1. Introduction

Applications of meteorological data often require information at times that do not match those at which the data are available. For example, a numerical wave model may require inputs in the form of hourly wind fields, while the numerical weather model from which it seeks outputs may only provide gridded wind fields every 6 h. Or, if data are taken from a satellite-borne scatterometer, winds are only available on swaths at intervals determined by the orbit. In such cases, some form of temporal interpolation will be required.

The standard approach is to apply linear interpolation in time. This is reasonably satisfactory if the variable concerned varies slowly compared to the sampling interval. Even in this case though, linear interpolation can smooth the variation excessively. For example, data interpolated linearly from monthly means to, say, daily values no longer have the original monthly means when reaveraged (Epstein 1991). This problem can, however, be addressed by relatively simple correction methods (Killworth 1996).

It is useful to consider three forms of temporal variation. In the first case, variation is “stationary,” changing with time in a uniform way over a large spatial region. A second possibility is that temporal variation at a given site is very weakly correlated with variations at other nearby positions. In both cases it is best to apply temporal interpolation (either linearly, or by some higher-order method) independently at each position, there being no extra information to be gained by taking spatial variation into account.

A third possibility, frequently seen in meteorological applications in particular, is that much of the temporal variation at each point is associated with the advection of features (e.g., weather systems) past the site. Within a suitable frame of reference moving with the feature, temporal variation of data variables (e.g., wind, pressure) can be much slower than that observed at a fixed point. This opens the possibility of developing more accurate interpolation schemes if such advective properties of the combined spatiotemporal variation can be exploited.

To some extent this is already the case. In the preparation of weather forecasts, methods for the interpolation of advected features are now used that rely on human intervention to identify and locate the features for numerical processing (Wier 1995). Similar methods are also used to prepare wind fields for numerical wave model inputs, particularly where intense, localized storm systems are present (Cox et al. 1995).

Another approach that can be used in meteorological studies (Van den Dool and Qin 1996) is to identify climatological phase speeds (as a function of latitude) of eastward and westward propagation of zonal harmonics, and apply these to extrapolate–interpolate motions between successive records of global-scale atmospheric fields. While this has proved successful, it requires a global domain to determine harmonics, and it would still be desirable to find a scheme that can be applied more generally (e.g., to smaller domains and to nonmeteorological systems).

Zavala-Hidalgo et al. (2003) have suggested an approach that uses a complex empirical orthogonal function (EOF) methodology to characterize advective patterns in the data. Having decomposed the data in this way, interpolation is done on the time-dependent coefficients of the modes, before reconstructing the interpolated fields. They report satisfactory initial results in tests in the Gulf of Mexico with wind fields combining Quick Scatterometer (QuikSCAT) data with Eta mesoscale atmospheric model outputs.

In this paper we introduce another method, based on fast Fourier transforms (FFTs), for comparison with the EOF and linear interpolation methods in two test cases. The first test involves interpolation from 6-hourly wind fields replicating a tropical cyclone in the southern Pacific Ocean (without background winds). The second case considers interpolation from 12-hourly wind fields over a domain extending over 120° of longitude and 70° of latitude in the Southern Hemisphere.

2. Wind field interpolation methods

Suppose we are given a sequence of wind fields U(u, t_n) at regularly spaced times t_n = t₀ + nδt, n = 0, … , N. We seek a method to provide an accurate estimate of the wind field U(x, t) at arbitrary intervening t₀ < t < t_N. We outline below three methods for approaching this problem.

a. Linear interpolation

A quantity a(x, t_n) varying in space and time can be estimated from its values at fixed input times by weighted interpolation from the tabulated values:

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The simplest such method is linear interpolation. That is, for times in the range t_n < t < t_n+1 we take linear weights

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with all other weights zero.

When interpolating vector fields, the simplest approach is to use linear interpolation of the vector components U = (u, υ) [i.e., applying the interpolation (1, 2) to both a = u and a = υ]. Such a vector interpolation can produce unrealistically low wind speeds in situations where the wind shifts direction through a large angle. In such a case it is generally more realistic for the wind speed to stay approximately constant while the direction changes. Hence it is usual to apply linear interpolation directly to the wind speed |U| = u² + υ² and the wind direction θ = arctan(u,υ). Providing that care is taken to treat jumps of 2π in wind direction appropriately, this is a robust procedure that is easy to implement, and is the default method in many applications.

A difficulty occurs interpolating between successive “snapshots” of a compact, rapidly moving storm system. If the displacement of the storm in the sampling time interval δt is large relative to a characteristic length scale of the system, a simple linear interpolation scheme will perform poorly.

b. FFT interpolation

Suppose that the spatiotemporal variation of a quantity a(x, t) consists solely of a single moving system that varies only through translation of the whole system, such that

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Then if we take the 2D Fourier transform in the (infinite) spatial domain,

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After a change of variables to x′ = x − x₀(t), we obtain

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The only time variation in the Fourier components A(k,t) is through variation of the phase. In particular, if the motion is at a constant velocity, that is, x₀(t) = x₀(0) + ct, then linear time interpolation of the amplitude and phase of the Fourier components will give exact results. This suggests the following interpolation procedure to estimate the value of a(x, t) from its values at fixed times t_n:

1. Take a 2D Fourier transform of the fields at the input times t_n:

   

   

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2. Interpolate the amplitude and phase linearly in time:

   

   

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3. Use the inverse Fourier transform to estimate a at the intermediate time:

   

   

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In step 2, we again must take care with phase jumps, as phase can vary rapidly with time for high wavenumbers. In this case, the phase shifts should be “unwrapped” by adding multiples of 2π in such a way that ϕ(k, t) is a continuous function of wavenumber (starting with the lowest nonzero values). This will preserve the linear variation of phase noted above in the case where the field is dominated by a system moving at uniform velocity.

If, at step 2, we interpolated the real and imaginary parts of A(k, t), instead of its amplitude and phase, the result would be identical to a direct linear time interpolation of a(x, t) due to the linearity of the Fourier transform.

c. EOF interpolation

Another approach suggested by Zavala-Hidalgo et al. (2003) uses a complex empirical orthogonal function methodology to capture moving patterns in the data.

To apply this method, we first perform a Hilbert transform in the time domain. Suppose the function a(x, t) can be expressed by a Fourier integral,

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over the frequency domain. Then the Hilbert transform of a is (Titchmarsh 1948)

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This can be used to provide an analytic signal

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We define the correlation between data at pairs of points by the matrix components

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Here the {x_m; m = 1, … , M} represent all the discrete cells of the 2D spatial grid (which may in principle be distributed arbitrarily), and Z* is the complex conjugate of Z. This correlation matrix will have at most P = min(M,N) nonzero eigenvectors S_px and eigenvalues λ_p:

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The eigenvectors can be scaled to satisfy an orthonormality condition,

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where S* is the complex conjugate of S, in which case the analytic signal is represented by a sum

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while S_p and T_p are complex spatial and temporal functions, respectively. The temporal summation coefficients are given by

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As noted by Winant et al. (1975), the variance in the data contributed by each term in the sequence (16) is proportional to the value of the corresponding eigenvalue λ_p. Thus the eigenvectors with large eigenvalues represent the dominant modes of variation of the data. Ideally these might be associated with physically meaningful phenomena, although generally a combination of modes is needed to form such a correspondence. In any case, often only a small number of modes are needed to closely approximate the data.

Having made the EOF decomposition, interpolation can be carried out on the complex temporal coefficients T_p, before reconstructing Z(x, t), of which the field of interest a(x, t) is the real part. If interpolation is applied directly to the complex components of T_p, EOF interpolation gives an identical result to direct time interpolation of the field. That is, if we use a weighted interpolation

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then the reconstructed analytic signal is

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which is exactly the same result as applying the same interpolation weights directly to the signal. The same result applies to a real EOF decomposition, in which the original field of interest a(x, t) is used directly, without a Hilbert transform.

There is, however, a potential benefit if the magnitude and phase of T_p are interpolated, rather than its real and imaginary parts, in a manner analogous to the FFT method. Zavala-Hidalgo et al. (2003) report some promising results on wind fields in the Gulf of Mexico derived from a combination of QuikSCAT data and the Eta atmospheric model, applying EOF interpolation to the wind stress components, and interpolating on magnitude and phase of T_p. They found that moving fronts were reproduced satisfactorily, and also noted better results applying EOF interpolation directly to wind stress fields than to scalar fields (divergence, vorticity, deformation) into which the vector fields can be decomposed.

When interpolating the phase of T_p, care must again be taken with phase jumps. In our applications, the phase shifts were adjusted to the range (−π,π) by adding multiples of 2π.

3. Application to a simulated tropical storm

The various interpolation methods were tested using a synthetic wind field constructed to simulate a moving tropical cyclone. A historic event (Tropical Cyclone Waka, December 2001–January 2002) was selected, and best-track data for this event were sourced from the archive of the Joint Typhoon Warning Center (U.S. Naval Pacific Meteorology and Oceanography Center). Tropical Cyclone Waka initially developed from a tropical depression between the Solomon Islands and Vanuatu moving over Fiji and the Wallis Islands. After reaching cyclone strength, Waka initially turned to a southwestward track before recurving southeastward, passing over Vava’u in the Kingdom of Tonga (Fig 1). Waka was an intense tropical cyclone with a peak intensity of about 95 kt, and followed a relatively smooth path during its most intense phase.

A rectangular spatial grid was established covering the domain shown in Fig 1, that is, longitudes 35°–5°S and longitudes 175°–200°E (160°W), at 0.1° resolution in both latitude and longitude.

Using the tabulated values of the eye position and the magnitude V_max and radius R_max of maximum winds, the wind field in the cyclone was constructed from an isotropic form of the Holland model (Holland 1983), with a wind speed

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at a distance r from the storm center. No background wind field was added. The Joint Typhoon Warning Center (JTWC) track data provided storm parameters at 6-hourly intervals. Cubic spline interpolation was used to derive parameter values from which the “correct” idealized wind fields were derived at intermediate times.

The various time interpolation methods described above were then applied to estimate interpolated wind fields at hourly intervals from the synthetic cyclone wind fields at the 6-hourly tabulated times, for comparison with the correct wind fields. This is a rather stern test, as in these 6-h periods the cyclone moved distances up to several times R_max, particularly in the latter part of its evolution (Figs. 1, 2).

Input wind fields were taken at 6-hourly intervals from 1200 UTC 28 December 2001 to 0000 UTC 2 January 2001 (i.e., a total of 19 input fields). The linear and FFT interpolation methods use only the two input times before and after a given output time, whereas the EOF method can use any number of input fields, up to the full record. It is also possible to limit the number of modes used in the interpolation and reconstruction, up to a number equal to the number of input fields. In most tests the maximum value of 19 modes from 19 input fields was used, but a test was also carried out using just the two adjacent times. Other variations of the EOF interpolation method were also tried; interpolating either the real and imaginary parts of the temporal coefficient T_p, or its magnitude and phase, using either linear or cubic spline interpolation.

Some examples of interpolated wind fields produced by the various methods are shown in Figs. 3 and 4. Between the input times of 1200 and 1800 UTC 30 December 2001, the cyclone moved by approximately twice the radius of maximum winds. Linear interpolation to intermediate times 1400 and 1600 UTC (Fig 3a) can only produce a weighted combination of two cyclonic wind fields centered on the initial and final locations, where winds remain artificially low. This interpolation actually produces some of the highest wind speeds near where the eye should be, and vorticity is also overpredicted in this region (Fig 4a). When the EOF method is used (Fig 3b), wind speeds are lower at the correct eye location, but only as part of a band of low wind speeds connecting the initial and final eye locations. The distributions of velocity and, particularly, vorticity (Fig 4b), are still more like a superposition of initial and final wind fields than a single advected cyclonic pattern.

Wind fields from the FFT interpolation method (Figs. 3c, 4c) do, however, closely resemble the intended single advected cyclonic form. The wind field is not exactly isotropic, with some variation in maximum wind speed around the radius of maximum wind. These small discrepancies are in fact spread across the full spatial domain in the x and y directions. Because of the time interpolation being carried out in the wavenumber domain, spatial localization of any features in the wind field depends on detailed matching of phases, so any numerical error (associated with discretization, for example) may not remain localized.

Results were evaluated in terms of a root-mean-square error in the velocity fields:

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where U(x_m, t) and U₀(x_m, t) are the interpolated and correct wind fields, respectively. Variation of ε(t) through the simulation period for the three interpolation methods is shown in Fig 5. All methods of course give zero error at the 6-hourly input times, with errors generally reaching maxima midway through the intervening times. As the storm changed direction sharply in the first 12 h of the simulation, all interpolation methods give large errors, but midinterval errors decrease up to 1200 UTC 29 December. In the following 6 h, when the storm radius contracted significantly, the FFT method gives poorer results than linear or EOF interpolation. Subsequently, as the storm maintains a relatively constant size, but steadily increases in ground speed, errors in all methods tend to increase, but the FFT method gives consistently lower error values than linear and EOF interpolation, which gives the highest error.

For reference, the ratio ξ = V_stormδt/R_max of the distance moved by the storm in each sampling interval to the storm radius was calculated, and its variation is also shown in Fig 5. Maximum interpolation errors appear to scale closely with ξ through the majority of the record, but with proportionally higher error values during the first 12 h when the storm track underwent large changes in direction.

Values of ε averaged over the full period are listed in Table 1. The FFT method gave a mean RMS error of 0.78 m s⁻¹, approximately half that of the “baseline” case of standard linear interpolation of vector wind components, which gave a mean RMS error of 1.44 m s⁻¹, and around 66% of the error from using direct interpolation of wind speed and direction.

Table 1 includes some further interpolation options not plotted in Fig 5. For example, some cases using the EOF method with T_p interpolated through its real and imaginary parts were tested to confirm that this gives identical results to standard linear interpolation of vector wind components. This was shown to be the case whether each EOF interpolation used 2 input times or the full set of 19 input times.

In fact, if only two input times are used, the EOF method also gives identical results if amplitude and phase of T_p are interpolated. This arises because the Hilbert transform of a length-two signal is zero, so the “analytic” signal (12) is real. Hence the orthogonal functions S_p(x) and temporal coefficients T_p(t) are also real, so the two interpolation methods are identical.

When using a larger set of input times (as intended) for the EOF method, slightly larger root-mean-square errors (ε = 1.62 m s⁻¹) were in fact produced. Changing the interpolation of the magnitude and phase of T_p from linear to cubic spline led to a further slight increase in errors.

4. Application to large-scale synoptic wind fields

The next application considered was interpolation from a sequence of wind fields covering a domain large enough to include multiple weather systems at any time. This was aimed at representing a task typically required in an operational forecasting context.

The wind fields were taken from the operational analysis dataset provided by the European Centre for Medium-Range Weather Forecasts (ECMWF), selected for a region of the southwest Pacific-Southern Ocean. A latitude–longitude grid at 1.125° × 1.125° resolution was used covering latitudes 81° to 9°S and longitudes 99° to 220.5°E (139.5°W). One month of wind fields (October 1998) was used, available in the dataset at 6-hourly intervals.

For the purpose of this comparison, wind fields U₀(x, t) were subsampled to 12-hourly records, then interpolated wind fields U(x, t) were computed at the original 6-hourly intervals using the various interpolation methods. Results were evaluated in terms of a root-mean-square error

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taking an average over all (12 hourly) output times in the 1-month test period. Three interpolation methods were tested:

1. linear interpolation of wind speed and direction;

2. EOF method, with linear interpolation of |T_P|, phase(T_p); and

3. FFT method, with linear interpolation of |A(k)|, phase [A(k)].

The RMS errors obtained for each method are shown in Fig 6, plotted as a ratio of ε(x) to time-averaged wind speed. Of the three methods, EOF interpolation generally produced the lowest errors, with ε ranging between 20% and 40% of mean wind speed across most of the domain. By comparison, linear interpolation gives RMS errors on the order of 50% or more over much of the domain. Both methods give smaller relative errors over the tropical oceans than over both higher-latitude waters and the Australian landmass (where mean wind speeds are lower than over surrounding waters).

If we restrict attention to winds over deep ocean waters, the FFT method gives RMS errors between results of EOF and linear interpolation. Unfortunately it gives very poor results in the vicinity of terrestrial regions associated with high topographic relief, notably Papua New Guinea, New Caledonia, Fiji, New Zealand, Tasmania, the southeast of mainland Australia, and the Transantarctic Mountain chain adjacent to the Ross Sea. In these areas, RMS errors can greatly exceed 100% of the mean wind speed. Such topographic features can produce strong local disruption to the smooth advection of weather systems, a significant violation of the assumption behind the FFT method.

Some elevated error values for the FFT method are also noted around the domain boundaries. Even for the ideal case considered in Eqs. (3)–(4), the resulting exact linear time dependence of the phase depends on the spatial domain being infinite. Errors will be introduced in the presence of finite domain boundaries, as information on a feature entering or leaving the domain is truncated at the boundary.

Table 2 lists spatial averages of error statistics over the whole domain, and separately over the sea and land portions. Averages are also taken over “coastal” regions, defined as all cells with at least one neighbor of the opposite type (i.e., sea cells with neighboring land, or vice versa). The EOF method gives relative RMSE values averaging 27% over the whole domain, compared to 54% and 53%, respectively, for the linear and FFT methods. Both the linear and EOF methods show increases of the order 10%–20% in mean relative errors due to land effects, while the FFT method is affected more severely: over sea it performs slightly better on average than linear interpolation, but considerably worse over land, and particularly near the land–sea interface.

5. Discussion

We have considered the problem of interpolating from data available as fields available on a 2D spatial domain at discrete times to arbitrary output times. For the particular applications considered, the data consisted of wind fields, the evolution of which can to some extent be described as the translation, growth, and decay of ensembles of weather systems. Other related meteorological fields may be expected to have similar properties.

Three methods were considered: simple linear interpolation in time, and methods based on interpolation of the coefficients of decomposition into either Fourier (FFT) or empirical orthogonal function (EOF) bases. The standard linear interpolation is a completely local method, which makes no use of any spatial structure in the data. As such, when sampling at a time interval δt, it can only accurately represent temporal variation at frequencies below the Nyquist frequency ½δt. The other methods aimed at making use of spatial structure to improve the accuracy of interpolation.

The FFT method, in particular, is intended to be highly accurate for fields that are stationary in a frame of reference moving with a constant translation velocity. The wind fields of a compact, fast-moving tropical cyclone would, at least in an idealized case, be a good example. The FFT method would interpolate such fields accurately even if the system moves several times its own radius between sample times, a condition under which standard linear interpolation will perform poorly. The FFT method relies on sampling at a spatial resolution δx sufficiently small that all important spatial Fourier components are below the Nyquist wavenumber ½δx. For the idealized case of a uniformly translating system, an exact temporal interpolation of these components can then be made due to the stationarity of the Fourier amplitudes and linear temporal variation of their phases. For wind fields differing slightly from the idealized translating systems, we would still expect small interpolation errors. Although theoretical limits on these errors are beyond the scope of the present paper, some empirical tests are helpful.

We therefore tested the interpolation methods for a realistic representation of a historic cyclone, which included variation in the size, strength, and translation velocity of the storm. It was found that the FFT method still provided substantial improvements in accuracy over standard linear interpolation in this case.

However, the FFT method was found to be insufficiently robust to interpolate more complex wind fields over a domain large enough to contain multiple evolving weather systems, a typical situation in numerical weather prediction. The method has particular difficulty with the interaction of systems moving with different velocities, as seen in our test case where moving weather systems are perturbed by (stationary) topography. It could also be said that, in contrast to standard methods, the FFT method is excessively nonlocal for such cases, with errors arising at one location being spread immediately across the spatial domain.

In these more complex situations the EOF method introduced by Zavala-Hidalgo et al. (2003) offers a more useful solution, being able to capture modes of variation associated with multiple moving features. Like the FFT technique, this method also relies on using a sufficiently fine spatial resolution to capture all significant modes of variation in the spatial structure of the wind fields. Provided also that the temporal variation of the dominant EOF coefficients be characterized by frequencies well below the Nyquist frequency, which may not be the case for the original data, accuracy superior to that of simple linear interpolation can be expected.