## 1. Introduction

Spectral analysis techniques of global atmospheric fields in the horizontal have been used for several years for diagnostic purposes (e.g., Boer and Shepherd 1983; Trenberth and Solomon 1993) as well as for the numerical simulation of the atmosphere (e.g., Bourke 1972; Daley et al. 1976). However, their popularity as diagnostics tools has been until now largely circumscribed to applications whose domain covers the sphere. On such a domain, a spectral decomposition in terms of triangularly truncated spherical harmonics is natural and very convenient for many reasons. For instance, one of the two transforms required for obtaining the spectral coefficients is the very well known discrete Fourier transform (DFT); this transform is well suited for the global atmosphere because its basis functions are periodic, as are the atmospheric fields, on latitude circles. Moreover, fast numerical algorithms for computing the DFT (called FFT for fast Fourier transform) are widely available.

Spectral analysis of atmospheric fields on limited-area grids using Fourier transforms is less popular for two reasons. First, fields on such grids are generally aperiodic and are, in most cases, dominated by large-scale features whose wavelengths are greater than the domain size. To represent such fields with periodic Fourier basis functions, Fourier coefficients corresponding to the high wavenumbers are highly solicited for matching the boundary conditions. As an example, let us consider Fig. 1, which shows a winter snapshot of the 925-hPa specific humidity taken from a simulation produced by the Canadian Regional Climate Model (CRCM; see Caya and Laprise 1999 for the model's description). As can be seen, this field contains variance at many scales but is largely dominated by an east–west gradient or trend. To quantify how much information or power is present in the different spatial scales we need to compute its variance spectrum. The solid curve on Fig. 2 shows the spectrum computed with the direct use of the periodic Fourier transform without any prior modification of the field of Fig. 1. We see that this spectrum has a tail that undergoes an upward distortion. This is caused by the aliasing or projection of the large-scale trend on the high-wavenumber components. This distorted spectrum is therefore misleading and it is not representative of the variance inside the domain since a considerable part of it comes from the lateral boundaries and not from the field in general.

To circumvent these deficiencies, Errico (1985) used a method that involves preprocessing the field to make it periodic by removing its linear trend before the application of a two-dimensional (2D) DFT (detrending-DFT method hereafter). The linear trend is calculated from the difference between the first and last gridpoint field values along each row and column of the grid. Fig. 3 shows the result of detrending the original field of Fig. 1. The benefit of this detrending technique is easily seen in Fig. 2 by comparing the tail of the new spectrum produced by the detrending-DFT method (dashed curve) with the original spectrum tail (solid curve). A side effect of the detrending is the removal of the large-scale gradient across the limited domain as seen in Fig. 3, which affects the large-scale components of the spectrum (Fig. 2). Another side effect of this technique is the pattern of lines generated by the linear detrending as can be seen in Fig. 3. Spectra computed from such massaged fields are sometimes questionable. As mentioned by Errico (1985, 1987) himself, this technique should not be used for fields that are noisier at their boundaries than in the interior of the domain.

Another technique, which is frequently employed for spectral analysis of time series, is the application of a weighting function on the physical field prior to the DFT. This technique, called windowing, involves multiplying the physical field by a weighting field of the same dimension but with values equal to unity in the central subdomain and diminishing to zero when approaching the boundaries. The weighting function used to make the transition from the central subdomain to the boundaries is often of cosine or Gaussian form. Examples of windowing with 2D spatial atmospheric fields can be found in Turner (1994) and Salvador et al. (1999). These techniques are to some extent similar to that of the detrending-DFT method because they massage the field at or near the boundaries prior to the application of the DFT. But the windowing technique is softer and does not generate a spurious pattern of lines. Although effective for removing the major part of the distortion of the spectrum tail, the application of such windows may not be desirable. In effect, it would modify the spectrum of an already periodic field. Such windowing technique is only appropriate for large-dimension domains, a criterion seldom met by regional climate models.

This paper explores the use of a special type of transform suitable for spectral analysis of data on a limited area. This transform is called the discrete cosine transform (DCT) and is the core of some data-compression algorithms for digital images. The remainder of this document is organized as follows: Section 2 includes definitions of the direct and inverse DCT, a study of its properties using basic test cases, and a comparison with the detrending-DFT method. Section 3 provides some examples of practical applications including kinetic energy spectra and spectral filtering. The performance, limitations, and usefulness of the DCT for atmospheric applications are discussed in section 4.

## 2. The discrete cosine transform and power spectra

### a. DCT definition

Besides the detrending/windowing techniques, there is a simple way to cure the problem of aperiodic boundary conditions when discrete Fourier-type of transforms are employed: making the field to be analyzed periodic by a symmetrization process. This process involves simply taking a mirror image of the original function prior to the application of the Fourier transform. It can be shown that this procedure leads to a special Fourier transform called the discrete cosine transform, first introduced by Ahmed et al. (1974) for digital image processing purposes.

A mathematical derivation of the 1D- and 2D-DCT, using the mirror effect, can be found in the appendix. Other derivations can also be found in Ahmed et al. (1974) and in textbooks dedicated to digital image processing. The DCT has become very popular in the field of image processing (Pratt 1991; Gonzalez and Woods 1992) and forms the core of digital image compression algorithms such as those of JPEG and MPEG formats (Pennebaker and Mitchell 1993). This is due to the capacity of the DCT to treat aperiodic fields while producing spectra where the variance is concentrated in the low-wavenumber components. Thus, only a small number of spectral coefficients are required to retain the major part of the visual information of the image. The compression algorithm preserves only these most important spectral coefficients, therefore reducing the size of the image in terms of computer memory without a dramatic loss of the image quality. There is an important difference between the compression algorithms mentioned above and the application of the DCT in this paper. Here, the DCT shall be applied on grids of the order of 100 by 100 grid points, but in usual image processing the DCT is applied on blocks of 8 by 8 grid points (pixels) forming images of the order of 1000 by 1000 grid points (pixels).

*f*(

*i,*

*j*) of

*N*

_{i}by

*N*

_{j}grid points, the direct and inverse DCT are respectively defined as,

*f*(

*i,*

*j*) is the field value at gridpoint integer numbers (

*i,*

*j*), and

*F*(

*m,*

*n*) is the spectral coefficient corresponding to the (

*m,*

*n*) adimensional wavenumbers. A 2D-DCT applied to a physical field

*f*(

*i,*

*j*) of

*N*

_{i}by

*N*

_{j}values produces an

*N*

_{i}by

*N*

_{j}array of

*F*(

*m,*

*n*) real spectral coefficients.

### b. Power spectra construction from the DCT

*f*〉 is the domain average. The total variance can also be computed from the spectral coefficients:

*f*〉 from

We see from (5) that the total variance can be decomposed into spectral variance components forming an *N*_{i} × *N*_{j} array [excluding the element (0, 0), which is related to the domain average]. Figure 4a shows an example of a 2D spectral variance *σ*^{2}(*m,* *n*) array computed by applying the 2D-DCT of the humidity field shown in Fig. 1. Figure 4b is a monthly mean of such arrays.

*m,*

*n*) needs to be associated with a single-scale parameter, namely a wavelength

*λ.*For a square domain,

*N*

_{i}=

*N*

_{j}=

*N*and we have

*k*the 2D wavenumber defined as

In the variance arrays of Fig. 4, a given *k* corresponds to a circle of radius *k* having for an origin the element (*m* = 0, *n* = 0). Each element (*m,* *n*) on a given circle has the same wavenumber *k.* It can be seen from these examples, for which the grid is a square and Δ*x* = Δ*y,* that the power decreases quasi-radially from the largest scale (small wavenumbers) components to the smallest scale (large wavenumbers) components. This is particularly evident for the monthly mean of variance arrays (Fig. 4b) and suggests that the horizontal humidity field has a quasi-isotropic turbulent nature since the variance components depend only on the magnitude of the 2D wavenumber *k* and not its direction.

*λ.*First, the lowest mode (

*k*= 1) supported along one axis of the domain corresponds to a half cosine wave (see Fig. A2). Therefore, from (7) the spectral variance of the data that projects onto this mode can be associated with a wavelength of twice the length of the domain along this axis,

*λ*

_{max}

*N*

*k*=

*N*− 1) corresponds to

*N.*Thus for simplicity, the variance of the highest wavenumber displayed in the spectra can be associated with a wavelength of nearly twice the gridpoint spacing. To include the possibility of analyzing data on rectangular domains in addition to square domains, Eqs. (7) and (8) are generalized as follows:

*α*is a normalized 2D wavenumber. Equation (12) can be seen as a normalization of the

*m*and

*n*wavenumber axes leading to an elliptic shape. With these new definitions, the variance components that should be retained for constructing a spectrum are those for which 0 <

*α*< 1. More specifically, the variance contributions of the

*F*(

*m,*

*n*) coefficients need to be binned accordingly to bands of

*α.*In the case of a square domain, each binning is accomplished by adding all spectral variances existing inside a ring formed by two circles centered on the lower-left corner of the variance array and having specific radii of

*α*and

*α*+ Δ

*α,*respectively. The thickness (Δ

*α*) of the ring defines a specific wavenumber band and, consequently, waves of specific wavelengths. If the domain is rectangular, the region of the variance array contributing to a specific wavelength band is the region between two ellipses. The limiting values of

*α*for each wavelength band are determined as follows:

*k*= {1, 2, 3 … min(

*N*

_{i}− 1,

*N*

_{j}− 1)}. For the particular case of a rectangular domain, the minimum operators ensure that the number of wavenumber bands is no larger than the number of wavenumbers of any individual variance array axis.

The algorithm for variance binning is as follows:

For a given

*k,*determine the limits of the contributing band defined by*α*(*k*) and*α*(*k*) + Δ*α*(*k*) using (13) and (14);for each element in the variance array, compute

*α*using (12) and add its contribution to the variance*σ*^{2}[*α*(*k*)] if*α*(*k*) ≤*α*<*α*(*k*) + Δ*α*(*k*); andrepeat the procedure to sweep all

*k*= {1, 2, 3, … min(*N*_{i}− 1,*N*_{j}− 1)}.

The wavelengths corresponding to each *k* can be computed from Eqs. (13), (14), and (11). The result of this procedure is a discrete variance distribution as a function of wavenumber *k* (or normalized wavenumber *α*) or wavelength *λ.* It is important to note that this construction of spectra, which is based on physical scales (true wavelength bands) through the use of normalized wavenumbers, is similar to that used in Van Tuyl and Errico (1989) but is different from Errico (1985), who used wavenumber bands defined by circles instead of ellipses for his rectangular domain. For a similar reason, a square (or rectangle) truncation is not used for constructing power spectra from a square (or rectangular) spectral variance array since it would mix variances pertaining to different spatial scales. Furthermore, the isotropy usually found in the atmosphere (see Fig. 4) suggests a circular (elliptical) truncation instead of a square (rectangular) one. The amount of variance left out by using a circular (or elliptical) truncation (upper-right corners in Fig. 4) is negligible and represents wavelengths smaller than 2Δ.

*σ*

^{2}[

*α*(

*k*)] of the DCT were gathered and summed two by two. The variances are computed as

### c. DCT properties and test cases

Before using the 2D-DCT for spectral analysis of meteorological fields, we need to make some basic tests for validating its use. First, we will look to a 1D theoretical spectral response using a continuous form and with a known input analytic function. Then, the same analytic input function will be used to test the discrete form [Eqs. (1), (2) and (3)] and the spectra produced using the procedure explained in the previous section. We will close this section with tests that will help to evaluate the spectral analyzing performance of the DCT when applied to input functions (including aperiodic ones) for which the power spectra are specified beforehand.

*h*

*x*

*px*

*ϕ*

*x*≤

*π,*

*p*is not necessarily an integer, and

*ϕ*is a phase shift. If we use a continuous form of the 1D-DCT and we apply it to

*h*(

*x*)

*k*, Eq. (19) reduces to

*p*≠

*k,*the transform responds by returning power at many

*k*wavenumbers. But the

*k*

^{4}factor in the denominator implies aliasing or “frequency leakage” with a steep slope (−4) for

*k*>

*p.*Secondly, the phase shift

*ϕ*also suggests aliasing; there is no provision in the DCT basis functions for a phase shift for a given wavenumber, as there is in those of the DFT.

*p*is not an integer. In this case, Eq. (20) reduces to

*k*and for some selected input wavenumbers

*p.*No two-by-two gathering of variances described by Eq. (15) or (16) has been done. We see that the maximum power (1/4) is reached when

*k*=

*p,*but some of the power spreads over other

*k*values when

*k*≠

*p*due to the aliasing. Maximum aliasing occurs when

*p*is a half integer. Notice that the aliased power decreases rapidly, at a rate of

*k*

^{−4}, on the right-hand side of the maxima (for

*k*>

*p*).

*p,*Eq. (20) becomes

*k*=

*p*is perfect when there is no phase shift, but drops as the phase shift tends to

*π*/2. Meanwhile, the other components of the spectrum, except for

*k*+

*p*even, receive some power such that the aliasing due to the phase shift also exhibits a

*k*

^{−4}power law. Figure 6 illustrates this behavior for an input wavenumber

*p*= 3.

Up to this point, we have looked at the basic behavior of the 1D-DCT in an approximated continuous form. The same input function, but discretized and including a one-half gridpoint shift, can be analyzed with the discrete form [1D version of Eqs. (1), (2), and (3)]. Figure 7 shows the variance spectra for such an input function having wavenumber *p* ranging from 6 to 8.2, for different phase shifts (none, *π*/8, and *π*/2). The discretized 1D domain has 100 grid points. Because the DCT wavenumbers are in fact made of full and half cosines, we have used the two-by-two gathering procedure described by Eq. (16) to get spectra as a function of a wavenumber similar to what would be obtained with a standard Fourier transform (as explained at the end of section 2b). It is this gathered wavenumber named pg (pg = *p*/2) that is displayed on each panel of Fig. 7, and varies from 3.0 to 4.1. We see that as the input wavenumber pg increases, the peaks gradually move with it in the wavenumber space, as it should. When the phase shift increases, the peak diminishes but still dominates, which shows how well the 1D-DCT captures the variance present in the input field with a particular wavenumber and phase shift. Finally, the −4 slope predicted from the analysis of the continuous form is still present, except at the extreme end of the spectra where it becomes markedly steeper.

Before applying the DCT to real meteorological cases, we have pushed the validating tests one step further. Since meteorological fields can show a wide continuous range of aperiodicity, we have tested the DCT on synthetic 1D fields having this characteristic. Each field was constructed from a sum of functions similar to that of Eq. (17) and for which each wavenumber and phase shift has a random component. Furthermore, the amplitude of each function contributing to the synthetic field has been scaled to give specific power spectra that were known a priori. The test is designed to evaluate the extent to which the DCT spectral analysis is able to return the correct spectra for aperiodic cases. Figure 8 shows spectra computed from 10 000 synthetic fields having specified spectrum slopes of 0, −1, −2, −3, −4, and −5 on a log–log scale. The spectra have been computed using the 1D-DCT and the 1D-DFT for comparison. The gathering procedure of Eq. (16) has been used for the DCT, and will be so used for the rest of the paper. Figures 8a and 8c show the results for all cases and Figs. 8b and 8d show only cases for which the aperiodicity was larger than 3 standard deviations of the field values. It can been seen that the DCT produces reasonable spectra for slopes shallower than −4; for steeper slopes, the aliasing effect discussed previously contaminates the calculated power distribution. On the other hand, the DFT shows a distorted tail for slopes equal to or steeper than −2, especially for strongly aperiodic cases (Figs. 8b-c). Figure 9 shows the results for positive slopes. In this case, the aliasing appears at the larger scales (small wavenumbers) in both the DCT and DFT. Results are clearly unacceptable for slopes larger than +1. All of these results indicate that the DCT can be safely used to spectrally analyze meteorological fields having spectral slopes between −4 and +1; outside these limits the calculated spectra would be too contaminated by the noise caused by aliasing.

### d. Comparison of the DCT and detrending-DFT method

To illustrate the spectral analysis capability of the DCT for 2D real meteorological fields, we show in Fig. 10 the spectra of the humidity field of Fig. 1 using the 2D-DCT, and also the detrending-DFT method proposed by Errico (1985). Given that the field of Fig. 1 is highly aperiodic, the use of the two methods is well motivated. From Fig. 10, we see that both methods removed similarly the distorted tail produced by the direct application of the DFT without detrending.

## 3. Examples of application of the 2D-DCT

In this section, other examples of applications of spectral analyses using the 2D-DCT will be presented. Most of the limited-area fields that will be analysed were simulated by the CRCM.

### a. Kinetic energy spectrum

The kinetic energy spectrum is one of the most fundamental spectra to examine in order to understand the dynamical behavior of the atmosphere. At large (global) scales, the stationary (time mean) component of kinetic energy is known to dominate the flow dynamics (Boer and Shepherd 1983; Trenberth and Solomon 1993). On the other hand, the transient kinetic energy component gradually dominates the spectrum at smaller scales. This can be seen in Fig. 11, which shows the vertically integrated kinetic energy spectrum for the total, stationary, and transient components, as computed from a set of European Centre for Medium-Range Weather Forecasts (ECMWF) global objective analyses for February 1993. The variance shown is expressed as a function of the total wavenumber *N* of triangularly truncated spherical harmonic basis functions (up to T213). The total component spectrum exhibits an approximate spectral slope of −3 over an energy cascading subrange between wavenumber *N* = 10 and *N* = 60. Somewhat unexpectedly, the spectrum gets damped at larger *N,* due to a reduction of the transient activity of the smallest scales.

Now, the 2D-DCT technique is used to produce kinetic energy spectra from the same ECMWF analysis but for a limited-area domain, as opposed to the spectra of Fig. 11, which were produced using spherical harmonics basis functions on the globe. The region encompassed a subdomain of 100 × 100 grid points, centered over the domain of Fig. 1. The gridpoint spacing is 45 km. Figure 12a shows that the kinetic energy spectra computed using the 2D-DCT exhibits a similar spectral slope behavior to that found in Fig. 11. The spurious tail beyond wavenumber *k* = 24 (*λ* = 187 km; *N* = 213) is an artifact caused by the interpolation of the data from the Gaussian transform grid to the higher-resolution polar-stereographic grid, and should be discarded. Figure 12b shows the corresponding spectra computed from a 45-km gridpoint spacing CRCM simulation of that month [driven by low-resolution NCEP (T32) analyses]. Comparing Figs. 12a and 12b, we see that the CRCM gives spectra that are generally similar to those computed from the ECMWF analysis, the difference being that the CRCM transient activities completely dominate the small scales, even at the extreme end of the spectrum. Finally, Fig. 12c displays the spectra of the total component of the CRCM and the ECMWF. It can be seen that even though the CRCM spectrum is also somewhat damped at its end, it contains more variance than does the ECMWF, probably due to the CRCM's higher resolution.

### b. Scale separation using DCT spectral filtering

An objective technique for separating horizontal meteorological fields into different scales can be easily designed based on the DCT. As we saw, the direct application of the 2D-DCT on a physical field produces an array of spectral variances in which the spatial scales are related to the 2D wavenumbers *k* (e.g., Fig. 4). Low-pass, high-pass, or any bandpass filtering can easily be performed by applying a 2D transfer function onto the 2D spectral variance components. This is done by multiplying, element by element, the spectral variance array by a transfer function array with values between 0 and 1. Thereafter, an inverse transform is applied to rebuild the filtered physical field.

The choice of the transfer function is very important. If the cutoff is too abrupt, Gibbs phenomena will appear. Sardeshmukh and Hoskins (1984) showed how this shortcoming may be minimized by choosing a soft cutoff, that is, a gradually varying transfer function. In this paper, we use a taper that follows a squared cosine. Figure 13 shows an example of a transfer function, also commonly called the amplitude response of the filter. Even though Fig. 13 shows the transfer function as a function of a unidimensional wavelength, it is applied in the 2D wavenumber space (*m,* *n*) with the amplitude response varying radially from the lower-left corner of the spectral variance array. For this low-pass filter, all scales larger than 1000 km are preserved and all scales shorter than 500 km are removed. Figure 14a shows the results of the application of this filter on the humidity field of Fig. 1. As expected, only the largest-scale features survived the filtering. The small-scale content (Fig. 14b) of the original field can be obtained by subtracting this filtered field from the original field (Fig. 1). We see that the method is effective in extracting the mesoscales from the original field. It is interesting to note that no Gibbs phenomena are apparent as would be the case if we had used the DFT. In the DFT case, the field (not shown) is largely dominated by Gibbs waves with their maximum amplitudes along the lateral boundaries. These spurious waves are simply the physical representation of the distorted tail seen in the DFT curve in Fig. 2.

For the second example of extraction of mesoscales by filtering, we chose a more difficult case for the DCT. We mentioned that the DCT circumvents the aperiodicity problem of the DFT that generates Gibbs waves. But it is still possible that these spurious Gibbs waves may appear with a periodic field if the slopes (derivatives) of that field are large at the boundaries. In effect, such a field, even though periodic in terms of field values, necessitates high-frequency spectral components to meet these slopes. This is even more apparent when small-scale variances inside the domain are not predominant. To illustrate this effect, let us first look at Fig. 15, which shows a simulated monthly mean of a mean sea level pressure field. As can be seen, this field has little visible small-scale variance because the time averaging procedure removed most of it. Nevertheless, stationary mesoscale features due to surface forcings should be present although they are not easy to see. Figure 16a shows how spectral filtering using the 2D-DCT isolates the mesoscale components; in this case, the filtering has also generated strong Gibbs oscillations that are very visible near the left boundary. An easy way to alleviate this Gibbs problem is to diminish the slopes in the direction normal to the lateral boundaries. Figure 16b shows how a simple smoothing, applied over only five points in normal directions starting from the boundaries, removed the spurious phenomenon while having little impact on the mesoscale features of interest further inside the domain. It should be noted that this periodicity issue in the slopes at the boundaries is not unique to the DCT but can also be seen with the use of the DFT, even after detrending the input field.

## 4. Discussion

### a. Variance spectra

Methods for avoiding the spectral tail distortion caused by the application of the DFT on atmospheric fields defined on limited-area domains have been the topic of several publications. These include papers (e.g., Errico 1985, 1987), appendices of papers (e.g., Barnes 1986; Van Tuyl and Errico 1989) and a thesis (Turner 1994). None of these have shown that the detrending technique was totally satisfactory. In the present paper, we abandoned the DFT in favor of the DCT as a way to solve the distortion problem. After providing the 1D- and 2D-DCT definitions and describing how a spectrum can be constructed from the DCT spectral coefficients, the reasons why the DCT can be beneficial for spectral analysis on limited-area domains were investigated methodically in section 2c. It was shown that, with the DCT, the aliasing caused by aperiodicity (or trend) projects either on the first (lowest mode) half-cosine basis function and/or on smaller scales following approximately a −4 slope. This implies that for atmospheric fields with natural spectral slopes no steeper than −4, the true spectral variance of the field remains above the noise level generated by the aliasing. On the other hand, fields possessing “blue” spectra (positive slopes), have the large-scale portion of the spectrum distorted, but not the small-scale portion. Therefore, it appears (from Figs. 8 and 9) that the use of the DCT for producing spectra is justified for fields having spectral slopes between +1 and −4. Because most of the atmospheric fields have their spectral slopes within this range, the use of the DCT seems appropriate and valuable.

In section 2d, we saw that, for a low-level simulated specific humidity field, the DCT prevented the occurrence of a spuriously distorted spectrum tail, as did the detrending-DFT method. In effect, the two approaches yielded very similar spectra for the small-scale components. But the detrending-DFT method reduces the spectral variance of the largest scale. This is not surprising since detrending acts as a damping agent on the largest scale present in the physical field.

### b. Spectral filtering

The advantage of a spectral filter, such as the one employed in this paper, is that the control on the wavelength band to be filtered can be easily obtained. This is not the case with gridpoint (digital) filters such as a simple moving-average box filter (e.g., Giorgi et al. 1993; Takle et al. 1999), a Shuman filter (Shuman 1957) or, to some extent, a Barnes filter [see Pauley (1990) for response analyses, and Maddox (1979) and Weygandt and Seaman (1994) for a scale separation usage of this filter]. With gridpoint filters, the scale selection, that is, the position *and* the sharpness of the amplitude response transition, is difficult to obtain at the same time without sacrificing one for the other. Some high-order digital filters such as those described in Raymond (1989) and Raymond and Gardner (1991) do possess the control on the amplitude response mentioned earlier, but they need matrix inversion algorithms because of their implicit nature. No such algorithm is needed for the DCT, and a fast DCT version can be used for the spectral filtering, thus making this approach computationally attractive (see the appendix for references). For the cases presented in this paper, the fast 2D-DCT filter was four times faster than a Shuman filter, which needed at least 100 iterations for removing scales smaller than 500 km. It should be noted that these relative performances may be dependent on the computer architecture.

To further compare spectral and digital filtering, there are two interesting points to note. First, high-order (greater than second-order) digital filters can, as spectral filters, create fictitious extrema (Gibbs effects) near strong discontinuities inside the domain unless a special constraint is included in the scheme (e.g., Xue 2000). For our spectral DCT filtering, a taper was used in the spectral space for attenuating these Gibbs ripples. Second, digital filters on limited-area domains necessitate special attention near and at the lateral boundaries. Either the order of the scheme, in the normal direction to the boundaries, must be diminished as the stencil approaches the border, or fictitious data has to be extrapolated for the part of the stencil that falls outside the domain. The first option is not optimal since a lower-order scheme means a lower-scale selective filter and can produce spurious effects deep inside the domain (e.g., Achtemeier 1986; Pauley 1990). Raymond (1989) chose the second option in employing a reflecting boundary condition to keep the high order of his filter throughout the domain. Interestingly, this reflecting boundary condition is analogous to the mirror effect implicit in the DCT.

In section 3b we showed examples of spectral filtering using the DCT. The DFT in conjunction with the detrending method can also be used for scale separation purposes (e.g., Errico 1985; Van Tuyl and Errico 1989). However, detrending can introduce fictitious small-scale features especially when the field is noisy at the boundaries. In effect, once the large-scale field has been produced by filtering the detrended field, it needs to be “retrended” in order to get the real large-scale component of the original field. During this process the small-scale noise created by the detrending is reintroduced into what should be a field containing only large scales. The use of the DCT avoids this shortcoming.

As for other spectral transforms using global (as opposed to local) basis functions, fields that contain few and localized features may be difficult to analyze. This is because they may (and probably will) project variance on many scales, when in fact the basis functions were simply not spatially compact enough to be representative of those features. For such particular fields other methods, such as a spatial 2D-wavelet analysis, might be more appropriate since wavelet basis functions are localized in space. The use of 2D-wavelet analysis for atmospheric purposes such as those described in this paper is still in its infancy and studies involving 2D spatial fields are rare. Grotjahn and Castello (2000) reported success in 2D-scale separation for sea level data but not for data at upper levels. Nevertheless, success has been reported in studies of other 2D geophysical fields (e.g., Bergeron et al. 1999) and the investigation of applications to turbulence has begun (Farge 1992).

### c. Other applications

The application of the DCT for the purpose of power spectra production and spectral filtering of 2D atmospheric fields was shown in section 3. The DCT has also been used for diagnosing the CRCM climate simulations (Denis et al. 2002), and in short-term predictability studies (Laprise et al. 2000; de Elía et al. 2002). The revised nesting scheme of the CRCM using nudging of large scales makes use of the DCT, following the initial work of Biner et al. (2000). The weather radar research group of McGill University is also exploring the use of the DCT in radar applications. The DCT has also been used for direct numerical solvers in limited-area model (LAM) with a C-grid in the horizontal and proper boundary conditions, and appears in the LAM version of the Canadian Global Environmental Model. Finally, it is worth noting that the use of cosine basis functions, although not exactly following the DCT definition presented here, can be found in some limited-area spectral model formulations (e.g., Juang and Kanamitsu 1994; Cocke and LaRow 2001).

## 5. Conclusions

The main objectives of this paper were to introduce and give a broad assessment of the use of the discrete cosine transform (DCT) for spectral decomposition of atmospheric fields on limited-area domains. The principal conclusions that can be drawn are the following.

*Variance spectra constructed with the 2D-DCT:*

avoid the aperiodicity issue encountered with the direct application of the discrete Fourier transform (DFT) (in effect, the DCT does not produce the characteristic distorted spectral tails as does the DFT when applied on aperiodic fields);

compare favorably with the detrending-DFT method (Errico 1985), which also removes the distorted spectral tail, but at the price of modifying the spectra throughout the wavenumber range;

are reliable only for fields having spectral slopes between −4 to +1.

*Spectral filtering with the 2D-DCT:*

is effective for scale separation purposes such as extraction of mesoscale features;

can be made very scale selective (a bandpass filter can be easily defined throughout the wavenumber range);

avoids the Gibbs phenomenon caused by the aperiodicity of the field values at the lateral boundaries but can still produce Gibbs oscillations due to the aperiodicity of the field slopes normal to the boundaries (these latter undesirable oscillations can be largely prevented by applying a simple smoothing near the boundaries before the spectral filtering).

As far as the authors are aware, the use of the DCT for spectral decomposition of two-dimensional atmospheric fields on limited-area domains is novel. This paper tentatively exposed its strengths and limitations but it is through a more widespread usage that its real value will be revealed.

## Acknowledgments

This work has been financially supported by funding from the USDOE Climate Change Prediction Program (CCPP) and the UQAM regional climate modelling group. We gratefully acknowledge the Meteorological Service of Canada for granting an education leave to the first author during this study and for providing access to its facilities. We are thankful to the UQAM regional climate modelling staff for their invaluable technical help with the Canadian RCM. Finally, we especially would like to thank Mr. Sébastien Riette for testing and improving the program for making spectra, and also Drs. Ramón de Elía, Bernard Dugas, and Daniel Caya for stimulating discussions.

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## APPENDIX

### Derivation of the Discrete Cosine Transform

In this appendix we show that the 1D-DCT can be derived by using a “mirror image” artifice. A visual representation of a 1D-DCT basis function set is also given.

*f*(

*i*) be a real function with

*i*= 0, … ,

*N*− 1, where

*N*is the number of grid points. The first step consists in building a periodic function

*f*

_{s}(

*i*) by symmetrizing

*f*(

*i*). This is done by taking the position

*j*= −1/2 as a mirror, that is,

*N*grid points.

*i*= −1/2:

*î*=

*F*

_{s}(

*m*) are the complex Fourier amplitudes. By using symmetry (antisymmetry) properties of the cosines (sines) functions, Eq. (A2) reduces to:

*m*= 0:

*πm*(

*i*′ + 1/2)/

*N*], summing over

*m,*using the orthogonality property of the cosine terms, and dropping the primes:

*N*real values of

*f*(

*i*) in the physical space and

*N*real values of

*F*(

*m*) coefficients in the spectral space.

*α*(

*m*) with

Figure A2 shows the various possible modes of the cosine functions (A7) evaluated at grid points *i* = 0 … *N* − 1 with *N* = 8. We notice that except for mode 0, the values at points 0 and *N* − 1 never reach +1 or −1. This would not have been the case without the half gridpoint shift in symmetrizing the function as reflected by the 1/2 term in (A9)–(A10). This is why these equations are sometimes called discrete shifted cosine transforms. The shift is visible in Fig. A2 because the evaluation of the cosine begins at *i* = −1/2 and ends at *i* = *N* −1/2.

*N*

_{i}by

*N*

_{j}grid points) are

A very useful property is the separability of these two-dimensional transforms. In effect, a 2D transform can be obtained conveniently by the successive application of two 1D transforms. A computationally fast version of the DCT can be built from existing 1D FFT computer code (Press 1992). But, as for the FFT, it is easy to find DCT code in the public domain on the Internet. Finally, computing and visualization programming environments such as MATLAB (1998) provides readily useable 1D-DCT and 2D-DCT routines.

Variance spectra of the 925-hPa specific humidity field. The solid curve corresponds to the direct use of periodic 2D-DFT and the dashed curve corresponds to the application of the 2D-DFT after the field has been detrended. The straight line is a reference line with a slope of −2. The units of the variance axis are (g kg^{−1})^{2}

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Variance spectra of the 925-hPa specific humidity field. The solid curve corresponds to the direct use of periodic 2D-DFT and the dashed curve corresponds to the application of the 2D-DFT after the field has been detrended. The straight line is a reference line with a slope of −2. The units of the variance axis are (g kg^{−1})^{2}

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Variance spectra of the 925-hPa specific humidity field. The solid curve corresponds to the direct use of periodic 2D-DFT and the dashed curve corresponds to the application of the 2D-DFT after the field has been detrended. The straight line is a reference line with a slope of −2. The units of the variance axis are (g kg^{−1})^{2}

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

(a) Trend component that is removed from the humidity field (Fig. 1) to get the (b) detrended field used as input for the detrending-DFT method. The line pattern is not generated by a printing problem but by the detrending itself

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

(a) Trend component that is removed from the humidity field (Fig. 1) to get the (b) detrended field used as input for the detrending-DFT method. The line pattern is not generated by a printing problem but by the detrending itself

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

(a) Trend component that is removed from the humidity field (Fig. 1) to get the (b) detrended field used as input for the detrending-DFT method. The line pattern is not generated by a printing problem but by the detrending itself

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Examples of a 2D variance array obtained by the application of the 2D-DCT on low-level specific humidity computed from (a) the field in Fig. 1, and (b) a monthly mean of such 2D arrays. A gray scale is used to display the logarithm of the square of the spectral coefficients. The spectral indices *m* and *n* follow the abscissa and ordinate axes, respectively

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Examples of a 2D variance array obtained by the application of the 2D-DCT on low-level specific humidity computed from (a) the field in Fig. 1, and (b) a monthly mean of such 2D arrays. A gray scale is used to display the logarithm of the square of the spectral coefficients. The spectral indices *m* and *n* follow the abscissa and ordinate axes, respectively

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Examples of a 2D variance array obtained by the application of the 2D-DCT on low-level specific humidity computed from (a) the field in Fig. 1, and (b) a monthly mean of such 2D arrays. A gray scale is used to display the logarithm of the square of the spectral coefficients. The spectral indices *m* and *n* follow the abscissa and ordinate axes, respectively

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Power responses of the continuous DCT applied to *h*(*x*) = cos(*px*) [see Eq. (21)]. The responses are shown as a function of wavenumber *k* and for selected input wavenumber *p.* A line with a slope of −4 is also drawn for comparison

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Power responses of the continuous DCT applied to *h*(*x*) = cos(*px*) [see Eq. (21)]. The responses are shown as a function of wavenumber *k* and for selected input wavenumber *p.* A line with a slope of −4 is also drawn for comparison

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Power responses of the continuous DCT applied to *h*(*x*) = cos(*px*) [see Eq. (21)]. The responses are shown as a function of wavenumber *k* and for selected input wavenumber *p.* A line with a slope of −4 is also drawn for comparison

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Example of the influence of phase shifts on power spectra. The input function is *h*(*x*) = cos(*px* + *ϕ*) with wavenumber *p* = 3 and phase shifts of *ϕ* = 0, *π*/8, *π*/4, 3*π*/8 and *π*/2. A line with a slope of −4 is also drawn for comparison

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Example of the influence of phase shifts on power spectra. The input function is *h*(*x*) = cos(*px* + *ϕ*) with wavenumber *p* = 3 and phase shifts of *ϕ* = 0, *π*/8, *π*/4, 3*π*/8 and *π*/2. A line with a slope of −4 is also drawn for comparison

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Example of the influence of phase shifts on power spectra. The input function is *h*(*x*) = cos(*px* + *ϕ*) with wavenumber *p* = 3 and phase shifts of *ϕ* = 0, *π*/8, *π*/4, 3*π*/8 and *π*/2. A line with a slope of −4 is also drawn for comparison

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Variance spectra of *h*(*x*) = cos (*px* + *ϕ*). The wavenumber *p* varies from (upper left) 6 (pg = 3) to (bottom right) 8.2 (pg = 4.1). On each panel, three phase shifts *ϕ* are shown; solid curve: *ϕ* = 0, dashed curve: *ϕ* = *π*/8, dotted curve: *ϕ* = *π*/2. The straight line at the upper-right corner is a reference line with a slope of −4

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Variance spectra of *h*(*x*) = cos (*px* + *ϕ*). The wavenumber *p* varies from (upper left) 6 (pg = 3) to (bottom right) 8.2 (pg = 4.1). On each panel, three phase shifts *ϕ* are shown; solid curve: *ϕ* = 0, dashed curve: *ϕ* = *π*/8, dotted curve: *ϕ* = *π*/2. The straight line at the upper-right corner is a reference line with a slope of −4

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Variance spectra of *h*(*x*) = cos (*px* + *ϕ*). The wavenumber *p* varies from (upper left) 6 (pg = 3) to (bottom right) 8.2 (pg = 4.1). On each panel, three phase shifts *ϕ* are shown; solid curve: *ϕ* = 0, dashed curve: *ϕ* = *π*/8, dotted curve: *ϕ* = *π*/2. The straight line at the upper-right corner is a reference line with a slope of −4

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Variance spectra from the DCT and DFT of 1D fields having prescribed spectral slopes of (a) (b) 0, −1, −2 and (c) (d) −3, −4, −5. (a), (c) All cases; (b), (d) only cases with large trends (aperiodicity larger than 3 std dev). The heavy solid lines are reference slopes with values marked at the right on each plot.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Variance spectra from the DCT and DFT of 1D fields having prescribed spectral slopes of (a) (b) 0, −1, −2 and (c) (d) −3, −4, −5. (a), (c) All cases; (b), (d) only cases with large trends (aperiodicity larger than 3 std dev). The heavy solid lines are reference slopes with values marked at the right on each plot.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Variance spectra from the DCT and DFT of 1D fields having prescribed spectral slopes of (a) (b) 0, −1, −2 and (c) (d) −3, −4, −5. (a), (c) All cases; (b), (d) only cases with large trends (aperiodicity larger than 3 std dev). The heavy solid lines are reference slopes with values marked at the right on each plot.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

As in Fig. 8, but for spectra of 1D fields having prescribed spectral slopes of (a)(b) 0, +1, +2 and (c)(d) +3, +4, +5.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

As in Fig. 8, but for spectra of 1D fields having prescribed spectral slopes of (a)(b) 0, +1, +2 and (c)(d) +3, +4, +5.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

As in Fig. 8, but for spectra of 1D fields having prescribed spectral slopes of (a)(b) 0, +1, +2 and (c)(d) +3, +4, +5.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Variance spectrum of the 925-hPa humidity field (shown in Fig. 1) computed using the DCT (bold solid curve), and the DFT with and without detrending (thin dashed and solid curves, respectively). The units of the variance axis are (g kg^{−1})^{2}

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Variance spectrum of the 925-hPa humidity field (shown in Fig. 1) computed using the DCT (bold solid curve), and the DFT with and without detrending (thin dashed and solid curves, respectively). The units of the variance axis are (g kg^{−1})^{2}

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Variance spectrum of the 925-hPa humidity field (shown in Fig. 1) computed using the DCT (bold solid curve), and the DFT with and without detrending (thin dashed and solid curves, respectively). The units of the variance axis are (g kg^{−1})^{2}

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Vertically integrated kinetic energy spectra for the ECMWF global analysis of Feb 1993. Shown are the total (solid curve), stationary (long dashed curve), and transient (short dashed curve) components. The straight line is a reference line with a slope of −3. The wavenumbers *N* are the total wavenumbers of spherical harmonics. The wavelengths corresponding to the wavenumbers *N* are indicated on upper axis. Units are J m^{−2}

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Vertically integrated kinetic energy spectra for the ECMWF global analysis of Feb 1993. Shown are the total (solid curve), stationary (long dashed curve), and transient (short dashed curve) components. The straight line is a reference line with a slope of −3. The wavenumbers *N* are the total wavenumbers of spherical harmonics. The wavelengths corresponding to the wavenumbers *N* are indicated on upper axis. Units are J m^{−2}

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Vertically integrated kinetic energy spectra for the ECMWF global analysis of Feb 1993. Shown are the total (solid curve), stationary (long dashed curve), and transient (short dashed curve) components. The straight line is a reference line with a slope of −3. The wavenumbers *N* are the total wavenumbers of spherical harmonics. The wavelengths corresponding to the wavenumbers *N* are indicated on upper axis. Units are J m^{−2}

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Vertically integrated kinetic energy spectra computed using the 2D-DCT over a limited area on (a) ECMWF analysis data and (b) simulated data produced by the CRCM. Both are shown on the same plot in (c) but for the total component only. The straight line is a reference line with a slope of −3. Units are J m^{−2}

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Vertically integrated kinetic energy spectra computed using the 2D-DCT over a limited area on (a) ECMWF analysis data and (b) simulated data produced by the CRCM. Both are shown on the same plot in (c) but for the total component only. The straight line is a reference line with a slope of −3. Units are J m^{−2}

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Vertically integrated kinetic energy spectra computed using the 2D-DCT over a limited area on (a) ECMWF analysis data and (b) simulated data produced by the CRCM. Both are shown on the same plot in (c) but for the total component only. The straight line is a reference line with a slope of −3. Units are J m^{−2}

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Response of the spectral filter used for low-pass filtering of the field shown in Fig. 1

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Response of the spectral filter used for low-pass filtering of the field shown in Fig. 1

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Response of the spectral filter used for low-pass filtering of the field shown in Fig. 1

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Band-pass filtering of the humidity field of Fig. 1 for (a) large and synoptic scales, and (b) mesoscales. Units are g kg^{−1}. (N.B., the color scales of the two panels are different)

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Band-pass filtering of the humidity field of Fig. 1 for (a) large and synoptic scales, and (b) mesoscales. Units are g kg^{−1}. (N.B., the color scales of the two panels are different)

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Band-pass filtering of the humidity field of Fig. 1 for (a) large and synoptic scales, and (b) mesoscales. Units are g kg^{−1}. (N.B., the color scales of the two panels are different)

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Monthly mean of mean sea level pressure. Contours are at 2-hPa intervals. Areas with values smaller than 1010 hPa are shaded.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Monthly mean of mean sea level pressure. Contours are at 2-hPa intervals. Areas with values smaller than 1010 hPa are shaded.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Monthly mean of mean sea level pressure. Contours are at 2-hPa intervals. Areas with values smaller than 1010 hPa are shaded.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Small-scale component of the monthly mean of mean sea level pressure (a) without and (b) with boundary smoothing before the 2D-DCT application. Contour intervals are every 0.1 hPa. The zero contour is omitted. Regions with values smaller than −0.1 hPa are shaded.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Small-scale component of the monthly mean of mean sea level pressure (a) without and (b) with boundary smoothing before the 2D-DCT application. Contour intervals are every 0.1 hPa. The zero contour is omitted. Regions with values smaller than −0.1 hPa are shaded.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Small-scale component of the monthly mean of mean sea level pressure (a) without and (b) with boundary smoothing before the 2D-DCT application. Contour intervals are every 0.1 hPa. The zero contour is omitted. Regions with values smaller than −0.1 hPa are shaded.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Fig. A1. New function *f*_{s}(*i*) created by symmetrization of the function *f*(*i*). In this example the original function *f*(*i*) had eight grid points (on the right-hand side)

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Fig. A1. New function *f*_{s}(*i*) created by symmetrization of the function *f*(*i*). In this example the original function *f*(*i*) had eight grid points (on the right-hand side)

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Fig. A1. New function *f*_{s}(*i*) created by symmetrization of the function *f*(*i*). In this example the original function *f*(*i*) had eight grid points (on the right-hand side)

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Fig. A2. The possible modes of the cosine basis functions with eight grid points. The gray bars represent the discrete form of Eq. (A7), that is, the values at grid points *i* = 0 … *N* − 1 with *N* = 8. For comparison, their continuous forms are shown by the thin curves.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Fig. A2. The possible modes of the cosine basis functions with eight grid points. The gray bars represent the discrete form of Eq. (A7), that is, the values at grid points *i* = 0 … *N* − 1 with *N* = 8. For comparison, their continuous forms are shown by the thin curves.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2

Fig. A2. The possible modes of the cosine basis functions with eight grid points. The gray bars represent the discrete form of Eq. (A7), that is, the values at grid points *i* = 0 … *N* − 1 with *N* = 8. For comparison, their continuous forms are shown by the thin curves.

Citation: Monthly Weather Review 130, 7; 10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2