of a particular observation data type represented the information content of that type of observations, and the relative importance of different observation types determined by information content was in good qualitative agreement with the observation impact from other studies. In addition, information content has also been used in channel selection in multi-thousand channel satellite instruments, such as IASI (Rabier et al., 2002). 3

4 Though the approximate method to calculate the analysis sensitivity in 4D-Var is computationally feasible, it introduces spurious values that are not within the theoretical range (between 0 and 1). In contrast with variational data assimilation schemes, Ensemble Kalman filters (EnKF) (Evensen, 1994; Anderson, 2001; Bishop et al., 2001; Houtekamer and Mitchell, 2001; Whitaker and Hamill, 2002; Ott et al., 2004; Hunt et al., 2007) generate ensemble analyses that can be used to calculate the analysis error covariance. Because of this, it would be more straightforward to calculate the analysis sensitivity in an EnKF than in a variational data assimilation system. In addition, the Cross Validation (CV) (Wahba, 1990) score can be calculated from the analysis sensitivity, observation and analysis values in EnKFs without carrying out data denial experiments (section 4), whereas CV cannot be computed in this way in variational data assimilation systems because of the approximations made in calculating the analysis sensitivity. In this paper, following Cardinali et al. (2004), we show how to calculate the analysis sensitivity and the related diagnostics in an EnKF, and further study the properties and possible applications of these diagnostics. This paper is organized as follows: section 2 describes how to calculate the analysis sensitivity in an EnKF. In section 3, with a geometrical interpretation method adapted from Desroziers et al. (2005), we show that the analysis sensitivity is proportional to the analysis errors and anticorrelated with the observation errors. With the Lorenz 40-variable model (Lorenz and Emanuel, 1998), section 4 verifies the self-sensitivity calculation procedure, and shows that the CV can be efficiently calculated from the self-sensitivity, observation and analysis values, and it can be used in observational quality control. In addition, the geometrical interpretation is experimentally tested in this section. In section 5, with a 4

5 primitive equation model and a perfect model experimental setup (no model error), we examine the effectiveness of the trace of the self-sensitivity in assessing the observation impact obtained from data denial experiments. Section 6 contains a summary and conclusions. 2. Calculation of the analysis sensitivity in an EnKF In this section, we first briefly derive the analysis sensitivity equation valid for all data assimilation schemes (equations (1), (2), (3), equivalent to equations (3.1), (3.4), and (3.5) in Cardinali et al., 2004) and then focus on how to calculate this quantity in an EnKF. In data assimilation, the analysis state n-dimensional vector a x is a combination of the background (an b x ) with the observations (a p-dimensional vector weighting matrix K. It can be expressed as: o y ) based on a x ) a o b = Ky + ( I n! KH x (1) The (n! p) weighting matrix K = P b H T (HP b H T + R)!1 is a function of the background error covariance P b and the observation error covariance R, and H(!) is the linearized observation operator, which transforms a perturbation from model space to observation space. The sensitivity of the analysis vector x a to the observation vector y o is: S o =!ya!y o = KT H T = R "1 HP a H T, (2) and the sensitivity with respect to the background is given by S b =!ya!y b = I p " K T H T = I p " S o (3) 5

6 where y a = Hx a = HKy o + (I p! HK)y b is the projection of the analysis vector x a on the observation space, and b b y = Hx is the projection of the background on the observation space. P a = (I p - KH)P b is the analysis error covariance. The matrix o S is called influence matrix, because the elements of this matrix reflect the sensitivity of the analysis state to the observations. The diagonal elements of the matrix o S are the analysis selfsensitivities, and the off-diagonal elements are cross sensitivities. Similarly!ya!y b reflects the sensitivity of the analysis state to the background. As shown in Cardinali et al. (2004), the sensitivity of the analysis to the observation and the sensitivity of the analysis to the corresponding background at that observation location are complementary (i.e., they add up to one), and the self-sensitivity has no units and its theoretical value is between 0 and 1 when the observation errors are not correlated (R is diagonal). EnKFs generate ensemble analyses at every analysis cycle, and the analysis error covariance can be written as products of the ensemble analysis perturbations, so in EnKFs the influence matrix S o can be written as: S o = R!1 HP a H T = 1 k! 1 R!1 (HX a )(HX a ) T (4) where HX a is the analysis ensemble perturbation matrix in the observation space whose th i column is k HX ai! h(x ai ) " 1 # h(x ai ), (5) k i=1 x ai is the i th analysis ensemble member, k is the total number of ensemble analyses, and h(!) is the observation operator, which can be linear or nonlinear. When the observation 6

7 operator is linear, the right hand side and the left hand side of Equation (5) are equal. Otherwise, the left hand side is a linear approximation of the right hand side. When the observation errors are not correlated, the self-sensitivity can be written as: S o jj =!y a j!y = # 1 & o j $ % k " 1' ( 1 k 2 + (HX ai ) j * (HX ai ) j (6) ) j i=1 and the cross sensitivity S o jl (see details in Appendix), which represents the change of the analysis y l a with respect to the variation of the observation y j o, can be written as: S o jl =!y a l!y = # 1 & o j $ % k " 1' ( 1 k 2 + [(HX ai ) j * (HX ai ) l ] (7) ) j i=1 where! j 2 is the j th observation error variance. Equation (5) indicates that the calculation of the self-sensitivity and cross sensitivity in EnKFs only requires applying the observation operator to each analysis ensemble member, and then doing a scalar product based on the equations (6) and (7). In section 4, this calculation procedure will be verified with the Lorenz 40-variable model. The calculation of the self-sensitivity based on equation (6) requires no approximations when the observation errors are not correlated, so that S o jj satisfies the theoretical limits (between 0 and 1). In 4D-Var, by contrast, the calculation of the analysis error covariance is based on a truncated eigenvector expansion with the vectors obtained through the Lanczos algorithm that can introduce spurious values larger than 1 (Cardinali et al., 2004). 3. Geometric interpretation of self-sensitivity Equation (6) shows that the self-sensitivity is proportional to the analysis error variance and inversely proportional to the observation error variance. Since in most cases, 7

8 the analysis and the observation error statistics used in data assimilation do represent the accuracy of the analyses and the observations respectively, equation (6) implicitly indicates that the analysis sensitivity increases with the analysis errors and decreases with the observation errors. In this section, we adapt the geometrical interpretation that Desroziers et al. (2005) used to explain the relationship among the background error, observation error and analysis error to show the relationship among the analysis sensitivity, the observation error and the analysis error. Following the same notation as Desroziers et al. (2005), we rewrite equation y a = HKy o + (I p! HK)y b by subtracting h(x t ), the true state in the observation space, from both sides and obtain y a! h(x t ) = HK(y o! h(x t )) + (I p! HK)(y b! h(x t )) (8) We define!y a = y a " h(x t ),!y b = y b " h(x t ) and!y o = y o " h(x t ), which are the analysis and the background errors in the observation space, and the observation errors respectively. Then equation (8) can be written as,!y a = HK!y o + (I p " HK)!y b (9) After eigenvalue decomposition, HK can be written as HK = V!V T, where! is a diagonal matrix whose elements are the eigenvalues of HK, and the columns of V are the eigenvectors of the matrix HK. Following Desroziers et al. (2005), the projections of!y a onto the eigenvectors V are given by V T!y a = V T V"V T!y o + V T V(I p # ")V T!y b, which can be further written as:! y! a = "! y! o + (I p # ")! y! b (10) 8

9 Here!! y a,!! y o and!! y b are the projections of!y a,!y o and!y b on the eigenvector matrix V respectively. When these vectors are projected on a particular eigenvector V i with the corresponding eigenvalue equal to! i, the above equation can be written as,!! y i a = " i!! y i o + (1# " i )!! y i b (11) Therefore, in the eigenvector V i space, the analysis sensitivity with respect to the observation is! i, and with respect to the background is (1-! i ). As shown in Cardinali et al. (2004), they are complementary when the observation error covariance matrix is diagonal. All the elements except! i in equation (11) are shown schematically in Figure 1. In the following, we will show how to represent! i as a function of the angle!, and how to interpret! i with the observation error!! y o i and the analysis error!! y i a. As shown in Figure 1, the observation error (!! y o i ) and the background error (!! y b i ) in the eigenvector V i space are perpendicular because the background error and the observation error are assumed to be uncorrelated, which implies that the inner product between!! y o i and!! y b i is zero. The analysis error (!! y i a ) is also perpendicular to the line connecting the observation and the background, reflecting that the analysis is a linear combination of the background and the observations closer to the truth (Desroziers et al., 2005). With these two relationships, the projection of!! y o i on!! y a i is,!! y i a "!! y i o = # i!! y i o "!! y i o $!! y i a!! y i o cos(90 % &) = # i!! y i o 2 (12) 9

10 Since sin(!) = "! y i a / "! y i o, and! i = "! y i a "! y i o cos(90 # $) / "! y i o 2, based on the above equation, then! i = sin 2 (") and (1-! i ) =cos 2 (!). This indicates that the analysis sensitivity with respect to the observation increases with the angle! (! is greater than 0º and smaller than 90º), and the analysis sensitivity to the background decreases with the angle!. This geometrical representation is consistent with equation (6) showing that the analysis sensitivity per observation increases with the analysis error and decreases with the observation error. Figure 1 Geometrical representation of the vector elements in equation (11) (see text). The analysis sensitivity with respect to the observations is sin 2! (adapted from Desroziers et al., 2005). 4. Validation of the self-sensitivity calculation method in EnKFs and cross validation experiments with the Lorenz 40-variable model Cardinali et al. (2004) showed that the change in the analysis value y i a that takes place with the deletion of the i th observation can be calculated from S ii o, y i o and y i a without calculating y i a(!i) (the estimate of y a i when the i th observation is deleted), and that 10

11 the same was true for the calculation of Cross Validation (CV) score (Wahba, 1990, p theorem 4.2.1), which is defined as "(y o i! y a(!i) i ) 2, where p is the total number of i=1 observations. The CV score represents the statistical difference between the actual observation y i o and the predicted observation y i a(!i) based on all the buddy observations. Therefore, to verify the self-sensitivity calculation method (Equation (6)), we will follow equations (2.9) and (2.10) in Cardinali et al. (2004) comparing the change of the analysis value y i a! y i a(!i) and CV scores based on the actual data denial results of y i a(!i) to those from the self-sensitivity: y i a! y i a(!i) = o S ii (1! S o ii ) (y o i! y a i ) (13) p p (y " o (y o i! y a(!i) i ) 2 = i! y a i ) " 2 (14) (1! S o ii ) 2 i=1 With a correct estimate of the self-sensitivity, the left-hand sides and the right-hand sides in equations (13) and (14) should be equal. Deleting each observation in turn is a computationally formidable task in a realistic NWP data assimilation system, so we use the Lorenz 40-variable for this purpose. i=1 4.1 Lorenz 40-variable model and experimental setup The Lorenz 40-variable model is governed by the following equation: d dt x j = j+ 1 j! 2 ) j! 1 j + ( x! x x! x F (15) 11

12 The variables ( x j, j = 1,!, J ) represent meteorological variables on a latitude circle with periodic boundary conditions. As in Lorenz and Emanuel (1998), J is chosen to be 40. The assimilation interval is F is the external forcing, which is 8 for the nature run, and 7.6 for the forecast, allowing for some model error in the system. Observations are simulated by adding Gaussian random perturbations (with a standard deviation equal to 0.2) to the nature run. The data assimilation scheme we use is the Local Ensemble Transform Kalman Filter (LETKF), which is one type of EnKF especially efficient for parallel computing (see Hunt et al. (2007) for a detailed description of this method). Since F has different values in the nature run and in the forecast run during data assimilation, the multiplicative covariance inflation method (Anderson and Anderson, 1999) has been applied to account for model error in addition to sampling errors. The covariance inflation factor is fixed to be 1.3 in this study, which means that the background error covariance P b is multiplied by 1.3 in each data assimilation cycle. In verifying the self-sensitivity calculation method, we use 40 ensemble members, and observe at every grid point. The analysis sensitivity S ii o is calculated after each analysis cycle based on equation (6). After full observation data assimilation, each observation is left out in turn to get y i a(!i) using the same background forecasts. To experimentally examine the relationship among analysis sensitivity per observation, observation coverage, and analysis accuracy, we carry out several experiments with uniform observation coverage, which are 20, 25, 30, 35 and 40 observations. Analysis accuracy is measured by the Root Mean Square (RMS) error, which is defined as the RMS difference between the analysis mean state and the state from the nature run. In order to ensure statistical significance of the results, we run each 12

13 experiment for 1000 analysis cycles, and then calculate the time average over the last 500 analysis cycles. 4.2 Results Comparison between y i a! y i a(!i) and o S ii (1! S o ii ) (y o i! y a i ) at a single analysis time (top panel in Figure 2) shows that they have the same value at every grid point. The differences (not shown) between the two quantities are on the order of The CV score based on deleting each observation in turn (plus signs in the bottom panel of Figure 2) and that from the self-sensitivity (open circles in the bottom panel of Figure 2) essentially also have the same value. These comparisons prove that the self-sensitivity calculation method we propose (equation (6)) is valid in EnKF, and that a complete cross-validation that would be unfeasible with the standard approach, can be efficiently computed in EnKFs based on the self-sensitivity. Further experiments with different observation coverage support this conclusion (not shown). The fact that y i o! y i a(!i) can be calculated without carrying out data denial experiments can be used in observational quality control. In current numerical weather prediction, one way to do observational quality control is to compare the actual observation y o i to the predicted observation y b i from numerical model forecast. When the difference between y o i and y b i is larger than a certain statistic, the observation is marked as a bad observation (Dee et al., 1999). In this method, we can replace y b i with y a(!i) i, which can be calculated from equation (13). Since y i a(!i) is based on not only the numerical forecast, but also on the buddy observations, it is more accurate than y i b, as 13

14 shown in Figure 3. Since the error difference between y i a(!i) and y i b is small compared to the observation error (standard deviation is 0.2), using y i a(!i) instead of y i b will not make much difference in observational quality control in the current case, and it may only make a significant difference when y i b is much worse. Note that since the calculation of y i a(!i) needs to know the self-sensitivity, which is a function of analysis ensemble perturbations (equation (6)), it is only possible to use it in examining the observation quality after the data assimilation or in reanalysis mode. Figure 4 shows that the self-sensitivity increases with the analysis RMS error, which is consistent with the geometrical interpretation in section 3. The analysis error is anti-correlated with the observation coverage, and so is the self-sensitivity (Figure 4). The analysis sensitivity per observation becomes larger when the observation coverage becomes sparser. The analysis sensitivity per observation is about 0.16 when all the grid points are observed, which indicates that 16% of the information in the analysis comes from the observation at each observation location. Since the analysis sensitivity with respect to the background is complementary to the analysis sensitivity with respect to the observation (section 3), 84% of the information of the analysis comes from the background forecast. When only half of the grid points have observations, the analysis self-sensitivity is about 0.32, which indicates that deletion of one observation in the dense observation coverage will do less harm to the analysis system than deletion of one observation in a sparse case. This is consistent with field experiments (e.g., Kelly et al., 2007). 14

15 Figure 2 Top panel: y a i! y a(!i) i (open circles) and (1! S o ii ) r i (plus sign) comparison at one analysis time as function of grid point. Bottom panel: comparison of CV score calculated p p (y from " o (y o i! y a(!i) i ) 2 (open circles) and that from i! y a i ) " 2 (plus sign) in the first 60 i=1 (1! S o ii ) 2 i=1 analysis cycles. S ii o 15

16 Figure 3 Comparison of the time-average of the absolute 6-hour forecast error ( y i b! y i T, black line) and of the absolute error of the predicted observation ( y i a(!i)! y i T, grey line). y i T is the truth (from the nature run) projected onto the observation space. Figure 4 Scatter plot of the time averaged analysis sensitivity per observation (y-axis) and the analysis RMS error (x-axis) (from the bottom to the top, the points correspond to 40 observations, 35 observations, 30 observation, 25 observation and 20 observations). 16

17 5. Consistency between information content and the observation impact obtained from data denial experiments Equation (13) indicates that the deletion of an observation with larger selfsensitivity will result in larger change in the analysis value compared to the deletion of the observations with smaller self-sensitivities. Assuming that assimilating an observation makes the analysis better, which is statistically true when the error statistics used in data assimilation are correct, the deletion of the observation with larger self-sensitivity will result in a worse analysis than the deletion of the other observations. Equation (13) is only valid when only one observation is left out. However, in most Observing System Experiments (OSEs) or Observing System Simulation Experiments (OSSEs), we need to examine the impact of a subset of observations on analysis, which is done by data denial experiments. In this section, we will explore whether the trace of the self-sensitivities of a subset of observations, which is referred to as information content, can qualitatively show the actual observation impact from the data denial experiments. 5.1 Experimental setup We use the Simplified Parameterizations primitive Equation DYnamics (SPEEDY, Molteni, 2003) model, which is a global atmospheric model with 96x48 grid points in the horizontal and 7 vertical levels in a sigma coordinate. We follow a perfect model OSSE setup, (e.g., Lord et al. 1997) in which the simulated truth is generated with the same atmospheric model as the one used in the data assimilation (identical twin experiment). Observations are created by adding Gaussian random noise to the truth. The observation error standard deviations assumed for winds and specific humidity are about 30% of their natural variability, shown in Figure 5. The specific humidity is only 17

18 observed in the lowest five vertical levels, below 300hPa. Since temperature variability does not change much with vertical levels, we assume the observation error standard deviation is 0.8K in all vertical levels. The error standard deviation for surface pressure is 1.0hPa. In such an experimental setup, the observation error statistics are fairly accurate, and so are the analysis error statistics estimated in each analysis cycle. We carry out 1.5- month data assimilation cycles using the LETKF data assimilation scheme. The results shown in this section are averaged over the last month. Figure 5 The observation error standard deviation for zonal wind (Unit: m/s, left panel), meridional wind (Unit: m/s, middle panel) and specific humidity (Unit: g/kg, right panel). In the data denial experimental setup, the control experiment (called all-obs) has full observation coverage (both the closed circles and the plus signs in Figure 6). In the data denial experiments, the denied variable is only observed at the rawinsonde locations (closed circles in Figure 6). For instance, in the no-u sensitivity experiment, zonal wind observations are still observed at the rawinsonde locations but not at the locations marked with plus signs. We compare the information content of the zonal wind observations over the locations with plus signs with the actual impact of these observations obtained as the analysis error difference between no-u and all-obs 18

19 experiment. Again, the information content is the summation of the self-sensitivity, which is calculated from equation (6) based on the ensemble analyses of the control run. We also do a similar comparison for a no-q sensitivity experiment. Ideally, the area with larger information content will correspond with the area with larger error differences between the sensitivity experiment and the control run. Figure 6 The full observation distribution (31% of the total grid points; closed dots: rawinsonde observation network; plus signs: dense observation network). Every observation location is made at the grid point. 5.2 Results Figure 7 shows zonal mean zonal wind analysis RMS error difference (contours) between no-u and all-obs and the information content (shaded) of the zonal wind observations. Here, the information content is the trace of zonal wind self-sensitivity at the observation locations with plus signs in each latitude circle (Figure 6). Quantitatively, the analysis RMS error difference (contour) between no-u and all-obs experiment is largest over the tropics, and is smallest over the mid-latitude Northern Hemisphere (NH). Qualitatively, the information content (shaded area) agrees with this RMS error difference, showing the largest values over the tropics and the smallest values in the midlatitude NH. Interestingly, the zonal wind RMS error difference over the mid-latitude 19

20 Southern Hemisphere (SH) is small, even though the zonal wind observation coverage is significantly different between no-u and all-obs over that region. This is because the observations other than zonal wind, such as the meridional wind observations and the mass related observations (such as temperature and surface pressure) are used by the LETKF to update the zonal wind analysis. The information content basically reflects this feature, also showing relatively small values over that region. Figure 8 shows the time averaged zonal wind self-sensitivity (filled grid points) at the observation locations with plus signs (Figure 6) and the zonal wind RMS error difference between no-u and the control run at the sixth model level (! = 0.2 in the sigma coordinate, around 200hPa). It shows that the self-sensitivity is larger between 30ºS and 30ºN, and so is the RMS error difference. However, we should not overly interpret this result. For any two single points, the relative magnitude of the selfsensitivity may not reflect the relative magnitude of the RMS error difference due to sampling errors. Self-sensitivity can only qualitatively show the observation impact when it is summed over some spatial domain. Since the sum of the self-sensitivity and the sensitivity of the analysis with respect to the background at the same observation location are equal to 1, Figure 8 indicates that more than 85% of the information in the analysis comes from the background forecasts, which is consistent with Cardinali et al. (2004), because the background forecast contains observation information from the previous analysis cycles. 20

21 Figure 7 Zonal wind RMS error difference (contour, unit: m/s) between no-u and the control experiment, and the zonal wind observation information content (shaded) over the locations with plus signs (Figure 6). Figure 8 Time averaged zonal wind RMS error difference (contour; unit: m/s) between no-u and the control experiment at the sixth model level (! = 0.2 in sigma coordinate, around 200hPa) and the self-sensitivity (filled grid point) of the zonal wind observations at the locations with plus signs (Figure 6) at the same level. 21

22 The high spatial-temporal variability and the nonlinear physical processes related with humidity make the observation impact study related with humidity a very challenging problem (e.g., Dee and DaSilva, 2003; Langland and Baker, 2004). In spite of these challenges, the spatial distribution of the specific humidity information content is consistent with the specific humidity RMS error difference between the no-q and all-obs experiments (left panel in Figure 9). It is important to note that the information content of the specific humidity observations qualitatively reflects not only the impact on the humidity analysis field, but also the impact on the other dynamical variables, such as the zonal wind (right panel in Figure 9). This originates from the multivariate assimilation characteristics in the all-obs experiment, in which the specific humidity observations linearly affect the wind analyses through the error covariance, and this effect is maximized in the tropical upper troposphere (right panel in Figure 9). This large impact of humidity observations on the wind analyses over the high tropical levels is consistent with the multivariate assimilation of the AIRS humidity retrievals in a low resolution NCEP Global Forecast System (Liu, 2007, Liu et al., 2009). The horizontal distribution of the specific humidity self-sensitivity (Figure 10) is also qualitatively consistent with the analysis RMS error difference between no-q and the control run for both the specific humidity and zonal wind analyses. Interestingly, the information content and the self-sensitivity of the specific humidity observations (Figure 9 and Figure 10) are larger than those of the zonal wind observations (Figure 7 and Figure 8). This indicates that the specific humidity analyses are more sensitive to the humidity observations than the zonal wind analyses to the zonal wind observations. However, this does not mean that assimilating specific humidity observations would 22

23 reduce more error overall than assimilating zonal wind observations. This is because specific humidity and zonal wind have different dynamical roles in the forecast system. The large-scale interaction between zonal wind and the other dynamical variables during the forecast may make zonal wind observations more important to reduce the overall analysis RMS error. Therefore, information content may not be applicable to examine the relative impact of the observations belonging to different dynamical variable types for which forecast sensitivity to observations may be a more appropriate measure (Langland and Baker, 2004, Liu and Kalnay, 2008, Cardinali, 2009). Nevertheless, the qualitative consistency between the information content and the actual observation impact from data denial experiments suggests that we can use it to examine the relative impact of the same dynamical variable type of observations without actually carrying out the much more expensive data denial experiments. Figure 9 RMS error difference (contour) between no-q and all-obs experiment and the specific humidity observation information content (shaded) (Left panel: specific humidity RMS error difference (Unit: 10-1 g/kg); right panel: zonal wind RMS error difference (Unit: m/s). 23

24 Figure 10 The time averaged RMS error difference (contour) between no-q and all-obs experiment and the self-sensitivity (filled grid point) of specific humidity observations at the locations with plus signs (Figure 6) at the fourth model level (! = 0.51 in the sigma coordinate, around 500hPa). The contours in the top panel are the specific humidity RMS error difference (Unit: 10-1g/kg), and the contours in the bottom panel are the zonal wind RMS error difference (Unit: 10-1m/s). 24

25 6. Conclusions and discussion The influence matrix reflects the regression fit of the analysis to observations, and the self-sensitivity (the diagonal values of influence matrix) gives a measure of the analysis sensitivity to the observations. Information content, defined as the trace of the self-sensitivities, reflects the amount of information that the analysis extracts from a subset of observations during data assimilation. These measures show the analysis sensitivity to observations, and can further show the relative impact of the same type of observations on the performance of the analysis system when the statistics used in the data assimilation reflect the true uncertainty of each factor. In this paper, we showed how to calculate the self-sensitivity and the cross sensitivity (off-diagonal elements of influence matrix) in an EnKF framework. Based on the ensemble analyses generated in each analysis cycle, the calculation of the selfsensitivity and cross-sensitivity in EnKFs only needs the application of the observation operator to each analysis ensemble member (equations (6) and (7) in section 2). By comparing cross-validation (CV) scores calculated from leaving out each observation in turn with those based on self-sensitivity, we verified the self-sensitivity calculation and the cross-validation method in EnKF with the Lorenz 40-variable model. Unlike the selfsensitivity calculation in 4D-Var (Cardinali et al., 2004), the self-sensitivity calculated in EnKFs satisfies the theoretical limits (between 0 and 1) when the observation errors are not correlated, which makes it possible to calculate the predicted observation based on all the buddy observations without carrying out data denial experiments. The predicted observation calculated in this way is more accurate than the predicted observation from the numerical model forecast, which makes it a better choice in observational quality 25

26 control when the background forecast is not very accurate. In agreement with geometrical interpretation (Desroziers (2005)), we showed experimentally that self-sensitivity is proportional to the analysis errors and anti-correlated with the observation coverage when the error statistics used in the data assimilation are fairly accurate. In an idealized experimental setup, the comparison between information content and the actual observation impact given by data denial experiments shows qualitative agreement. This implies that the spatial distribution of information content can be utilized in examining the relative importance of the observations belonging to the same dynamical variable type without actually carrying out much more expensive data denial experiments. Since the impact of different dynamical variable types is also related with the dynamical role they play in the forecast system, information content may not reflect the relative importance of the observations belonging to different dynamical variable types during a forecast: the sensitivity of the forecasts to observations (Langland and Baker, 2004; Liu and Kalnay, 2008, Cardinali, 2009) would be a more useful measure for this purpose. Acknowledgements We are grateful to the Weather-Chaos group in Maryland for suggestions and discussions, and to Prof. John Thuburn and to anonymous reviewers whose valuable suggestions greatly improved the manuscript. This work has been supported by NASA grants NNG06GB77G, NNX07AM97G, NNX08AD40G, and DOE grants DEFG0207ER Appendix: Cross sensitivity derivation 26

27 Equations (4) and (5) in section 2 show that the influence matrix S o can be written as: S o = R!1 HP a H T = 1 k! 1 R!1 (HX a )(HX a ) T (A1) In the case of a two-observations ( j th and l th ) scenario, the influence matrix S o is:! S o = # " S o jj o S lj o S jl o S ll k k! $ $ & % =! 1 $ " # k ' 1% & ( 2 '1! j 0 $ #*[(HX ai ) j ) (HX ai ) j ] *[(HX ai ) j ) (HX ai ) l ] & i=1 i=1 2 " # 0 ( l % & # & k k # *[(HX ai ) l ) (HX ai ) j ] *[(HX ai ) l ) (HX ai & ) l ] " # % & i=1 i=1 (A2) Therefore, the cross sensitivity is: S o jl =!y a l!y = # 1 & o j $ % k " 1' ( 1 k 2 + [(HX ai ) j * (HX ai ) l ] (A3) ) j i=1 References Anderson JL An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129, Anderson JL and Anderson SL A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Wea. Rev., 127, Bishop CH, Etherton B and Majumdar SJ Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Wea. Rev., 129, Cardinali C, Pezzulli S and Andersson E Influence-matrix diagnostic of a data assimilation system. Quart. J. Roy. Meteor. Soc. 130,

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