Assimilation of Equatorial Waves by Line-of-Sight Wind Observations

Nedjeljka Žagar Department of Meteorology, Stockholm University, Stockholm, Sweden

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Abstract

This paper investigates the potential of line-of-sight (LOS) wind information from a spaceborne Doppler wind lidar to reduce uncertainties in the analysis fields of equatorial waves. The benefit of LOS winds is assessed by comparing their impact to that of a single wind component, full wind field information, and mass field data in three- and four-dimensional variational data assimilation.

The dynamical framework consists of nonlinear shallow-water equations solved in spectral space and a background error term based on eigenmodes derived from linear equatorial wave theory. Based on observational evidence, simulated wave motion fields contain equatorial Kelvin, Rossby, mixed Rossby–gravity, and the lowest two modes of the westward-propagating inertio–gravity waves. The same dynamical structures are included, entirely or partially, into the background error covariance matrix for the multivariate analysis. The relative usefulness of LOS data is evaluated by carrying out “identical twin” observing system simulation experiments and assuming a perfect model.

Results from the experiments involving a single observation or an imperfect background error covariance matrix illustrate that the assimilation increments due to LOS wind information rely more on the background error term specification than the full wind field information. This sensitivity is furthermore transferred to the balanced height field increments.

However, all assimilation experiments suggest that LOS wind observations have a capability of being valuable and need supplemental information to the existing satellite mass field measurements in the Tropics. Although the new wind information is incomplete, it has a potential to provide reliable analysis of tropical wave motions when it is used together with the height data.

Corresponding author address: Dr. Nedjeljka Žagar, Department of Meteorology, Stockholm University, SE-106 91 Stockholm, Sweden. Email: nedjeljka@misu.su.se

Abstract

This paper investigates the potential of line-of-sight (LOS) wind information from a spaceborne Doppler wind lidar to reduce uncertainties in the analysis fields of equatorial waves. The benefit of LOS winds is assessed by comparing their impact to that of a single wind component, full wind field information, and mass field data in three- and four-dimensional variational data assimilation.

The dynamical framework consists of nonlinear shallow-water equations solved in spectral space and a background error term based on eigenmodes derived from linear equatorial wave theory. Based on observational evidence, simulated wave motion fields contain equatorial Kelvin, Rossby, mixed Rossby–gravity, and the lowest two modes of the westward-propagating inertio–gravity waves. The same dynamical structures are included, entirely or partially, into the background error covariance matrix for the multivariate analysis. The relative usefulness of LOS data is evaluated by carrying out “identical twin” observing system simulation experiments and assuming a perfect model.

Results from the experiments involving a single observation or an imperfect background error covariance matrix illustrate that the assimilation increments due to LOS wind information rely more on the background error term specification than the full wind field information. This sensitivity is furthermore transferred to the balanced height field increments.

However, all assimilation experiments suggest that LOS wind observations have a capability of being valuable and need supplemental information to the existing satellite mass field measurements in the Tropics. Although the new wind information is incomplete, it has a potential to provide reliable analysis of tropical wave motions when it is used together with the height data.

Corresponding author address: Dr. Nedjeljka Žagar, Department of Meteorology, Stockholm University, SE-106 91 Stockholm, Sweden. Email: nedjeljka@misu.su.se

1. Introduction

Recent observational studies have shown that a significant part of the tropical time and space variability can be described in terms of the equatorially trapped wave solutions of the linearized shallow-water equations on an equatorial β plane. Wheeler and Kiladis (1999), for example, identified modes such as Kelvin, mixed Rossby–gravity (MRG), equatorial Rossby (ER), and westward-propagating equatorial inertio–gravity (WEIG) waves by analyzing 18 yr of satellite observations of outgoing longwave radiation (OLR), a proxy for deep tropical cloudiness. Wave signals with similar characteristics are reported by Yang et al. (2003), for a brightness temperature record of about the same length.

Presented evidence of equatorial waves relies on satellite observations of the convection, that is, the mass field information. Information about the dynamical structure of the waves, that is, the wind field, is currently not available from the global observing system (GOS). Instead, various methods have been utilized to identify the dynamical structure of convectively coupled equatorial waves from tropical analysis fields. Wheeler et al. (2000) used linear regression between OLR data and reanalysis fields of the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) while Yang et al. (2003) projected the reanalysis fields of the European Centre for Medium-Range Weather Forecasts (ECMWF) onto the theoretical horizontal structure functions with a best-fit trapping scale. Both studies indicate consistency between reanalysis datasets and equatorial wave theory.

At the same time, intercomparison studies (e.g., Lim and Ho 2000; Hodges et al. 2003) identify uncertainties in the existing reanalyses for the Tropics. The seriousness of the problem is clearly illustrated in Fig. 17 of Kistler et al. (2001), which shows that the differences between the tropical wind fields in the NCEP–NCAR and the ECMWF reanalyses are of the order of the full mean wind variations. Although the differences are influenced by the different use of observations, the main differentiating factors are the assimilation models and, in particular, the statistical and dynamical properties related to a priori (background) information for the assimilation; in data-sparse areas, such as the Tropics, a priori information become crucial for the analysis result.

In this paper, we study the impact of wind field information on the analysis fields of convectively coupled equatorial waves. The type of observed information investigated here is not the full wind field, but the wind component measured along a single line of sight (LOS). This particular choice is motivated by the Atmospheric Dynamics Mission (ADM) of the European Space Agency (ESA), which will provide the first direct observations of atmospheric wind profiles from space (ESA 1999).

While a lack of direct wind measurements has been recognized as the main missing component of the current GOS (e.g., Baker et al. 1995), the extent to which LOS wind measurements can fill the gap is not clear. Classical adjustment theory (e.g., Haltiner and Williams 1980, Chapter 2) provides arguments that wind field information has greater importance for numerical weather prediction (NWP) than information about the mass field at small horizontal scales and low latitudes; global observing system experiments (OSEs) and observing system simulation experiments (OSSEs) suggest that the argument applies globally. For example, Atlas (1997) has shown that the global error reduction in the initial state was larger and more rapid with the assimilation of wind profiles as compared to temperature data; Graham et al. (2000) found that wind profiles provided more benefit than temperature profiles, especially at levels above ∼400 hPa, in 60 studied significant data-impact events.

The launch of the ADM mission, planned for 2007, will result in a spaceborne Doppler wind lidar (DWL) providing global measurements of LOS winds. LOS wind information is a projection of the wind vector onto the direction of the satellite's line of sight; it cannot be directly compared with full wind information and it can presumably not be utilized as the single observation source. It is the data assimilation procedure which, by applying observation operators to compare a LOS wind observation with its model equivalent, ensures usefulness of this particular type of information. Therefore, the usefulness of LOS winds cannot be studied by OSSE with complete wind field information.

A key question we ask here is: how well can the LOS wind observations constrain the analysis solution in the Tropics? In particular, are LOS winds a useful complement to the existing mass field observations? The answers to these questions are presented by comparing the usefulness of LOS winds to that of single wind component data, full wind field information, and mass field data.

The methodology and the model employed for this purpose are presented in section 2. The analysis method is variational data assimilation (e.g., Le Dimet and Talagrand 1986), nowadays the operational method for data assimilation at many NWP centers. The numerical model framework is, however, significantly simplified with respect to a NWP model; we utilize a nonlinear shallow-water system and idealized tropical motion simulations as observations. While a simplified framework restricts the physical validity of our conclusions for full-scale NWP models, it allows us to emphasize the dynamical aspects of tropical data assimilation and to avoid confusion due to a poor representation of physical processes in the Tropics and their interaction with dynamics, as is often the case with more complete NWP models. In an idealized framework, the assessment of the potential of LOS wind data and its comparison with other data types is much easier. This is important since the potential usefulness of LOS wind information has been questioned (Riishøjgaard et al. 2002). In addition, a simplified framework should more easily illuminate the relevance of the background error term in the Tropics and, in particular, the mass–wind coupling; these aspects of data assimilation have been difficult to demonstrate in more comprehensive OSSE studies (Hoffman et al. 1990; Gauthier et al. 1993; Marseille et al. 2001), although an improvement in the analysis accuracy with DWL data has been demonstrated.

To interpret the ADM data, information about the wind component perpendicular to the line of sight needs to be inferred from the background error covariance matrix for the assimilation. Dynamical and statistical relationships built into this matrix determine the spatial structure of the analysis increments from a single LOS wind observation. Furthermore, in the case of multivariate data assimilation, a LOS wind observation will also produce a balanced increment in the height field. This is discussed in detail in section 3.

In section 4, a typically observed equatorial wave is assimilated by using different data types and various LOS viewing azimuth angles. The analysis results with multiple waves and simulated ADM observations are described in section 5. The experiments utilize both idealized conditions of a faithful background error covariance matrix as well as less idealized conditions of an incorrect one. A further discussion of the results, including the possible relevance of presented simulations for NWP, is given in section 6, and main conclusions are stated in section 7.

2. Numerical model and observations

a. Assimilation method and model

The variational data assimilation method searches for a solution which minimizes the distance to both the observations and the background model state, taking into account statistics of their respective errors. In three-dimensional variational data assimilation (3DVAR), all observations are valid at the same time instant and the analysis increments obtained at the end of the minimization process satisfy, in a statistical sense, the balances implied by the background error covariance matrix. When observations are available at multiple time instants, admissible solutions also fulfill the dynamical model equations [four-dimensional variational data assimilation (4DVAR) e.g., Le Dimet and Talagrand 1986]. An optimal analysis solution xa minimizes the following distance (cost) function (J):
i1520-0469-61-15-1877-e1
Here, the vector y contains observations distributed among K different times while the model state vector x consists of two wind components (u, υ) and the height of the fluid-free surface (h). Since observation errors are assumed to be statistically independent, the covariance matrix of the observation errors, 𝗥, is a diagonal matrix including the variance values for the wind components and the height. The background model state vector and the matrix of its error covariances are denoted by xb and 𝗕, respectively. The model integration is implicitly included in (1) via the observation operator H.

The 4DVAR system applied to the Tropics represents a new methodology, developed in two previous studies, Žagar et al. (2004a, hereafter ŽGK1) and Žagar et al. (2004b, hereafter ŽGK2). The first study provides a detailed description of the spectral solution method for the shallow-water equations and the experiments involving the observational part of the cost function, Jo. The second paper describes experiments with a new approach to modeling of the background part of the cost function, Jb.

The new approach consists of a set of suitable transformations, applied to the analysis increments (δx = xxb), which allows us to redefine the variational optimization problem (1) in terms of a new variable χ, for which 𝗕 is an identity matrix. A sequence of transformations leading to the diagonalization of the 𝗕 matrix, described thoroughly in ŽGK2, is based on the idea that forecast error covariances can be approximated by using the normal modes of the model (Phillips 1986; Daley 1993). In the tropical case, these are eigensolutions of the linearized shallow-water equations on the equatorial β plane, represented by a parabolic cylinder function expansion (Matsuno 1966; Gill 1982).

The observation operator H consists of interpolation of the model state to positions with observations and computation of the model equivalent of the LOS wind measurements. At any step of the minimization process, a background model state at an ADM measurement point (xob, yob), H(x), is obtained by summing up contributions from all waves in the Fourier series solution for a particular variable. The model counterpart of the observed LOS wind, VLOS, is obtained by a projection of the model wind vector at the measurement point, knowing the corresponding azimuth angle of the DWL instrument, θ, relative to the local y direction:
i1520-0469-61-15-1877-e2

b. Numerical model setup

Assimilation experiments are carried out on a domain made of 180 × 65 points, with a 1° resolution in both horizontal directions. Seven grid points are added in the meridional direction for the purpose of biperiodization (see ŽKG1 for details). Periodic boundary conditions are applied in the zonal direction. In the meridional direction, the waveguide effect, in addition to the relaxation towards flat boundaries, ensures that the motion is confined to the inner model area.

All experiments are performed for an equivalent depth H = 25 m. The phase velocities of the corresponding linear wave solutions lie in the range typical for convectively coupled equatorial waves, found through observations; the corresponding meridional (trapping) scales are within 5°–15° latitude (Wheeler and Kiladis 1999).

Both 3DVAR and 4DVAR experiments are performed; their difference reveals the impact of the internal model adjustment. The adjustment between the mass and the wind fields in 4DVAR occurs through the model dynamical equations, supported by the a priori wave structures built into the background error covariance matrix.

The assimilation time window for 4DVAR is taken to be 48 h. This is a longer window than usually used in NWP. However, for this study it is entirely appropriate since its purpose is to allow adjustment to take place. A window as short as 12 h, commonly used for NWP purposes, does not make tropical adjustment possible because of slow phase speeds as determined from observations. This applies especially to the use of the height field information, as discussed in ŽGK1. Nonlinearities, primarily convection, are likely to prevent a longer 4DVAR window in NWP; these are not taken into account here.

c. Estimation of the background field error statistics

The spectral variance model (see ŽKG1 for details) distributes the errors in the background field among different equatorial eigenmodes. Following the observational evidence presented in Wheeler and Kiladis (1999), the variance is divided (in a descending order) between the ER, Kelvin, eastward and westward MRG, and n = 1, 2 WEIG waves. As demonstrated in ŽGK2, including Kelvin and MRG waves effectively reduces the coupling between the wind and the height fields near the equator, as compared to the case when only ER waves are applied; in addition, it makes the horizontal structures more zonal.

Values of the background error standard deviations in the gridpoint space are obtained by carrying out a randomization experiment which takes into account the actual Jb formulation (Fisher 1996). The method takes a random vector in the space of control variable χ, generated from a Gaussian population with zero mean and unit variance, and transforms it into the increment fields for u, υ, and h by applying the background transformation operator of the assimilation system. Dynamical fields obtained in this way represent therefore error variances of equatorial waves present in the 𝗕 matrix. Averaging over a large number of realizations, a background error variance distribution representative of the model is obtained. In the case of LOS winds, the error statistics calculation is completed by using the relevant observation operator: horizontal interpolation to ADM locations and projection operator for different azimuth angles.

A distribution of background error variance, based on mass field observations, and the utilized background constraint define the variance of the wind field in our assimilation system. The resulting structure of the background error variance fields is homogeneous in the zonal direction for all three variables. In the meridional direction, the maximum of the variance for all fields is located on the equator and the maximal errors in the zonal wind field are larger than in the meridional wind component. Farther off the equator, the variance field is mainly due to ER modes and its magnitude is larger for the height field than for the wind field components.

d. Observations

The ADM observations are simulated at a single horizontal level. Locations of measurements and corresponding azimuth angles are calculated by an orbitography simulator. The orbit geometry follows that of the proposed ADM mission (ESA 1999): the inclination angle is 97.2° while the LOS azimuth is 90° with respect to the subsatellite track. A duty cycle of 25% results in measurements taking place over a 50-km range, followed by 150 km without measurements. In this way, a wind vector projection along the direction of the line of sight is provided every 28 s. Over a 48-h period, four global scans are carried out. At the equator, the azimuth angle, defined as an angle between a line of sight and the y axis (north), is about 97°. Traveling from the equator southward, the azimuth is increasing up to 112° at 30°S. All together, the ADM measurements in the Tropics are close to zonal.

The satellite trajectory, shown in Fig. 1, is starting from the equator at time zero and during 48 h about 6200 LOS wind measurements are made globally, out of which 1114 are found within the predefined inner model domain. This corresponds to about 10% of the number of degrees of freedom per model variable. The same volume of information is assumed accessible for 3DVAR. So many satellite winds will hardly ever be available for 3DVAR at a single point in time. Moreover, this is not the common way data assimilation comparison experiments are done within the NWP framework, where 3DVAR cycling is performed over the 4DVAR time window. But, our choice is justified by an intention to illustrate the dynamical effect of the time dimension in data assimilation, as compared to the potential of the background error covariance matrix; consequently, it is of interest to have the same number of degrees of freedom for observations in 3DVAR as in 4DVAR.

It can be seen in Fig. 1 that the distance between adjacent observations in the zonal direction is over 1000 km at most places, whereas the resolution for observations is much higher in the meridional direction. An additional experiment is therefore constructed with two DWL instruments in the space, both measuring along the same single line of sight. A second satellite is defined to start 6° longitude east of the first one, which places it 24 min ahead (for the sun-synchronous orbit). In this way the zonal spacing between LOS wind observations is reduced to about 600 km (figure not shown). Adding another satellite results in doubling the number of degrees of freedom for observations of each scalar variable measured along the satellite track.

Simulated observations are constructed by using the randomization experiment, that is, from a population that has the same flow statistics as the background field errors. A complete dynamical consistence of the simulated truth is achieved by carrying out a few days of forecast simulation. This “nature run” field is shown in Fig. 2; it illustrates features of the tropical circulation such as strong cross-equatorial flow, zonal winds along the equator and geostrophically balanced structures off the equator. Two-thirds of the energy is kinetic, two-thirds of which come from the zonal wind component. All perturbations decay to zero as the meridional model boundaries are approached. Final observations are constructed by adding to a simulated variable a random error from a distribution of zero mean and variance equal to the variance of the background error field at the same point, N(0, σb); that is, it is assumed that σo = σb.

Besides the multiple wave field shown in Fig. 2, assimilation experiments are also carried out with observations from an analytical linear equatorial wave solution, perturbed in the same manner. Locations for observations in this case coincide with the model grid points and they are randomly distributed within the inner model area, starting from 50 observations and increasing up to 3000 observations (every 50 up to 250, then every 250 up to 1000, and every 500 thereafter). When adding more observations, positions of those already included are not changed. Azimuth angles range from 0° (measuring υ wind) to 90° (measuring u component), with a 10° step.

3. Single LOS observation experiments in 3DVAR

This section compares analysis increments due to various single LOS wind observations with increments from a full wind vector observation. First we present the structures of the increments due to a single observation of the height, and zonal and meridional wind fields, all located at the equator (Fig. 3).

The analysis increment in the height field due to a single height observation is nearly isotropic (Fig. 3a). Presence of a coupling between the mass and wind fields leads to increments in the u wind; the whole structure resembles a Kelvin wave. A quite similar shape results from a single zonal wind observation (Fig. 3b). However, in this case the increment is not isotropic; on the contrary, the zonal scale of the wind increments is almost twice as large as the meridional scale. The coupling between increments in u and υ wind components is very weak. Similarly, the assimilation increment due to a northward wind observation at the equator is also characterized by a very weak coupling of the increments in the meridional wind with both the zonal wind and the height fields. The structure of the balanced height increments is asymmetric about the equator, suggestive of an MRG wave (Fig. 3c).

When the full wind field information is reduced to an LOS component, the structure of the analysis increments is inevitably altered. The degree of modification will depend on the departure of the LOS from the direction of the actual wind vector. Examples, illustrating primarily bad scenarios, are shown in Fig. 4. Wind observations are the same as those employed in Figs. 3b,c, but observed along two lines of sights, with azimuths 30° and 75°.

When the zonal wind is observed with an azimuth angle of 30°, the resulting analysis increments in the wind field are characterized by small-amplitude winds and very small increments in the height field. The whole structure is rotated towards the satellite-view direction (Fig. 4a). When the azimuth is 75°, an LOS which is much closer to the direction of the true vector, the increment is only slightly distorted (Fig. 4b; cf. Fig. 3b). A similar behavior is obtained in the case of LOS observations of a meridional wind. With an azimuth 30° off the true direction, the analyzed wind field is weaker and rotated clockwise as compared to the case with the complete information (Fig. 4c; cf. Fig. 3c). A balanced height increment is centered at the equator, as in the zonal wind case. When the azimuth increases to 75°, far from the true wind direction, assimilation increments look as those due to a zonal wind observation of a much smaller amplitude (Fig. 4d). All together, Fig. 4 highlights the fact that a distortion of assimilation increments due to incomplete wind field information is not limited to the wind field but, in the case of a multivariate analysis, affects also balanced increments in the height field.

The impact of changing the azimuth angle between 0° and 90° on the amplitude of the analyzed wind is summarized in Fig. 5. It shows the difference between the amplitudes of increments at the observation point of full wind information and an LOS observation, scaled by the amplitude of increments in the case of full wind field information. Two features can be noticed. First, the error in the observed wind component (e.g., zonal wind in the case when the observed wind is zonal) decreases approximately linearly with the deviation of the azimuth angle from the direction of the true wind vector. Second, the error amplitude in the other component, which is known to be zero in the case of the complete wind observation, is quadratic; that is, the error is largest when the azimuth is approximately 45° off the true direction. In addition, the error can be twice as large for the zonal wind when the simulated true wind is meridional (up to −0.65 in Fig. 5b) than for the meridional wind when the wind is zonal (up to −0.3 in Fig. 5a). This is because the dominant a priori flow component in our tropical assimilation system is zonal.

The way the background error covariances are formulated in the model attributes a greater weight to the u than to the υ component. This is because a significant part of the background error variance, assigned to the Kelvin waves, favors a projection onto the zonal wind for a single LOS observation. This anisotropy is illustrated in Fig. 6, which presents assimilation increments for a LOS observation of a single southwesterly wind observation. The azimuth angle is 45°, that is, it coincides with the direction of the wind vector. While the amplitudes of the increment fields for u and υ are the same when the complete wind information is provided, there is a significant difference in the case of LOS wind information. The υ increment in this case has an amplitude about 70% of that in u increment. It is both due to a smaller increment in υ (−20%) and a larger increment in u (+20%). The balanced h increment is also by some 20% larger as compared with the standard case. This feature needs to be kept in mind when discussing results of assimilation with multiple observations.

4. Assimilation of a single equatorial wave

As often inferred from observations (Wheeler and Kiladis 1999, and references therein), the ER wave represents a relevant structure for addressing the issue of information content of LOS winds for analysis of real equatorial waves. An additional interesting aspect of the n = 1 ER wave is that the coupling between its mass and wind fields near the equator is opposite to that in the Kelvin wave, that is, the signs of dominant correlations built into the 𝗕 matrix and the ER wave are opposite. In other words, choosing the ER wave makes the analysis problem more difficult. By doing so, we intend to highlight an important aspect of tropical data assimilation, absent in the midlatitude case—the uncertainty related to the existence of an effective correlation between the mass and wind fields, to the sign of this correlation as well as its length scale.

The n = 1 ER solution, representing the truth, is shown in Fig. 7. A distinct feature in this figure is the concentration of the wave kinetic energy near the equator. The potential energy, on the other hand, is found mainly off the equator, where the height and wind fields are geostrophically balanced.

a. Random observations and azimuths

The results of 3DVAR experiments are summarized in Fig. 8 in terms of root-mean-square errors (rmse's) of h, u, and υ fields, normalized by their corresponding background field errors. Rmse values are presented as a function of an increasing number of observations of various types and for different azimuthal angles.

The most noticeable feature of the results is the inefficiency of wind field information to recover the height field (Fig. 8a) and the height field to recover the zonal wind component (Fig. 8b). Adding height data, besides the zonal wind observations, makes the rmse for u even worse than assimilating LOS winds with azimuths 90° (equals u) and 80°; this is clearly a consequence of opposite directions for the zonal wind in its observations and the balanced increments due to h observations. It illustrates the fact that the a priori balance, built into the background error covariance, represents an average mix of wave components that may not be optimal for the assimilation of a single particular wave component. Furthermore, LOS wind information of any azimuth appears in this case to be more valuable than height field information. Notice also that adding the υ observations, in addition to the h, contributes little to the analysis solution for u, in agreement with the single observation experiments (Fig. 3).

On the other hand, adding any wind information besides the height data improves the solution for h (Fig. 8a). Wind alone, either as the full vector or only the LOS component, performs fairly badly in recovering the mass field. An explanation for a significant part of the error is the poor a priori information about the height–wind coupling, as illustrated in Fig. 9a. It can be seen that fallacious height field increments are centered at the equator, in accordance with the Kelvin wave type of balance provided by 𝗕.

Nevertheless, the wave wind field—that is, its kinetic energy—is recovered from the LOS winds very well. The height field away from the equator is restored to some degree and the associated geostrophic circulation is also included (Fig. 9a). On the other hand, assimilation of height data (Fig. 9b) is incapable of reconstructing the wind field near the equator. The winds are not only far too weak, but they also have the wrong direction, provided through the multivariate information in 𝗕. The part of the wind field successfully restored is that which is geostrophicaly balanced with the height field off the equator. The same feature can be seen in the rmse scores for υ (Fig. 8c): height data have a better capability of restoring the meridional wind than LOS winds for azimuths larger than 40°.

A lack of kinetic energy in the analyzed wave field can be detrimental since a subsequent forecast will tend to adjust the mass field to the wind field; in other words, it is expected that the missing part of the wave is recovered better by the model internal adjustment in the case shown in Fig. 9a than that in Fig. 9b. Simple forecast experiments starting from these analyses confirm this, in particular for the first 12 h (not shown).

b. ADM assimilation experiments

Next the analysis solution for the ER wave is examined by using simulated ADM observations, described in section 2b, in 3DVAR and 4DVAR.

In general, 3DVAR results are somewhat better than those with 1000 randomly distributed observations and LOS azimuth 80°. Rather than repeating a figure quite similar to Fig. 9a, we present a result of assimilation where both height field and ADM wind data are available (Fig. 10a). This is a realistic scenario since numerous satellite mass field measurements are routinely available. In comparison to Fig. 9b (which can be considered as representing the current GOS), Fig. 10a provides strong evidence that measurements such as those provided by DWL could significantly improve tropical analysis fields. It is not only the wind field analysis, but the height field is also reconstructed better when LOS winds are added to the assimilation. (A figure based on only height data on ADM locations is not shown, but it is very similar to Fig. 9b).

In particular, it is worthwhile to emphasize the value of ADM winds in the equatorial belt 10°S–10°N, where the geostrophic balance ceases to apply. The ER wave winds there are mainly zonal and cannot be recovered by assimilating the height and meridional wind component data (Fig. 10b). Nevertheless, presence of some wind information helps to reverse the weak winds along the equator in the right direction. At the same time, the solution farther off the equator has become less balanced than in the case of assimilation with only height data (Fig. 9b).

Adding another wind component, that is, the component perpendicular to the line of sight, besides height and LOS wind information, is not important for a further improvement of the analysis. This applies both to 3DVAR and 4DVAR solutions, as illustrated in the next figure.

Figure 11 compares values of different observation types in 3DVAR and 4DVAR, with both solutions valid at the beginning of the assimilation window. It is noticeable that the rmse for h (Fig. 11a) is relatively large for all 3DVAR experiments without height observations. Adding the time dimension, that is, allowing for the internal model adjustment, results in a major improvement. At the same time, the analysis result with height data becomes somewhat worse as a consequence of a loss of valuable height information during 4DVAR, as discussed in ŽGK2. Also, it can be noticed that the 4DVAR solution for h is better with full wind information than with height data.

Figure 11b reveals that assimilating height data in 3DVAR makes hardly any contribution to the zonal wind field of the ER wave. This is not only due to the smaller importance of height observations near the equator, but in this case it can also be attributed to the opposite signs of height–wind coupling in the Kelvin and ER waves. Adding υ observations, besides h, helps very little since the coupling between u and υ is very weak. In these cases 4DVAR considerably improves the analysis result.

Behavior of υ scores (Fig. 11c) is particularly interesting as it shows an almost systematic worsening of the results in 4DVAR, in particular for the assimilation experiment with υ data alone. With either u, h, or υ in 3DVAR, the result is already good since the reconstruction is correctly handled by the balance relationships built into 𝗕. In 4DVAR, however, there is an adjustment to the dominant balance structure from the a priori information, and that one is dominated by the zonal flow; as a result, a part of the good solution for υ is lost.

5. Assimilation of multiple waves

For a more general case, simulated observations are constructed by using the model spectra, as described in section 2d. In this case, with error covariances perfectly known, the wind field information in 4DVAR (perfect model assumed) could be sufficient for recovering both the tropical wind and the mass fields (ŽGK2). The question asked here is to what extent this finding is valid when only part of the full wind field information is available.

Assimilation results are summarized in Fig. 12 for 3DVAR and 4DVAR experiments with observations taken along the ADM track, and, in addition, the result of the 3DVAR experiment with two satellites is included in the same figure.

As in the case of the ER wave, adding height data to LOS winds in 3DVAR hardly makes any contribution to the analysis scores for u and υ, and opposite, the rmse for h is little improved by adding any wind field information on top of the height data. There are, however, some distinguishing features in Fig. 12, as compared to Fig. 11, primarily due to the different relationship between observed structures and a priori information. First of all, rmse scores for h (Fig. 12a) reveal that the improvement of 4DVAR over 3DVAR for various wind observations is much smaller than in the ER wave case since, in the present case, a priori information strengthens the use of observations. A similar conclusion applies to the zonal wind field (Fig. 12b): information provided through the 𝗕 matrix supports the observations; therefore, the internal model adjustment is almost not needed. The same behavior does not apply to the meridional wind field (Fig. 12c). In this case the 4DVAR solution is clearly superior to 3DVAR unless assimilated data include also υ information. Even the assimilation of both h and u observations together in 3DVAR is bringing less information about the υ field than the LOS winds.

Comparing the rmse's for LOS and u and υ winds, it can be seen that the behavior of assimilation of LOS winds is neither that of u nor υ component, but rather like assimilation of the full wind information; the difference is that rmse scores for assimilation of LOS wind information are worse. The rmse of 3DVAR analysis in Fig. 12 suggest that LOS winds are better than the u data in recovering the υ field, besides being much more efficient than the υ data in restoring the u and h fields. In other words, LOS wind information deduces more useful information about the υ field from the 𝗕 matrix than the u data do, and extracts more information about the u field from 𝗕 than the υ data succeed in. This is an important property since the ADM setup favors observations of zonal winds in the Tropics. As will be shown later, making use of this particular feature of LOS winds' assimilation assumes that the 𝗕 matrix is modeled truthfully.

Results from the experiment with two DWL instruments in the space illustrate the impact of the increased horizontal resolution in the zonal direction on the 3DVAR analysis result. The main conclusion from this experiment is that improving adequately the observation coverage results in a larger improvement than obtained by propagating dynamical information from the background error covariances and observations in time (4DVAR). This applies in particular to the wind field since its rms errors respond less to the increased amount of height data. Among different types of information, LOS winds seem to profit the most from the increased number of observations in 3DVAR.

The output fields from different experiments are compared in Fig. 13. A detail of the simulated truth, with which the rest of the figures should be compared, is presented in Fig. 13a. The subsequent four figures illustrate how the analysis solution gradually improves starting from the height data in 3DVAR (Fig. 13b) and 4DVAR (Fig. 13c), over LOS wind information in 4DVAR (Fig. 13d) up to height and LOS wind observations assimilated by 4DVAR (Fig. 13e).

The poorest analysis is the one obtained with only height observations in 3DVAR (Fig. 13b). The analysis solution contains in this case mainly large-scale features of the height field. Only the zonal winds are restored, but the velocities are too weak. This is not surprising, having in mind the structure of the increments due to a single height observation, shown in Fig. 3a. A lack of meridional flow is corrected to some extent by the time propagation of observational information in 4DVAR, as illustrated in Fig. 13c. On the other hand, the height field structure is made worse by 4DVAR.

Analyzing the LOS wind information instead of the height data restores the wind field much better, at the expense of the height field (Fig. 13d). Recovered height field features have smaller scales than those reconstructed from height observations, a feature noticed in ŽGK2 and explained by the different spectral properties of the mass and the wind field response in the assimilation (Daley 1991, Chapter 5). Adding another wind component greatly improves the 3DVAR solution for υ (figure not shown), as previously indicated by the rmse scores in Fig. 12c; it also somewhat deteriorates the 4DVAR result due to adjustment toward the predominantly zonal structures of the 𝗕 matrix.

Finally, Fig. 13e ensures that the analysis solution using LOS winds, together with height field information, provides a reasonably reliable approximation of the truth. The result is only a little affected by the time dimension, except for the υ component.

The meridional wind observations have, on average, the greatest difficulties to recover the other two fields (u and h). For 3DVAR this is explained by the very weak correlation between υ and both h and u. In 4DVAR, the solution is not significantly improved because the adjustment to balanced structures in 𝗕 favors zonal winds and the adjustment through the model equations goes in the same direction, due to the trapping effect in the Tropics whereby the motion is channeled along the equatorial waveguide. As a result, the 4DVAR solution with υ observations (Fig. 13f) remains poor except for the cross-equatorial flow between the equator and 10°N.

An error-free forecast model and a perfect background error covariance matrix are never found in real NWP systems. It is interesting to discuss the extent to which the results would be changed in case the earlier assumptions were relaxed. Experiments are therefore carried out where the model error statistics are gradually changed from perfect ones toward rather incorrect ones. The change is introduced by disturbing the model spectra, that is, the spectral variance distribution among various equatorial eigenmodes. The resulting background error term does not support the observations to the same extent any longer.

In the first experiment (Fig. 14), imperfect background error spectra are produced by increasing the weight given to WEIG waves at the expense of Kelvin and MRG waves, whereby their spectral variance density is reduced to one-third of its previous value. The resulting negative impact in 3DVAR is largest for the unobserved variables, that is, the fields reconstructed through the balance relationships built into 𝗕. Consequently, allowing for internal model adjustment substantially improves the scores for the balanced variables. For example, the rmse for the h field, in case of 4DVAR analysis with LOS winds and a faithful 𝗕 matrix, is about 6.7% improved as compared to the corresponding 3DVAR solution. An imperfect background error covariance matrix results in a 15% worse rmse for h (than in the ideal 3DVAR case); this score is, however, improved by as much as 17% by 4DVAR (Fig. 14a).

The most important feature in Fig. 14 is that LOS winds are more sensitive to the changes in the 𝗕 matrix than the height or the full wind data. This is not a surprising outcome, having evaluated the larger dependence of LOS winds on the a priori information in section 3. On the other hand, measuring the LOS wind component may still be a better choice, on the average, than observing any particular wind component. This can be seen by comparing the rmse from the los experiment with those from the u and v experiments.

At last, we present results from the experiment with a much more dramatic change of the 𝗕 matrix. In this case, only ER and WEIG waves are included, making the background error covariance matrix better suitable for midlatitude motions than for equatorial flows. This can be considered as really poor statistics for the flow shown in Fig. 2; on the other hand, it can be argued that such a situation is not less likely than the ideal situation with perfectly known 𝗕. The two extremes are compared in Fig. 15.

There are several distinctive features in this figure, when compared to Figs. 12 and 14. First of all, all scores are significantly worse. Furthermore, referring to the previous discussion, we can notice that the advantage of the LOS winds over the u or υ wind components is not seen any longer. Other remarkable features of Fig. 15 include larger deterioration of 4DVAR with respect to 3DVAR results for variables which are provided by observations (e.g., h and experiments in Fig. 15a, u, los, and los2 experiments in Fig. 15b), even smaller relevance of the 4DVAR assimilation in cases when both mass and wind data are available (hlos and hlos2 experiments). The exception are the υ field scores (Fig. 15c), where practically no improvement over the first-guess field is obtained, not even when the υ field itself or the full wind field observations are assimilated.

The poor rmse in Fig. 15c may be explained by the filtering effect of the background error covariance matrix, which removes even the useful observational information. Namely, the balance relationships are in this case dominantly nondivergent, that is, given by the ER modes; this projects the increments due to the meridional wind observations near the equator onto the height and its geostrophically balanced wind field. As a result, very little energy is left in the meridional wind component (an example is included in Fig. 3 of ŽGK2). In addition, the largest part of the υ field used as the truth (see Fig. 2) is not in the ER, but rather in the MRG modes, which are completely absent from the 𝗕 matrix.

6. Discussion

A direct motivation for this study has been an ongoing ESA project to build a spaceborne DWL instrument, providing global LOS wind profiles in regions free of clouds. The most useful DWL measurements are expected in the upper troposphere and the lower stratosphere (ESA 1999), altitudes where important equatorial wave motions are located. At the same time, dynamical processes at these altitudes in the Tropics are less well understood, and their better modeling is generally considered as critically important for improvements of the extended-range and seasonal forecasting.

Although motivated by the DWL project and its potential for NWP, the present study is done within an idealized framework. This prevents us from comparing directly our results with results from OSSEs performed with NWP models. On the other hand, in a simplified framework the essential aspects of the tropical assimilation problem can be captured and illuminated more directly. Involving convection, land–sea contrasts, and radiative processes introduces additional couplings that may hide the main issue. For example, the impact of a spaceborne DWL assessed by OSSE experiments in some previous studies (e.g., Hoffman et al. 1990; Gauthier et al. 1993) resulted in a smaller impact than may be expected in the Tropics. A closer investigation of the properties of the background error covariance matrix in the Tropics, taken up in this study, can hopefully contribute to a better understanding of the reasons for this.

Presented assimilation experiments employ OSSE methodology; it provides only a qualitative assessment on observations' impact (Atlas 1997) and tends to overestimate the impact of sparse data. Moreover, the disadvantage of OSSE is that the observation error variance is unknown for nonexisting observing system like the ADM/DWL. Other weaknesses of ADM in the Tropics are uncertainty due to cloud cover and unavailability of vertical velocities.

Nevertheless, from the dynamical point of view, large-scale tropical motions should benefit most from the ADM, due to the importance of the wind field for tropical dynamics as well as the sparsity of direct wind observations. At the same time, the tropical region exhibits greater sensitivity than midlatitudes with respect to the formulation of the background error term for variational assimilation. Since an LOS wind measurement will contribute to both wind components, depending on its azimuth angle, information built into the background error term becomes even more important than usual; it affects more strongly both the wind components and the mass field.

The present investigation is limited to horizontal structures and deals primarily with the linear dynamics. Convection, in particular, introduces nonlinearities on short time scales that may make a 4DVAR assimilation over a long time window much more difficult than we have found it to be with a shallow-water model. Also, the inclusion of additional vertical length scales will make the assimilation problem more involved. Nevertheless, we believe that starting with a simplified framework is a necessary step to obtain improved understanding of the tropical data assimilation problem.

In section 3, some features of analysis increments due to LOS winds, relevant also for NWP systems, are illustrated by single observation experiments. It is shown that the horizontal structure of wind field increments is more dependent on the background information for LOS wind than for the full wind field information. Furthermore, this clearly negative effect on wind increments is transferred to the increments of the height field. One may ask whether this is an argument in favor of a univariate wind analysis in the Tropics, as has traditionally been the case. Observations, however, support the linear theory for tropical waves; it is therefore reasonable to attempt to incorporate their balance relationship into data assimilation systems for the Tropics.

The potential value of LOS winds is investigated by carrying out “identical twin” OSSE experiments for both a single equatorial wave and a complex flow pattern. It is maybe an unfair test for the assimilation system to assimilate a single wave, as exactly the same structure is available through the 𝗕 matrix. For example, a simple harmonic oscillation, such as a Kelvin wave, requires only two ideally located observation points in order to reconstruct the complete wave solution by 3DVAR. The two points can provide either mass field or wind field information, since the other field is obtained from the multivariate relationships built into 𝗕. The maximal amplitudes of the recovered wind and height fields depend in this case on the ratio between the variances of the background error and the observation error. However, recovering other equatorial waves from few observations is more difficult than for the Kelvin wave case; this is due to their more complex structure and due to a strong projection of the analysis increments for a single observation onto the Kelvin wave.

The choice of an analytical ER wave solution is found interesting for an additional reason; it illustrates the peculiarity of the mass–wind coupling at the equator. This is a nonexisting problem for midlatitude planetary and synoptic scales for which the geostrophic balance is the backbone of data assimilation systems. In a midlatitude large-scale case there is essentially only one candidate for the balanced wind (height) solution, when the height (wind) data are assimilated. A lack of a similar balance relationship for the Tropics leaves data assimilation procedures with greater uncertainty. The background error covariance matrix for the present study is dominated by a Kelvin wave type of balance near the equator. The mass–wind coupling in the ER wave is opposite to that; therefore, the recovery of the wind field from height observations, handled by 𝗕, results in an erroneous direction of the winds in the equatorial belt. Our results indicate that assimilation of LOS winds, together with height data, restores correctly the whole wave structure.

A comparison of the results from various multiple wave experiments demonstrates the importance of a reliable Jb term. Our assimilation system for the Tropics makes, on the average, better use of the wind component measured along the line of sight than of any particular wind component, unless the 𝗕 matrix is very unreliable. When the height and the LOS wind information are assimilated together, an almost negligible difference in rmse between 3DVAR and 4DVAR (except for υ in Fig. 12c) is a striking result. This does not suggest, of course, that 4DVAR is not needed. Rather, it illustrates the power of appropriate multivariate relationships built into the 𝗕 matrix and the importance of having both mass and wind information available. Using ADM data over a 48-h interval as if they were valid at the analysis time is a highly unrealistic scenario. In the NWP framework, a 3DVAR cycling by using, for example, a 6-h window over the 4DVAR time window would be carried out instead. Such an experiment is more difficult within a simple system like ours without taking into account the nonstationarity of the forecast error covariances, which is out of scope for the present study.

Using another set of satellite data in 3DVAR might seem to be just another unrealistic scenario; an operational ADM scenario must, however, include several satellites. In any case, it is an interesting result that assimilating more data in 3DVAR instead of using 4DVAR has a large positive impact on the variable which is observed; nonobserved variables still benefit more from the internal adjustment in 4DVAR.

Lastly, this study deals primarily with the Tropics, where expectations from the ADM data are most optimistic. We have not tackled the problem of including the appropriate tropical horizontal structures in global NWP background error covariance matrices, dominated by the extratropical structures. We have also found that it is not well understood how the tropical wave solutions are currently represented in the 𝗕 matrices of state-of-art models, an issue raised in ŽGK2. A study is currently under way to investigate in more detail the statistical structures of the forecast errors of the ECMWF model in the Tropics based on equatorial waves, as a step towards a full, three-dimensional tropical background error covariance matrix. Whether and how the present findings are modified when a more realistic model is used remains to be seen.

7. Conclusions

We have considered a highly simplified version of the problem of assimilating simulated LOS wind measurements in the Tropics. Our study, motivated by the future DWL instrument in space, applies the shallow-water dynamical framework and the perfect model assumption for carrying out single-observation and OSSE experiments to isolate the impact of the new wind information on the analysis of equatorial waves that are coupled to convection. Based on broad observational evidence as reported in the literature (Wheeler and Kiladis 1999, and references therein), simulated wave fields contain the Kelvin, ER, MRG, and n = 1, 2 WEIG waves; the same dynamical structures are utilized in building the background error covariance matrix for the multivariate analysis. The information content for different datasets is estimated by 3DVAR and 4DVAR, including also experiments with unreliable background covariance matrices.

Results from the single-observation experiments illustrate that the distortion of the assimilation increments due to incomplete wind field information is not limited to the wind field but also affects the balanced increments in the height field. Second, balance relationships and approximation of the background error covariance matrix reduce the information content of the LOS winds more than that of the full wind field data.

Nevertheless, measuring the LOS wind component is better than observing any particular wind component in the Tropics, provided the background error covariance matrix for the assimilation is reliable to a reasonable extent. Moreover, a faithful 𝗕 matrix and a combined use of height and LOS wind data in our idealized system could make the 4DVAR assimilation unnecessary.

However, ideal conditions are never met in reality. Moreover, in the tropical region, due to its particular balance (or rather lack of balance) conditions, situations where an average mix of wave components, built into the 𝗕 matrix, may produce erroneous analysis increments, are more likely to occur than in midlatitudes. This has been illustrated by the example of an ER wave.

Intercomparison of analyses with different data types shows that LOS wind measurements, such as the ones provided by the ADM, have the capability of being a valuable complement to the existing satellite mass field measurements. Although incomplete, the new wind information has the potential to reduce the uncertainty in the tropical dynamical fields, if it is used together with the height data, and in this way to contribute towards the improvement of NWP and climate modeling in the Tropics.

Acknowledgments

The author would like to thank Erland Källén and Nils Gustafsson for fruitful discussions during the course of this study, for carefully reading the manuscript, and for their relevant suggestions. Special thanks to Erik Andersson and Carla Cardinali of ECMWF for the valuable comments on the earlier version of the manuscript. The author is also grateful to David Tan of ECMWF who provided the program for calculation of satellite orbits and to Branko Grisogono for reading the manuscript. The constructive comments of the two anonymous referees helped to improve the quality of the revised version of the paper.

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Fig. 1.
Fig. 1.

ADM orbit coverage of the model domain during a 48-h period. The orbit simulation starts at the equator and longitude zero at time zero, with the satellite moving southward. Successive locations of the measurements are shown by circles, which increase in size as the time progresses. Vectors are aligned along the instrument's line of sight

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Fig. 2.
Fig. 2.

Tropical wave field (u, υ, h) representing the “nature run” for the assimilation experiments. Contours are used for the perturbation height, with a contour interval 1 m, starting from ±1 m. Thick lines represent positive values and thin lines negative values. The zero contour is omitted. Wind vectors are shown in every second grid point in the meridional and every third point in the zonal direction

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Fig. 3.
Fig. 3.

Analysis increments for (a) a single height observation, (b) a zonal wind observation, and (c) a meridional wind observation at the equator. The height observation is 4 m higher than the background, while the zonal wind and the meridional wind observations have amplitude 4 m s−1. Observational errors are taken equal to the errors in the background field. Presented are the resulting assimilation increments for the height field and the wind vectors. Isoline spacing is every 0.4 m in (a) and (b), starting from ±0.4 m, and every 0.1 m in (c), starting from ±0.1 m. Thick lines represent positive values and thin lines negative values. The zero contour is omitted. Wind vectors are shown as in Fig. 2. Only a part of the computational model domain is shown

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Fig. 4.
Fig. 4.

Analysis increments for an LOS observation of the zonal wind of Fig. 3b, seen with the azimuth angle (a) 30° and (b) 75°, and for an LOS component of a meridional wind of Fig. 3c with azimuth angles equal to (c) 30° and (d) 75°. Isoline spacing is everywhere 0.4 m, starting from ±0.4 m. The observational error is the same constant in all cases

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Fig. 5.
Fig. 5.

Difference between the amplitudes of analysis increment at the observation point of full wind information and an LOS observation of (a) a (true) zonal wind and (b) a (true) meridional wind. Differences are scaled with the magnitude of increment in the case of full wind information. Results are shown as a function of the azimuth angle, between 0° and 90°, with a 5° spacing. The magnitude of observation is 4 m s−1, and the error is the same (0.3 m s−1) for all azimuth angles

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Fig. 6.
Fig. 6.

Analysis increments for a single southwesterly wind observation of magnitude 4 m s−1 and the error σo = 0.32 m s−1. (left) Wind vectors and the height field increments, increments in (middle) zonal and (right) meridional wind fields. Isolines have spacing 0.25 m for the height field and 0.25 m s−1 for the wind components. Results for the case (top) when the full wind information is available (u, υ components), and (bottom) when only the LOS wind information is available, with a viewing azimuth of 45°. Percentages in bottom row indicate the relative change of the amplitude of increments at the observation point with respect to the top

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Fig. 7.
Fig. 7.

The analytical n = 1 ER wave solution used for the assimilation experiments. A zonal wavenumber is k = 3, which corresponds to a global value of k = 6. Shading represents kinetic energy and dashed contours potential energy, with a 40-unit interval between successive levels. Contouring and wind vectors are the same as in Fig. 2

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Fig. 8.
Fig. 8.

Rmse's of the (a) height field, (b) zonal wind, and (c) meridional wind components, as a function of the number of observations used in 3DVAR of the ER wave. Values are scaled by the rmse of the background field. Lines with markers correspond to standard 3DVAR solutions with height observations (circles), wind observations (squares), both wind and height (asterisks), height and u component (asteriks), and height and υ wind component (diamonds). Lines without markers are the solution of assimilation of LOS wind observations for azimuths varying between 0° (equivalent to υ wind component) and 90° (equivalent to u wind component). Line thickness increases with the azimuth

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Fig. 9.
Fig. 9.

As in Fig. 7 but from the 3DVAR assimilation experiment with (a) 1000 randomly distributed LOS wind observations and (b) 1000 height observations. (a) Azimuth angle is 80°, that is, 10° off the zonal axis

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Fig. 10.
Fig. 10.

As in Fig. 7 but from the 3DVAR experiment using ADM setup for observations. (a) Height and LOS winds; (b) height and the meridional wind component

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Fig. 11.
Fig. 11.

Rmse's of the (a) height field, (b) zonal wind, and (c) meridional wind components for various 3DVAR and 4DVAR experiments. Values are scaled by the rmse of the background field. Experiments with different observations are marked by symbols: height data (h), zonal wind (u), meridional wind (υ), ADM LOS winds (los), ADM LOS winds and wind components perpendicular to LOS (los2), height and zonal wind (hu), height and meridional wind (), height and LOS winds (hlos), height, ADM LOS winds, and component perpendicular to LOS (hlos2). 3DVAR solutions are shown by circles, while 4DVAR results are shown by filled squares

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Fig. 12.
Fig. 12.

As in Fig. 11 but for the multiple wave experiment. 3DVAR solutions for the experiment with two satellites are shown by asterisks

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Fig. 13.
Fig. 13.

Detail of (a) the truth field for the assimilation experiments and analysis solutions for assimilation of (b) height observations in 3DVAR, (c) height observations in 4DVAR, (d) LOS wind observations in 4DVAR, (e) both height and LOS wind observations in 4DVAR, and (f) meridional winds in 4DVAR. Isolines and spacing are as in Fig. 2

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Fig. 14.
Fig. 14.

As in Fig. 11 but for the multiple wave experiment with an imperfect background error covariance matrix, such that the weight given to WEIG is increased at the expense of Kelvin and MRG waves. Results from Fig. 12 are also shown (gray symbols) for the purpose of comparison

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Fig. 15.
Fig. 15.

As in Fig. 14 but the background error covariance matrix includes only ER and WEIG waves. Notice that the horizontal axis is not the same as in Fig. 14

Citation: Journal of the Atmospheric Sciences 61, 15; 10.1175/1520-0469(2004)061<1877:AOEWBL>2.0.CO;2

Save
  • Atlas, R., 1997: Atmospheric observations and experiments to assess their usefulness in data assimilation. J. Meteor. Soc. Japan, 75 , 111130.

    • Search Google Scholar
    • Export Citation
  • Baker, W., and Coauthors, 1995: Lidar-measured winds from space: A key component for weather and climate prediction. Bull. Amer. Meteor. Soc, 76 , 869888.

    • Search Google Scholar
    • Export Citation
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  • Fig. 1.

    ADM orbit coverage of the model domain during a 48-h period. The orbit simulation starts at the equator and longitude zero at time zero, with the satellite moving southward. Successive locations of the measurements are shown by circles, which increase in size as the time progresses. Vectors are aligned along the instrument's line of sight

  • Fig. 2.

    Tropical wave field (u, υ, h) representing the “nature run” for the assimilation experiments. Contours are used for the perturbation height, with a contour interval 1 m, starting from ±1 m. Thick lines represent positive values and thin lines negative values. The zero contour is omitted. Wind vectors are shown in every second grid point in the meridional and every third point in the zonal direction

  • Fig. 3.

    Analysis increments for (a) a single height observation, (b) a zonal wind observation, and (c) a meridional wind observation at the equator. The height observation is 4 m higher than the background, while the zonal wind and the meridional wind observations have amplitude 4 m s−1. Observational errors are taken equal to the errors in the background field. Presented are the resulting assimilation increments for the height field and the wind vectors. Isoline spacing is every 0.4 m in (a) and (b), starting from ±0.4 m, and every 0.1 m in (c), starting from ±0.1 m. Thick lines represent positive values and thin lines negative values. The zero contour is omitted. Wind vectors are shown as in Fig. 2. Only a part of the computational model domain is shown

  • Fig. 4.

    Analysis increments for an LOS observation of the zonal wind of Fig. 3b, seen with the azimuth angle (a) 30° and (b) 75°, and for an LOS component of a meridional wind of Fig. 3c with azimuth angles equal to (c) 30° and (d) 75°. Isoline spacing is everywhere 0.4 m, starting from ±0.4 m. The observational error is the same constant in all cases

  • Fig. 5.

    Difference between the amplitudes of analysis increment at the observation point of full wind information and an LOS observation of (a) a (true) zonal wind and (b) a (true) meridional wind. Differences are scaled with the magnitude of increment in the case of full wind information. Results are shown as a function of the azimuth angle, between 0° and 90°, with a 5° spacing. The magnitude of observation is 4 m s−1, and the error is the same (0.3 m s−1) for all azimuth angles

  • Fig. 6.

    Analysis increments for a single southwesterly wind observation of magnitude 4 m s−1 and the error σo = 0.32 m s−1. (left) Wind vectors and the height field increments, increments in (middle) zonal and (right) meridional wind fields. Isolines have spacing 0.25 m for the height field and 0.25 m s−1 for the wind components. Results for the case (top) when the full wind information is available (u, υ components), and (bottom) when only the LOS wind information is available, with a viewing azimuth of 45°. Percentages in bottom row indicate the relative change of the amplitude of increments at the observation point with respect to the top

  • Fig. 7.

    The analytical n = 1 ER wave solution used for the assimilation experiments. A zonal wavenumber is k = 3, which corresponds to a global value of k = 6. Shading represents kinetic energy and dashed contours potential energy, with a 40-unit interval between successive levels. Contouring and wind vectors are the same as in Fig. 2

  • Fig. 8.

    Rmse's of the (a) height field, (b) zonal wind, and (c) meridional wind components, as a function of the number of observations used in 3DVAR of the ER wave. Values are scaled by the rmse of the background field. Lines with markers correspond to standard 3DVAR solutions with height observations (circles), wind observations (squares), both wind and height (asterisks), height and u component (asteriks), and height and υ wind component (diamonds). Lines without markers are the solution of assimilation of LOS wind observations for azimuths varying between 0° (equivalent to υ wind component) and 90° (equivalent to u wind component). Line thickness increases with the azimuth

  • Fig. 9.

    As in Fig. 7 but from the 3DVAR assimilation experiment with (a) 1000 randomly distributed LOS wind observations and (b) 1000 height observations. (a) Azimuth angle is 80°, that is, 10° off the zonal axis

  • Fig. 10.

    As in Fig. 7 but from the 3DVAR experiment using ADM setup for observations. (a) Height and LOS winds; (b) height and the meridional wind component

  • Fig. 11.

    Rmse's of the (a) height field, (b) zonal wind, and (c) meridional wind components for various 3DVAR and 4DVAR experiments. Values are scaled by the rmse of the background field. Experiments with different observations are marked by symbols: height data (h), zonal wind (u), meridional wind (υ), ADM LOS winds (los), ADM LOS winds and wind components perpendicular to LOS (los2), height and zonal wind (hu), height and meridional wind (), height and LOS winds (hlos), height, ADM LOS winds, and component perpendicular to LOS (hlos2). 3DVAR solutions are shown by circles, while 4DVAR results are shown by filled squares

  • Fig. 12.

    As in Fig. 11 but for the multiple wave experiment. 3DVAR solutions for the experiment with two satellites are shown by asterisks

  • Fig. 13.

    Detail of (a) the truth field for the assimilation experiments and analysis solutions for assimilation of (b) height observations in 3DVAR, (c) height observations in 4DVAR, (d) LOS wind observations in 4DVAR, (e) both height and LOS wind observations in 4DVAR, and (f) meridional winds in 4DVAR. Isolines and spacing are as in Fig. 2

  • Fig. 14.

    As in Fig. 11 but for the multiple wave experiment with an imperfect background error covariance matrix, such that the weight given to WEIG is increased at the expense of Kelvin and MRG waves. Results from Fig. 12 are also shown (gray symbols) for the purpose of comparison

  • Fig. 15.

    As in Fig. 14 but the background error covariance matrix includes only ER and WEIG waves. Notice that the horizontal axis is not the same as in Fig. 14

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