1. Introduction

The statistical analysis of innovation (observation minus forecast) vectors has been widely used for estimating observation and forecast error covariances in large-scale data assimilation (Gandin 1965; Rutherford 1972; Hollingsworth and Lönnberg 1986; Lönnberg and Hollingsworth 1986; Thiebaux et al. 1986; Bartello and Mitchell 1992). The method was recently refined by Xu et al. (2001, henceforth referred to as Part I) for the analysis of height innovation vectors and by Xu and Wei (2001, henceforth referred to as Part II) for the analysis of wind innovation vectors. As a follow-up study, this paper applies the method to estimate the cross covariance and cross correlation between the height and wind forecast errors and to evaluate the related geostrophy (that can be measured statistically by a single parameter introduced in the next section). As in Part I and Part II, the height and wind innovation data were collected from the Navy Operational Global Atmospheric Prediction System (NOGAPS; Hogan and Rosmond 1991) over North America (between 25°–65°N and 60°–130°W) during the period from 1 March to 31 May 1999. The data quality control and assimilation system were briefly described in section 2 of Part I. The basic assumptions and formulations required by the wind–height correlation analysis are described in the next section. The method is illustrated for the single-level analysis in section 3 and then extended to the multilevel analysis in section 4. Conclusions follow in section 5.

2. Basic assumptions and forecast error geostrophy

Following Lönnberg and Hollingsworth (1986, henceforth referred to as LH), it is convenient to introduce the pseudostreamfunction, ψ, and pseudovelocity potential, χ, defined in pressure coordinates (x, y, p) by

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where f_o = 0.91 × 10⁻⁴ s⁻¹ is the Coriolis parameter at the central latitude of the domain, (u′, υ′) is the vector wind forecast error as in Part II, and (u′, υ′)f/f_o is the pseudovelocity. Clearly, ψ and χ are not exactly the same as those defined in (2.1) of Part II. To measure how well the geostrophy is satisfied statistically by the forecast error fields, it is also convenient to introduce the following nondimensional parameter:

aψϕ²σ_ψσ_ϕ⁻¹(2.2)

where 〈( · )〉 denotes the ensemble mean of ( · ) and will be computed as the time mean under the ergodicity assumption, ϕ = gz′/f_o is the geopotential forecast error scaled by f_o, z′ is the height forecast error, σ²_ψ = 〈ψ²〉 is the variance of ψ, and σ²_ϕ = 〈ϕ²〉 is the variance of ϕ. Clearly, if a = 0, then the forecast error fields satisfy the geostrophy exactly. The smaller the value of a, the better the geostrophy is satisfied statistically by the forecast error fields.

The random fields of ϕ, ψ, and χ are assumed to be jointly homogeneous and isotropic in the horizontal and so should be the pseudovelocity. This assumption is consistent with that made for the height forecast error in Part I, but they are slightly different from that made for the velocity (u′, υ′) itself in Part II. With the above assumption, the cross covariances between ϕ and (ψ, χ) are functions of only three independent variables (r, p_m, p_n), where r = [(x_j − x_i)² + (y_j − y_i)²]^1/2 is the horizontal distance between the two points, (x_i, y_i) and (x_j, y_j), along the great circle of the earth's surface (see appendix A of Daley and Barker 2000). Their general forms are

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For simplicity the covariances can sometimes be written as functions of r only as long as their implicit dependence on (p_m, p_n) is well understood.

The cross-covariance functions in (2.3a) and (2.3b) cannot be directly estimated from the data, but they can be related to the following cross covariances between z′ and the tangential and radial components of the pseudovelocity as follows [(5.3.7) of Daley (1991)]:

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where

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with l′ denoting the tangential component and t′ the radial component of the pseudovelocity with respect to the r direction from point (x_i, y_i) to point (x_j, y_j). It is easy to verify that C_ϕχ = C_χϕ, C_ϕψ = C_ψϕ, C_zt = −C_tz, and C_zl = −C_lz.

The Hankel transformations of C_ϕψ and C_ϕχ give the following cross spectra:

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The inverse transformations are

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Substituting (2.6a) and (2.6b) into (2.4a) and (2.4b) gives

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where

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and J₁( · ) = J^′₀( · ) ≡ ∂J₀( · )/∂( · ) is used. The inverses of (2.9a) and (2.9b) are given by

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Truncated discrete forms of (2.10a) and (2.10b) will be used to estimate C_zt(r) and C_zl(r) over a finite radial range in the later sections.

In addition to the above basic assumption for the forecast errors, the (radiosonde) observation errors are assumed to be independent of the forecast errors and to be uncorrelated between different stations and between different variables. With these assumptions, the forecast error cross covariances in (2.5a) and (2.5b) can be estimated directly by innovation cross covariances. The detailed analysis and results are shown in the following sections.

3. Single-level analysis

Over the finite radial range of r ≤ D (=3000 km), (2.10a) and (2.10b) reduce to the following truncated spectral expansions:

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where the summation Σ₁ is from i = 1 to i = M. The wavenumbers k_i(i = 1, 2, … , M) are zeros of J₁(kD) or, equivalently, zeros of J^′₀(kD). Since J₁(k₀r) = 0 for k₀ = 0, there is no zero-order spectral component (i = 0) although the summation can be formally extended to Σ₀ (from i = 0 to i = M) as in (4.1b) of Part II. Since the observation errors are assumed to be uncorrelated between different variables and also independent of the forecast errors, the forecast error cross covariances C_zt(r) and C_zl(r) should be the only nonzero part of their associated innovation cross covariances. Thus, the spectral coefficients S_zt(k_i) and S_zl(k_i) can be estimated by least squares fitting of (3.1) directly to the binned innovation cross covariances.

Note that (p_m, p_n) are implicit for the function forms in (3.1a) and (3.1b). For the single-level analysis considered in this section, we have p_m = p_n. In this case, the spectral coefficients S_zt(k_i) and S_zl(k_i) are constrained by the following conditions:

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Here, the spectral coefficients S_zz(k_i) are given by least squares fitting of (5.1) in Part I. The spectral coefficients S_ll(k_i) and S_tt(k_i) can be obtained by least squares fitting of the same spectral expansions as in (4.1a) and (4.1b) of Part II but applied to the pseudovelocity (instead of the original velocity). The condition in (3.2a) is derived from the semidefinite positiveness of the autocovariances: 〈(ζ_i + τ_i) (ζ_j + τ_j)〉 and 〈(ζ_i − τ_i) (ζ_j − τ_j)〉, where ζ = βz′ and τ = β⁻¹t′ for any positive number β. The condition in (3.2b) is derived from the semidefinite positiveness of the autocovariances: 〈(ζ_i + λ_i) (ζ_j + λ_j)〉 and 〈(ζ_i − λ_i) (ζ_j − λ_j)〉 where ζ = βz′ and λ_i = β⁻¹l′ for any positive number β. The detailed derivations follow the same steps as in (6.1) and (6.2) of Part I.

The estimated cross-covariance functions are plotted for p_m = p_n = 850, 500, and 200 mb in Fig. 1, where the binned innovation covariances are shown by symbols. For p_m = p_n = 850 mb, the structure of C_zt(r) in Fig. 1 is similar to that in Fig. 16 of LH, and the structure of C_zl(r) in Fig. 2 is flatter than that in Fig. 17 of LH. As shown by the cross spectra plotted in Figs. 2a and 2b, S_zl(k_i) is also very flat and negligibly small in comparison with S_zt(k_i). Thus, according to (2.8b), the cross-covariance function S_ϕχ(k_i) and thus C_ϕχ(r) = Σ₁ S_ϕχ(k_i)J₀(k_ir) are very small. Because C_ϕχ(r) is small and irrelevant to the geostrophy, we will only examine the cross covariance between ϕ and ψ; that is, C_ϕψ(r) = Σ₁ S_ϕψ(k_i)J₀(k_ir), where S_ϕψ(k_i) is obtained from S_zt(k_i) using (2.8a).

Since the large-scale component (i = 0) is not contained in the above spectral expansions, C_ϕψ(r) is the synoptic-scale part of the cross covariance between ϕ and ψ. Similarly, we denote by C_ϕϕ = (g/f_o)²C^S_zz the synoptic-scale part of the height forecast error covariance [see (6.7a) of Part I], and by S_ψψ = S_tt(k_i)/k²_i the synoptic-scale part of the pseudostreamfunction forecast error covariance [see (2.11a) and (4.5) of Part II]. The perfect geostrophy corresponds to a = 0 in (2.2). One can show that this condition (a = 0) is equivalent to C_ϕψ(r) = C_ϕϕ(r) = C_ψψ(r) (see the appendix). Thus, by comparing C_ϕψ(r) with C_ϕϕ(r) and C_ψψ(r), we can see how closely the geostrophy is satisfied by the forecast error fields. The three covariance functions are compared in Figs. 3a–c at the vertical levels of p_m = p_n = 850, 500, and 200 mb. As shown in Fig. 3a, C_ϕψ (solid) is flatter than C_ϕϕ (dashed) and C_ϕϕ is flatter than C_ψψ (dotted), indicating that the geostrophy is not well satisfied and the variance of the error field ψ is larger than that of ϕ at 850 mb. In Fig. 3b, C_ϕψ nearly coincides with C_ϕϕ and C_ψψ, so the geostrophy is well satisfied at 500 mb. In Fig. 3c, C_ϕψ is very close to C_ϕϕ but much flatter than C_ψψ, so the geostrophy is not well satisfied and the variance of the error field ψ is much larger than that of ϕ at 200 mb.

Associated with the above three covariance functions, three correlation functions can be defined by R_ϕψ(r) ≡ C_ϕψ(r)(σ_ϕσ_ψ)⁻¹, R_ϕϕ(r) ≡ C_ϕϕ(r)σ⁻²_ϕ, and R_ψψ(r) ≡ C_ψψ(r)σ⁻²_ψ, where σ_ϕ and σ_ψ are as defined in (2.2). The normalized cross-correlation function, defined by R_ϕψ(r)/R_ϕψ(0), is compared with the two autocorrelation functions R_ϕϕ(r) and R_ψψ(r) in Figs. 4a–c at the vertical levels of p_m = p_n = 850, 500, and 200 mb. As shown, the three functions are quite similar especially for p_m = p_n = 200 mb (Fig. 4c). For p_m = p_n = 850 and 500 mb (Figs. 4a,b), R_ϕψ(r)/R_ϕψ(0) is not very close to R_ϕϕ(r) but nearly coincides with R_ψψ(r). Since these correlation functions have similar shapes, how well the geostrophy is satisfied by the forecast error fields can be largely described by the relative amplitudes of the three covariance functions or, equivalently, by the following two parameters:

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where the dependence on p_m (=p_n) is implicit. Here, μ is the correlation coefficient as in LH and ν may be called the relative amplitude between the error fields ψ and ϕ. It is easy to see that

aνν⁻¹μ,(3.4)

where a is defined in (2.2). Note that ν + ν⁻¹ ≥ 2 and μ ≤ 1, so a = 0 if and only if μ = 1 and ν = 1. This means that a = 0 is equivalent to μ = ν = 1, which is consistent with the result in the appendix. If μ = 1 but ν ≠ 1, then ψ and ϕ are fully correlated but do not satisfy perfect geostrophy.

The computed parameter values of μ, ν, and a are plotted as functions of p_m (=p_n) in Fig. 5. As shown, both μ and ν are close to unity and, consequently, a is small in the middle troposphere. Clearly, the geostrophy is well satisfied by the forecast error fields in the middle troposphere and this is consistent with the weak divergence of the wind forecast error in the middle troposphere (see Fig. 6 of Part II). The geostrophy is not well satisfied in the boundary layer as indicated by the jump of a from 0.34 at 700 mb to 1.06 at 850 mb in Fig. 5. The geostrophy is also not well satisfied around the tropopause as shown by the local maximum of a (=0.88) at 200 mb (where μ = 0.65 and ν = 1.52). Around the tropopause, air masses come from two different origins (troposphere and stratosphere) and the divergent part of the wind forecast error reaches its maximal intensity (see Fig. 6 of Part II). This may explain why the geostrophy is not well satisfied around the tropopause in the pressure coordinates.

Above 100 mb, the spectral expansion of C_ϕϕ(r) is severely truncated under the semidefinite positive constraint [see (3.6) of Part I] and consequently the spectral expansion of C_ϕψ(r) is also severely truncated under the constraint (3.2). The truncated high-wavenumber components are negative for C_ϕϕ(r) but are not so for C_ϕψ(r). Thus, σ_ϕ could be overestimated and μ could be underestimated above 100 mb. The spectral expansion of C_ψψ(r), however, is not severely truncated under the semidefinite positive constraint [see (2.11) and (4.2) of Part II], so σ_ψ is unlikely to be overestimated, at least, not as much as σ_ϕ. This implies that ν = σ_ψ/σ_ϕ could be underestimated above 100 mb. Thus, the decreases of μ and ν and associated rapid increase of a above 100 mb in Fig. 5 could be spurious features.

The gross structures of the correlation functions—R_ϕψ(r), R_ϕϕ(r), and R_ψψ(r)—can be measured by their associated horizontal length scales. As in Hollingsworth and Lönnberg (1986) [or see (6.8) of Part I], these length scales are defined by

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where the summation Σ₁ is from i = 1 to i = M for the synoptic-scale forecast error components resolved by the current analysis. Here, (2.11a) of Part II is used in the derivation of (3.5c). Clearly, L_ϕϕ is essentially the same as L_z defined in (6.8) of Part I. The computed length scales are plotted as functions of p_m (=p_n) in Fig. 6. Above 50 mb, the spectral expansion of C_ϕϕ(r) is truncated to only one component under the semidefinite positive constraint [see (3.6) of Part I] and so does the spectral expansion of C_ϕψ(r) under the constraint (3.2). Because of this, L_ψϕ and L_ϕϕ cannot be reliably estimated and thus are not plotted above 50 mb. As shown in Fig. 6, the ranges of these length scales and their general increases with height are similar to those (from 925 to 100 mb) in Fig. 18b of LH. The detailed structures and values, however, are different. For example, in the middle troposphere, we have L_ϕϕ > L_ψϕ > L_ψψ in Fig. 6, while LH obtained L_ψψ > L_ψϕ > L_ϕϕ in their Fig. 18b, although the three length scales are close to each other in both figures.

4. Multilevel analysis

When p_m ≠ p_n, the analysis can be done similarly to the above single-level analysis. However, since p_m ≠ p_n, the spectral coefficients S_zt(k_i) = S_zt(k_i, p_m, p_n) and S_zl(k_i) = S_zl(k_i, p_m, p_n) in (3.1a) and (3.1b) are not constrained by (3.2a) and (3.2b). Instead, they are constrained by

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These conditions can be derived similarly to that in (3.2a) and (3.2b) by using the identities C_zt(r, p_m, p_n) = C_tz(r, p_n, p_m) and C_zl(r, p_m, p_n) = C_lz(r, p_n, p_m). The details are omitted.

The computed correlation function R_ψϕ(r, p_m, p_n) is plotted in Fig. 7a for fixed p_m = 850 mb (vertical level of ψ) and different p_n (vertical levels of ϕ). The plotted correlation function has a local maximum of 0.52 at r = 0 and p_n = p_m (=850 mb). The ψ field at 850 mb is not only maximally correlated to the ϕ field at 850 mb but also positively correlated to the ϕ fields between 950 and 300 mb, although the correlation decreases gradually as the vertical level p_n is shifted away from 850 mb. Note that R_ϕψ(r, p_n, p_m) = R_ψϕ(r, p_m, p_n) ≠ R_ψϕ(r, p_n, p_m) = R_ϕψ(r, p_m, p_n). The latter is plotted in Fig. 7b for fixed p_m = 850 mb (vertical level of ϕ) and different p_n (vertical levels of ψ). As shown, the correlation function in Fig. 7b has a local maximum of 0.68 at r = 0 and p_n = 700 mb, indicating that the ϕ field at 850 mb is correlated more closely to the ψ field at 700 mb than the ψ field of the same (850 mb) level. This is different from that in Fig. 7a. As shown in Fig. 5, the single-level geostrophic coupling, measured by μ = R_ϕψ(0, p_m, p_n) = R_ψϕ(r, p_n, p_m) with p_n = p_m, drops sharply from 0.84 to 0.52 as the vertical level (p_n = p_m) decreases toward the boundary layer from 700 to 850 mb. In comparison with this, the multilevel geostrophic coupling, measured by R_ϕψ(0, p_m, p_n) = R_ψϕ(0, p_n, p_m), drops more sharply from 0.84 (see Fig. 5) to 0.43 (see Fig. 7a) as the vertical level of ψ decreases from 700 to 850 mb while the vertical level of ϕ is fixed at 700 mb, but it drops less sharply from 0.84 (see Fig. 5) to 0.68 (see Fig. 7b) as the vertical level of ϕ decreases from 700 to 850 mb while the vertical level of ψ is fixed at 700 mb.

Plotted in Figs. 8a and 8b are the correlation functions R_ψϕ(r, p_m, p_n) and R_ϕψ(r, p_m, p_n) for fixed p_m = 500 mb and different p_n. As shown, for 0 ≤ r ≤ 950 km, R_ψϕ(r, p_m, p_n) in Fig. 8a is positive for all the vertical levels of p_n, but R_ϕψ(r, p_m, p_n) in Fig. 8b is positive mainly for the vertical levels of p_n below 200 mb. As the vertical level p_n is shifted away from 500 mb, the vertical profiles of R_ϕψ(r, p_m, p_n) in Fig. 8b drop more rapidly than those in Fig. 8a, and their values decrease to virtually zero as the vertical level of p_n (for the ϕ field) moves up to 200 mb. The major difference between Figs. 8a and 8b is that the ψ field at 500 mb is positively correlated with the ϕ fields at all the vertical levels but the ϕ field at 500 mb is positively correlated with the ψ fields mainly at the vertical levels within the troposphere (below 200 mb).

Figures 9a and 9b are the correlation functions R_ψϕ(r, p_m, p_n) and R_ϕψ(r, p_m, p_n) for fixed p_n = 200 mb. As shown, for 0 ≤ r ≤ 950 km, R_ψϕ(r, p_m, p_n) in Fig. 9a is positive only when p_n (the vertical level of ϕ) is between 500 and 50 mb, while R_ϕψ(r, p_m, p_n) in Fig. 9b is positive as long as p_n (the vertical level of ψ) is above 850 mb. This is the major difference between Figs. 9a and 9b. From the symmetry of R_ψϕ(r, p_m, p_n) = R_ϕψ(r, p_n, p_m), it is easy to see that the difference between Figs. 9a and 9b is consistent with the difference between Figs. 8a and 8b.

5. Conclusions

The method of statistical analysis of wind innovation (observation minus forecast) vectors is applied to the height and wind innovation data collected over North America for a 3-month period from NOGAPS to estimate the height–wind forecast error correlation and to evaluate the related geostrophy. The method and related analysis are extended upon the work of LH in three aspects: (i) new constraints are derived for the spectral representations of height–wind forecast error covariance functions [see (3.2) and (4.1)], (ii) the single-level analysis of cross covariance is extended to multilevel analysis [see section 4], and (iii) a new parameter a [see (2.2)] is introduced to measure the geostrophy. Zero value of this parameter (a = 0) defines the perfect geostrophy and this condition is shown to be the same as the equivalence condition between the cross covariance and two autocovariances of the pseudostreamfunction and geopotential forecast error [see the appendix]. It is also equivalent to μ = ν = 1, where μ is the correlation coefficient and ν is the standard deviation ratio between the pseudostreamfunction and geopotential forecast error fields [see (3.3) and (3.4)]. The smallness of a measures the closeness to perfect geostrophy and requires both μ and ν be close to unity. Conventionally, however, only μ is used for the geostrophic coupling in the multivariate optimal interpolation (Bergman 1979; Lorenc 1981; Daley 1991).

The single-level analysis performed in this paper shows that the height forecast error is correlated to the tangential wind forecast error but not (at least, not obviously) to the radial wind forecast error. This implies that the geopotential forecast error is correlated to the pseudostreamfunction but not to the pseudovelocity potential [defined in (2.1)]. This result is consistent with LH. It indicates that the geostrophy is satisfied, to a certain degree, by the forecast error fields. The degree to which the geostrophy is satisfied is examined quantitatively and the results can be summarized as follows (also see Figs. 5 and 6).

1. The geostrophy is well satisfied by the forecast error fields in the middle troposphere, and this is consistent with the weak divergence of the wind forecast error in the middle troposphere.

2. The geostrophy is not well satisfied in the boundary layer where the eddy diffusivity is strong.

3. The geostrophy is not well satisfied around the tropopause where air masses come from troposphere and stratosphere, and this is consistent with the maximal divergence of the wind forecast error at the tropopause (see Fig. 6 of Part II).

4. Above 100 mb, the geostrophy is not well satisfied by the synoptic-scale forecast error components resolved by the current analysis, but it may be well satisfied by the large-scale forecast error components not included in the current analysis (due to the limited data coverage).

5. The horizontal length scale of the cross correlation between the pseudostreamfunction and geopotential forecast error is found to be close to and mostly between the length scales of the two autocorrelation functions. The ranges of these length scales and their general increases with height are similar to those in LH, but the detailed structures are quite different.

When the National Meteorological Center (NMC) method (Parrish and Derber 1992) was extended and used to estimate the European Centre for Medium-Range Weather Forecasts (ECMWF) global model forecast error correlation structures, Derber and Bouttier (1999) found that a part of the divergent wind error can be explained by the rotational wind error in the middle latitudes near the surface and near the jet streams (at small scales) around the tropopause. A similar correlation between the vorticity and divergence forecast errors was also detected by Berre (2000) when the NMC method was further extended and applied to a limited-area model. Thus, the correlation between the pseudovelocity potential and geopotential forecast error may not be negligibly small. This was also suggested by Polavarapu (1995), based on divergent wind analyses in the marine boundary in the vicinity of cyclones. Nevertheless, the correlation between the pseudovelocity potential and geopotential forecast error appears to be very subtle and cannot be detected by the analysis performed in this paper due to the limited resolution of the radiosonde data. Apparently, the radiosonde and associated innovation data used in this paper are too coarse to detect the aforementioned small-scale correlation between the vorticity and divergence forecast errors near the jet streams around the tropopause and the radiosonde standard vertical levels are too coarse to detect the aforementioned correlation near the surface.

The multilevel analysis shows that the pseudostreamfunction and geopotential forecast error fields are correlated between different vertical levels, although the correlation decreases gradually as the two vertical levels are shifted away from each other. It is found that the pseudostreamfunction at the middle troposphere (500 mb) is positively correlated with the geopotential forecast error fields at all the vertical levels, but the geopotential forecast error field at the middle troposphere is positively correlated with the pseudostreamfunctions only at the vertical levels below 200 mb. The pseudostreamfunction around the tropopause (200 mb) is positively correlated with the geopotential forecast error fields between 500 and 50 mb, while the geopotential forecast error field around the tropopause is positively correlated with the pseudostreamfunctions above 850 mb. These features are consistent with the cross-correlation symmetry [i.e., R_ψϕ(r, p_m, p_n) = R_ϕψ(r, p_n, p_m)]. The results obtained from the multilevel analysis suggest that the geostrophic coupling between different vertical levels are very significant and should not be neglected. Multilevel geostrophic coupling has complex vertical structures. When these vertical structures are described by the vertical normal modes, the geostrophic coupling between different vertical modes may remain to be significant. Thus, considering multilevel geostrophic coupling or multimode geostrophic coupling may improve the conventional multivariate optimal interpolation and three-dimensional variational data assimilation.