a. Hypothesis testing

Our method for testing the Taylor hypothesis is developed based on the asymptotic joint normality of sample space–time covariance estimators derived by Li et al. (2008). Assume Z(x, t) is a strictly stationary space–time random field with covariance function C(r, τ) = Cov[Z(x, t), Z(x + r, t + τ)], where r and τ denote an arbitrary spatial lag and time lag, respectively. Let Λ be a set of space–time lags such as Λ = [(r₁, τ₁),…, (r_m, τ_m)], where m denotes the number of its elements. Let Ĉ(r, τ) denote an estimator of C(r, τ). For simplicity, we choose Ĉ(r, τ) as the moment estimator defined by Ĉ(r, τ) = (1/N)Σ_xΣ_t[Z(x, t) − Z][Z(x + r, t + τ) − Z], where Z denotes the mean of Z(x, t) and N is the total number of summands. This choice of estimator works well in testing properties of the covariance function (see Li et al. 2007). Let 𝗚 = [C(r, τ),(r, τ) ∈ Λ], and let = [Ĉ(r, τ), (r, τ) ∈ Λ denote the estimator of 𝗚. Li et al. (2008) derived that the appropriately standardized and centered has an asymptotic multivariate normal distribution in a variety of space–time contexts.

b. Test with a given v

We write Taylor’s hypothesis as

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Observe that H₀ is a contrast of covariances and thus can be rewritten in the form of 𝗔𝗚 = 0, where 𝗔 is a contrast matrix of row rank q, say.

For example, if

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then

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Define

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then we have 𝗔𝗚 = 0 under the null hypothesis. Replacing C(·) with the estimator Ĉ(·) in (2), we obtain a contrast vector for testing H₀ as the estimated left-hand side of (2), C = Ĉ(0, τ) − Ĉ(vτ, 0). Apparently, C can be rewritten into the form of AĜ. We form the test statistic (TS) based on the contrasts of and obtain the distribution of TS under the null hypothesis (Li et al. 2008) as

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in distribution as N → ∞, for a matrix 𝗔 with row rank q, and an appropriate sequence of normalizing constants a_N. We follow Li et al. (2008) and estimate Σ using subsampling techniques. The choice of subblock size is described in Carlstein (1986). In terms of our precipitation data, the rich temporal replicates allow us to consider the asymptotics in time dimension and form overlapping subblocks using a moving subblock window along time. Specifically, let N be the total number of time steps, a_N = N in (6), and the optimal block length for subblocks is [2γ / (1 − γ²)]^2/3 (3N/2)^1/3 where γ can be estimated by = Ĉ(0, 1)/Ĉ(0, 0). Covariance estimates obtained from each subblock constitute the sample to estimate Σ. The statistical significance of the resulting test statistics can be assessed based on the large sample χ² distribution of the test statistic.

a. Isotropic case

To provide some physical insight into the covariance function for precipitation, we develop a simple mathematical model of an evolving precipitation field following Cahalan et al. (1982), North and Nakamoto (1989), Bell et al. (1990), and Bell and Kundu (1996). The model includes propagation (advection), damping, diffusion, and a white-noise stochastic forcing. The evolution of the dependent variable R(x, t), which represents rain rate, is given by the equation

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where v is a constant advective velocity, D is the diffusion coefficient, b is the damping rate, and f is a stochastic forcing term. The analytical solution to this model can be used directly to compute the covariance function.

The equation is solved in the spectral domain (k, ω), where k = (k_x, k_y) is the spatial wavenumber and ω is the temporal frequency. The random forcing f (x, t) is a stationary Gaussian random variable that is white in both space and time; that is,

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where angle brackets indicate the ensemble mean.

If we define the Fourier transform of R(x, t) as

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and take the Fourier transform of (7), we get

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where = f₀ is a constant that specifies the magnitude of the white-noise forcing. Solving for yields

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The variance or power spectrum density S is the magnitude of the complex solution

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where * indicates the complex conjugate. Note that the variance in (12) is positive for all (k, ω) and hence is consistent with the definition of a valid covariance function (Gneiting et al. 2007). Finally, the covariance function in the physical domain is obtained by taking the inverse Fourier transform of S:

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where r = (r_x, r_y) and τ are appropriate lags in space and time. We solve the ω integral by contour integration by first making a substitution ω′ = ω + k · v and then dropping the prime to simplify the notation. The final integral is obtained as

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We first explore the above integral analytically by expanding the complex exponential in (14). We then test the validity of TH by examining whether C(0, 0, τ) = C(u |τ|, υ|τ|, 0) for a specific v = (u, υ) and τ. Evaluating each term separately yields

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and

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The TH holds if (15) and (16) are equal, for instance,

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The Taylor “frozen-field” hypothesis requires that the turbulent structure of the field being advected (rain in our case) evolves slowly compared to the advective time scale τ. In the model given in (7), the evolution of the rain field is controlled by the diffusion operator and the damping. We investigate the covariance function in the limit of negligible diffusion (D → 0). In this limit, the evolution of the rainfall field is controlled entirely by the damping and the stochastic forcing. Taking the limit and canceling like terms, we see that

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These two expressions are identical, except for the factor of e^−b|τ| on the left-hand side; that is, the TH is not satisfied for this model because the damping changes the magnitude of the correlation field, even if the shape does not change as the correlation is advected downwind. Thus, for a model like this it is not possible to satisfy TH, and the space and time covariances will differ in proportion to the exponential of the ratio of the temporal lag τ and the damping time scale 1/b.

b. Extension to anisotropic case

The degree of anisotropy is controlled by the magnitude of the diffusion term, which in the general case has directional dependence and consists of four components:

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For simplicity, we assume that the diffusion or stretching is only along the two major axes, that is, D_xy = D_yx = 0. Thus in this case (7) can be rewritten as

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We simplify the above equation by dividing throughout by b and scale the resulting equation with the following nondimensional variables:

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Following an analysis similar to the one outlined in the previous section, we obtain the covariance function for the anisotropic case in terms of nondimensional parameters as

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where r_x′ = r_x / D_xx/b, r_y′ = r_y / D_yy/b, and τ′ = bτ. The form of the above covariance function closely resembles that of (14); not surprisingly, the TH does not hold for this case either.

a. Covariance calculations

The time series of instantaneous area-averaged reflectivity for the study region is plotted in Fig. 4. Several periods of heavier rain are apparent, and a period of little rain can be seen on 5 May. The analysis methods described above are applied to the entire 4-day period and to the three subintervals of heavier rain indicated by vertical lines and labeled periods 1 through 3 in Fig. 4. For each of the periods of analysis, the time mean for that period at each point

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is removed, and all calculations are done with reflectivity anomalies dBZ′ = dBZ − .

The time-lagged covariance between the reflectivity anomaly dBZ_i′(t) at a reference point (designated by the subscript i) and the anomaly at a test point (designated j) for a given time lag τ is computed using

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where the space lag r_ij = x_j − x_i is the vector from point i to point j, and N is the number of observations in the time series. For a given reference point i, C(r_ij, τ_n) is calculated for all test points j on the data grid within a 2° × 2° rectangular region centered on the reference point (107 × 113 grid points) and additionally for all time lags (τ_n) from −3 to +3 h. With a time step size of 15 min, this amounts to discrete time lag indices (n) between ±12.

To estimate the average covariance function for the entire domain Ĉ(r, τ) we start off by selecting 2000 random but uniformly distributed reference points throughout the entire domain. The number 2000 is somewhat arbitrary, but it accurately represents the dataset spatially. Then, for each reference point, we prescribe the 2° × 2° rectangular region around it and calculate the mean of this region. If this temporally and spatially averaged value is greater than 4 dBZ, then we treat the point as a valid reference location; otherwise it is rejected. This omits from consideration any reference points where little or no rain fell during the period of analysis. By repeating this process for all 2000 reference points, we end up with around 300 valid reference locations. Then the calculation in (24) is repeated for these 300 locations and averaged over all locations. Reference points are chosen to be at least 1° away from the boundaries of the domain to avoid edge effects.

Correlations ĉ are estimated using

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where ² denotes the estimated variance of the entire precipitation anomaly field. Error bars for Ĉ and ĉ are roughly estimated by computing the standard deviation of the individual estimates of C or c for the 300 reference points. These correlation estimates based on ĉ are then tested for the validity of the TH and are subsequently compared with those from the method described in section 3, which takes into account the correlations between the individual covariance estimates.

b. Structure of the covariance field

The mean space–time covariance function Ĉ(r, τ) and correlation ĉ(r, τ) are estimated for the entire 4-day study as described in section 5a. The spatial structure of the empirical space–time correlation ĉ estimated over all pairs that have the same spatial and temporal lag is shown in Figs. 5, 6. Figure 5 shows contours of ĉ as a function of spatial lag at τ = 0, while Fig. 6 shows ĉ for τ = 0, 15, 30, and 45 min. Figure 5 reveals that the ĉ has an approximately elliptical shape with the major axis oriented somewhat north of east. The elliptical shape (anisotropy) of ĉ indicates the mean orientation of the precipitation areas during this period on the scale shown. As in the theoretical model, the peak observed at τ = 0 (Fig. 6a) decays with time as it moves upwind (downwind) with decreasing (increasing) lag (Figs. 6b–d).

To qualitatively compare the general shape and structure of ĉ with that of the anisotropic model (20) described in section 4b, we also evaluate the model correlation function (22) numerically for different values of the parameters D, b, u, υ, r, and τ. The goal is not to exactly reproduce the observed values, but to understand the nature of correlation functions with similar shape. Figure 7 illustrates the shape of (22) for b = 0.25 min⁻¹, u = 1.6 × 10⁻⁴ deg min⁻¹, υ = 2.5 × 10⁻⁵ deg min⁻¹, D_xx = 0.20 deg² min⁻¹ and D_yy = 0.075 deg² min⁻¹ for spatial lags (r_x, r_y) that range from −1.0° to +1.0° and temporal lags τ = 0, 15, 30, 45 min, respectively. While solving (14) numerically, the limits of the integration were truncated to get a finite correlation function. The shape of the correlation function is similar to the observations. For fields that have this type of covariance structure, with a localized peak in the covariance that decays as it is advected downstream, we generally would not expect the field to satisfy the Taylor hypothesis.

In Fig. 5 the maximum correlation at τ = 0 occurs at the origin, as expected. As the magnitude of τ increases from zero in the positive or negative direction, the peak of the correlation function shifts upstream or downstream, respectively, depending on the sign of the lag; the maximum correlation values decrease with increasing lag, as shown in Fig. 6. The plus symbols in Fig. 5 represent the locations of the peak correlations at time lags of ±15, 30, 45, and 60 min. The plus signs are approximately colinear and equally spaced, indicating that the advective velocity is approximately independent of lag. The average velocity vector v can be estimated by using the vectors from the origin to the plus signs, divided by the corresponding time lag τ. The mean velocities are found to have magnitudes between 25 and 30 m s⁻¹ (∼25.6° day⁻¹) oriented ∼83° from north. This is consistent with the mean wind plotted in Fig. 3 and with the motion of precipitation features visible in an animation of the radar reflectivity maps. We use this value of v = 25.6° day⁻¹ (83° east of north) to compute ĉ(vτ, 0) for the entire 4-day period of study. Due to some variability of v with time, we use individual velocity estimates obtained from the correlation functions for each subperiod when testing the TH for the three subperiods. Note that the propagation velocity v and the principal axis of the correlation function c are not oriented in the same direction, nor is there any reason to expect them to be.

c. Testing the Taylor hypothesis

Figure 8 shows the space-lagged correlations [ĉ(vτ, 0), triangles] and time-lagged correlations [ĉ(0, τ), diamonds] as a function of time lag for the entire 4-day period plotted on a logarithmic scale. These curves show that the TH in general does not hold for the 4-day time period and hence for large space and time scales. It is important to mention here that although the curves do appear close (especially at smaller time lags), the error bars (not shown) around the mean correlations corresponding to the shortest time lag of 15 min are small and do not overlap, thereby indicating that the curves are indeed statistically different. Figure 9 shows curves of correlation plotted as a function of time lag for the precipitation model (20). The correlation curves in Fig. 9 are similar to those in Fig. 8, with the rate of decay being controlled by the magnitude of the damping term b.

In addition, we test the TH for each of the smaller time periods illustrated in Fig. 4, during which the heaviest rainfall occurs. Figure 10a demonstrates a subperiod for which the TH held up to 15 min (35 km), while Figs. 10b,c show subperiods for which it did not hold for even short space and time scales.

To assess the TH using both methods, a table containing probability values (p values) pertaining to the statistical significance of the difference between the averaged correlations, that is, ĉ(0, τ) − ĉ(vτ, 0), is constructed for each testing period at various time lags using both a standard Student’s t test and the method from section 3. These tables serve to reinforce the results illustrated in the Figs. 8, 10a–c in that the TH does not hold for such systems. Table 1 lists p values generated using the Student’s t test, while Table 2 lists those generated using the method of Li et al. (2007). The convention used here is that a p value p* ≥ 0.05 implies that the difference is not significant at the 5% level, and the TH cannot be rejected, while p* < 0.05 indicates that the difference is significant at the 5% level and the TH is rejected. The results from both statistical methods show that the TH does not hold for the full 4-day period, but it is admitted for up to 15 min (35 km) for one subperiod (period 1), as indicated by large p* values (in boldface). The TH is not rejected in only one of the 16 period-and-lag combinations tested. Although the results from the two methods agree in all cases, it is important to note that the results from the Student’s t test (Table 1) tend to overestimate the significance at a particular τ compared to the method of Li et al. (2007; Table 2). This can be attributed to the inability of the correlation method to account for the correlation between the correlation estimates or, equivalently, to overestimating the number of degrees of freedom. Using the standard NEXRAD Z–R relationship (Z = 300R^1.4), space–time correlations were also computed using rain rates to gauge whether the choice of variable affects the conclusions. The statistical significance of the results has the same pattern as in Table 1. In only one case is the TH not rejected (period 1 at 15 min), from which we conclude that the TH does hold for this period up to 15 min (35 km). In Table 1, p* values for time periods 1 and 2 have not been shown for τ = 60 min because the velocity estimates for either case resulted in computing locations (and corresponding correlation estimates) that were beyond the 2° × 2° moving window in the spatial domain. The hypothesis-testing approach (Table 2), however, used a slightly larger moving window and thus reported correlations (and hence p* values) at τ = 60 min as well.