#### 2.1. Multiresolution Analysis Based on Orthonormal Wavelet Bases

[9] The aspects of wavelet theory that are relevant to the work presented in this paper are introduced in this section. A more thorough coverage on this topic may be found in the work of *Daubechies* [1988], and *Mallat* [1989], among many others. *Percival and Walden* [2000] also provide an extensive introduction to the subject and some of the applications of wavelet-based methods in time series analysis.

[10] Let *f*(*x*, *y*) denote an image or two-dimensional field having *N* × *N* square pixels, each with side length ℓ. Two-dimensional orthonormal wavelet transforms provide bases for decomposing the information imbedded in *f*(*x*, *y*) in terms of a coarse-scale or mean field approximation plus fluctuations about this mean level in the horizontal, vertical and diagonal directions at various resolutions. Given 2^{J} ≤ *N* < 2^{J+1} for some integer *J* ≥ 0, the coarsest scale approximation or mean level is given at spatial scale 2^{J} ℓ, and it is indexed by *j* = *J*. The fluctuations or details about the mean level are given at progressively finer spatial resolutions 2^{j−1} ℓ, for *j* = *J*, …, 1. Specifically, the fluctuations indexed by *j*, denoted *D*_{j}*f*(*x*, *y*), represent the new details of *f*(*x*, *y*) that become available as we move from spatial scale 2^{j} ℓ to finer scale 2^{j−1} ℓ, and they may be associated to spatial resolution 2^{j−1} ℓ.

[11] Let *A*_{j}*f*(*x*, *y*) denote the approximation or expansion of *f*(*x*, *y*) in terms of orthonormal wavelet bases at scale 2^{j} ℓ. Such a representation at the finest scale (ℓ = 2^{j=0} ℓ) is given by:

In (1), *n* and *m* index positions or spatial locations within *f*(*x*, *y*), and 〈·, ·〉 denotes the inner product in the *L*^{2} norm [*Boggess and Narcowich*, 2001]. *D*_{j}*f*(*x*, *y*) is composed of directional fluctuations or details along the vertical (*k* = 1), horizontal (*k* = 2), and diagonal (*k* = 3) directions, which are expressed in terms of the so-called wavelet functions in each of these directions, Ψ_{j,n,m}^{k}(*x*, *y*) for *k* = 1, 2, 3. The {*d*_{j,n,m}^{k}} coefficients are called detail or wavelet coefficients and capture the fluctuations or incremental amount of information contained in *f*(*x*, *y*) along direction *k*, at spatial scale 2^{j−1} ℓ, and at the spatial location with indices *n* and *m*.

[12] The discrete domain implementation of wavelet transforms is given in terms of real-valued finite filters [*Mallat*, 1989]. In particular, the wavelet coefficients at scale *j* are obtained by filtering the approximation image at scale *j* − 1, *A*_{j−1}*f*(*x*, *y*), with a bandpass filter with nominal bandpass range [2^{−(j+1)}, 2^{−j}] [*Percival and Walden*, 2000], and then downsampling. *Daubechies* [1988] showed that it is possible to define real-valued wavelet functions with arbitrary degrees of differentiability and maximum number of vanishing moments. Loosely speaking, a function is said to have *R* vanishing moments if it annihilates polynomial trends of order less than or equal to *R* − 1 [*Bogges and Narcowich*, 2001].

[13] In this study, we focus on the family of wavelet functions known as Daubechies wavelets. A Daubechies wavelet filter of order *R* is such that it has 2*R* × 2*R* nonzero terms, *R* vanishing moments, and approximately *R*/*5* continuous derivatives for large *R* [*Bogges and Narcowich*, 2001]. A tradeoff exists in improving the localization properties of wavelet filters in the space and frequency domains. A higher-order Daubechies filter provides better localization in the frequency domain at the expense of deteriorating localization in the space domain.

#### 2.3. Long Memory 1/*f* Processes and Their Wavelet Representation

[16] In this section, we introduce long memory 1/*f* processes for characterizing self-affine stochastic fields. We also show that the second-order moments of the wavelet coefficients from long memory 1/*f* processes exhibit scaling, which constitutes the foundation for our proposed downscaling scheme. The material presented in this section is largely based on and motivated by the work of *Wornell* [1993] and *Wornell and Oppenheim* [1990, 1992a, 1992b]. *Beran* [1994] and *Percival and Walden* [2000] provide a comprehensive review of the topics presented here as well as a vast number of references on them. The work of *Brillinger* [e.g., 1974, 2001] constitutes an excellent source on time series analysis concepts and their frequency domain implementation.

[17] Consider a two-dimensional image, *f*(*x*, *y*), having *N* × *N* pixels, with 2^{J} ≤ *N* < 2^{J+1} as in section 2.1. The power spectrum of *f*(*x*, *y*) is given by the Fourier transform (denoted by “ˆ”) of its autocovariance function, *c*_{ff}(*u*, *v*):

In (3), *i* = , (*u*, *v*) index lags in the horizontal (*x*) and vertical (*y*) directions, respectively, and (α, β) is referred to as wave number. It can be shown that the integral of (3) equals the variance of *f*(*x*, *y*), which we represent as σ_{ff}^{2} [*Brillinger*, 2001]. Let “¯” and superscript “^{T}” denote complex conjugation and matrix transpose, respectively. An estimate of the power spectrum is provided by the periodogram [*Brillinger*, 2001]:

where

The periodogram is always greater than or equal to zero [*Brillinger*, 2001]. It is also even and has period *N*. Therefore it suffices to display it for α < 0.5 and β < 0.5 in (4d). An improved estimate of the power spectrum is given by the smoothed periodogram, which is obtained by applying a moving average filter to the estimate provided in (4a) [*Brillinger*, 2001].

[18] A long memory 1/*f* process is characterized by having a power spectrum that goes to infinity as the wave number approaches zero [*Beran*, 1994; *Percival and Walden*, 2000] as:

where *C* > 0, λ ≤ 0, γ ≤ 0, and ∣·∣ connotes absolute value. This in turn implies that the autocovariance function of such a process must approach zero at a hyperbolic rate as the separation distance increases, i.e.,

where *C*_{c} > 0, and η = −λ − 1, ς = −γ − 1 for −1 < λ < 0 and −1 < γ < 0. Thus observations separated by large lags may still display nonnegligible dependence and the process is said to exhibit long memory. Standard time series models, such as autoregressive moving average (ARMA) processes [*Brillinger*, 2001; *Shumway and Stoffer*, 2001], and models commonly used in geostatistics, such as exponential models, have autocovariance functions that approach zero much faster (exponentially) as the separation distance approaches infinity [*Beran*, 1994].

[20] Since the multiresolution analysis based on wavelet theory given in (1) is constructed from orthonormal bases, Parseval's theorem [*Percival and Walden*, 2000; *Boggess and Narcowich*, 2001] tells us that the wavelet representation of *f*(*x*, *y*) preserves its energy. Hence from (1):

where ∥ · ∥ denotes distance in the *L*^{2} norm [*Boggess and Narcowich*, 2001]. On the basis of (7), a decomposition of the variance of *f*(*x*, *y*) across scales is given by:

where σ_{j,k}^{2} denotes the variance of the wavelet coefficients associated with Ψ_{j,m,n}^{k} (see (1)). Since the wavelet coefficients at scale *j* are related to a bandpass filter with nominal bandpass range [2^{−(j+1)}, 2^{−j}] and the integral of the power spectrum equals the variance of *f*(*x*, *y*), we may relate the power spectrum of a long memory 1/*f* process to the variance of the wavelet coefficients by equating the integral over (5) to (8):

[21] Expression (9) describes the evolution with scale of the sum of the variances of the wavelet coefficients for the horizontal, vertical, and diagonal directions. To obtain a relationship like the above for the wavelet coefficients associated to each of the wavelet functions in (1), namely Ψ^{k}(*x*, *y*) for *k* = 1, 2, 3, we make a closure hypothesis and let the variance of the coefficients for each direction scale as:

(10) is consistent with (9). Moreover, the evolution across scales of the variances of the horizontal, vertical, and diagonal wavelet coefficients for the radiobrightness temperature observations retrieved during SGP97 and used in this study seem to abide to the hypothesis made (refer to the discussion in section 3.2). In the work by *Perica and Foufoula-Georgiou* [1996a], the normalized wavelet coefficients corresponding to fields of rainfall intensity also display variances that evolve in a similar manner for all directions.

[22] If we take the natural logarithm of (10) and simplify, we obtain:

where β now denotes the frequency associated to the coefficients at scale *j*. We also designate λ + γ as the *scaling exponent* corresponding to the wavelet coefficients. It can be further shown that the autocorrelation structure of the wavelet coefficients from long memory 1/*f* processes is stationary and scale invariant provided the wavelet bases chosen for analysis have more vanishing moments than the maximum of ∣λ/2∣ and ∣γ/2∣ [*Flandrin*, 1992; *Tewfik and Kim*, 1992; *Wornell*, 1993; *Percival and Walden*, 2000]. According to the notation utilized in section 2.1, the condition imposed on the number of vanishing moments corresponds to utilizing a Daubechies wavelet of order *R* > *max* (∣λ∣, ∣γ∣)/2, where *max*(·) connotes the maximum of all arguments in the parenthesis.

[23] Expression (11) and the discussion following it indicate that the wavelet coefficients corresponding to the vertical, horizontal, and diagonal directions exhibit scaling of second-order moments at coarse scales as summarized below:

where cov_{j,k}(*u*_{j}, *v*_{j}) represents the autocovariance of the wavelet coefficients {*d*_{j,n,m}^{k}} (see (1)) at horizontal and vertical lags *u*_{j} and *v*_{j}, respectively. This implies that as long as (11) holds, it suffices to characterize the wavelet coefficients at a single resolution to infer the second-order statistics of the wavelet coefficients at any other scale. Particularly, it suggests the possibility of generating wavelet coefficients for scales finer than the observational one (i.e., for *j* ≤ 0 in (1)) and hence of obtaining approximations of *f*(*x*, *y*) at subgrid scales.

[24] Regarding the autocorrelation structure of wavelet coefficients for long memory 1/*f* processes obtained via Daubechies wavelets having *R* > max (∣λ∣, ∣γ∣), *Wornell* [1993] showed that, for any given scale, these are not necessarily uncorrelated, but display weak dependence, in contrast to the high autocorrelation exposed by the original image *f*(*x*, *y*). This implies that low parameter ARMA models [*Brillinger*, 2001; *Shumway and Stoffer*, 2001] may be used to describe the spatial dependence of the wavelet coefficients from long memory 1/*f* processes. Correlation across scales can also be shown to exist [*Wornell*, 1993]. This arises because wavelet filters do not constitute ideal bandpass filters. However, the correlation across scales decreases quickly at a rate dictated by the number of vanishing moments of wavelet bases. Recall that higher-order Daubechies wavelets have improved frequency domain localization, and hence they approach in behavior to ideal bandpass filters. These results were also derived by *Flandrin* [1992] and *Tewfik and Kim* [1992] in the context of fractional Brownian motion.

#### 2.4. Proposed Downscaling Framework

[25] Figure 1 shows a pictorial representation of the proposed downscaling framework. This scheme takes as inputs: (1) an approximation to the process of interest at scale *J*, denoted *A*_{J}*f*(*x*, *y*); (2) an estimate of λ + γ, which may be obtained as outlined in section 3.2.3 and further described by *Parada and Liang* [2003b]; and (3) the ARMA models fitted to the horizontal (*C*_{H} (*x*, *y*)), vertical (*C*_{V} (*x*, *y*)), and diagonal (*C*_{D} (*x*, *y*)) wavelet coefficients at a single scale, denoted *ARMA*_{H}, *ARMA*_{V}, and *ARMA*_{D}, respectively. Note that pure white noise, autoregressive (AR), and moving average (MA) models are considered special cases of ARMA models. In a downscaling setting, *J* indexes the resolution of coarse scale observations from a long memory 1/*f* process (e.g., radiobrightness temperature), and *A*_{J}*f*(*x*, *y*) denotes the observation for a single such pixel.

[26] Images of independent identically distributed (i.i.d.) Gaussian noise with mean zero and variances in agreement with expression (11) are used as building blocks for the horizontal (subscript *H*), vertical (subscript *V*), and diagonal (subscript *D*) wavelet coefficients at scales *j* = *J*, *J* − 1, …,1. Specifically, the i.i.d. input fields corresponding to the wavelet coefficients at all resolutions are generated such that their variances show a power law increase with increasing scale (i.e., increasing *j*) dictated by ∣λ + γ∣ as given in equation (12a). These i.i.d. input fields are denoted as ξ_{H} (*x*, *y*), ξ_{V} (*x*, *y*), and ξ_{D} (*x*, *y*), respectively. ARMA filters, which correspond to processes having exponentially decaying autocorrelation structure, are applied to the i.i.d. input fields so as to capture the spatial autocorrelation structure of the horizontal, vertical, and diagonal wavelet coefficients as needed. Optimal ARMA filters may be designed as described in section 3.2.2 and need only be fitted to a single level of wavelet coefficients for each direction. By virtue of the scaling properties of the wavelet coefficients discussed in section 2.3 (see (11) and (12)) and as long as these are valid, the same filters may be applied to the i.i.d. fields at all desired scales. Finally, (1) is used to produce a sample image realization of the long memory 1/*f* process at scale *j* = 0 possessing the desired long memory characteristics. It can be shown that within this framework all subgrid pixels in a realization have ensemble mean values equal to *A*_{J}*f*(*x*, *y*), and thus the coarse-scale approximation or mean field is preserved [*Percival and Walden*, 2000]. Note that the key implicit assumption within our downscaling scheme is that (11) holds for the wavelet coefficients at scales *j* = *J*, *J* − 1, …,1. Moreover, the proposed downscaling framework can easily accommodate wave number regimes with distinct λ + γ values as long as estimates of these can be provided.

[27] The proposed downscaling scheme for long memory 1/*f* processes has several conceptual, theoretical, and practical advantages. Intuitively, the representation of processes that exhibit dependence on scale provided by orthonormal wavelets is natural since it is based decomposing the information imbedded in them that becomes available as we move from coarse to fine resolutions. Practically, orthonormal wavelets yield a convenient transformation of long memory 1/*f* processes such that we may rely on the assumptions of stationarity and weak autocorrelation to characterize them via their wavelet coefficients. These assumptions enable us to use many powerful statistical tools that are not directly applicable to long memory 1/*f* processes, such standard time series (e.g., ARMA) models or geostatistical techniques. Above all, the proposed downscaling framework allows us to capture not only the variance of long memory 1/*f* processes, but also their overall structure of spatial dependence.