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Keywords:

  • downscaling;
  • subgrid variability;
  • brightness temperature

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Downscaling Framework
  5. 3. Application to L Band Microwave Radiobrightness Temperature Imagery
  6. 4. Results
  7. 5. Conclusions and Future Work
  8. Acknowledgments
  9. References
  10. Supporting Information

[1] In this paper we introduce a general downscaling framework and apply it to L band microwave radiobrightness temperature fields retrieved from electronically scanned thinned array radiometer (ESTAR). The gist of the downscaling scheme presented in this paper is the statistical characterization of scale-invariant properties of the wavelet coefficients or fluctuations from long memory 1/f processes. We test the proposed downscaling framework with the radiobrightness temperature images collected during the Southern Great Plains hydrology experiment of 1997. We produce realizations of radiobrightness temperature at 800-m resolution given a mean-area value at approximately 30-km resolution (the near-future expected operational scale). The results obtained evince that the proposed downscaling methodology is capable of accurately preserving the variability and overall structure of spatial dependence of the observed radiobrightness temperature fields.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Downscaling Framework
  5. 3. Application to L Band Microwave Radiobrightness Temperature Imagery
  6. 4. Results
  7. 5. Conclusions and Future Work
  8. Acknowledgments
  9. References
  10. Supporting Information

[2] Radiobrightness temperature observations retrieved from electronically scanned thinned array radiometer (ESTAR) are of vital importance for validation of radiation transfer models, as well as to derive fields of volumetric soil moisture content for the top 5 centimeters of soil [Jackson et al., 1995]. Soil moisture has a significant impact on processes occurring at a wide range of scales, such as ecosystems, hydrologic, and land-atmosphere dynamics. At the scale of soil pores, for instance, drier soil particles exert increasingly higher tension forces on water thus limiting its availability to plants and bio-organisms residing in the soil column. Soil moisture is also a key determinant in the partitioning of incoming radiation into latent, sensible, and ground heat fluxes, which must be characterized for the joint modeling of water and energy budgets from the canopy to continental scales. Hence there is much current interest in assimilating radiobrightness temperature or the derived near-surface soil moisture images into land-surface models to attempt to improve their predictability [e.g., Walker and Houser, 2001; Reichle et al., 2002, Crow and Wood, 2003; Parada and Liang, 2003a], and, as a result, to increase the accuracy of forecasts from numerical climate models (http://ldas.gsfc.nasa.gov).

[3] The National Aeronautics and Space Administration (NASA) and the U.S. Department of Agriculture (USDA) have jointly sponsored several large-scale hydrologic field measurement campaigns, in which L band microwave radiobrightness temperature observations (1.4 GHz) were collected by flying an ESTAR on board an airplane over the Little Washita watershed and surrounding areas. These include Washita '92 [Jackson et al., 1995] and the Southern Great Plains hydrology experiment of 1997 (SGP97) [Jackson et al., 1999]. While the final product resolutions of the radiobrightness temperature fields retrieved during Washita '92 and SGP97 are 200 m and 800 m, respectively, it is expected that the near-future spaceborne operational resolution of the observations will be on the order of 10 to 30 km [Jackson et al., 1999].

[4] Significant efforts have been devoted in the past to characterize the statistical structure and evolution across scales of the near-surface soil moisture fields derived from ESTAR and other L band remote sensing sources. The motivation underlying these efforts resides in the highly nonlinear parameterizations of land-surface schemes with respect to soil moisture [e.g., Nykanen and Foufoula-Georgiou, 2001; Crow and Wood, 2002] and hence in the need to describe subpixel variability of this state. In this context, numerous exploratory data analyses have reported that near-surface soil moisture images derived from ESTAR measurements and other remote sensing sources possess scaling properties over resolutions ranging from 800 m to within 10 km [Rodriguez-Iturbe et al., 1995; Hu et al., 1997; Cosh and Brutsaert, 1999; Crow et al., 2000; Oldak et al., 2002, and references therein]. While near-surface soil moisture images were found to exhibit multifractal scaling [e.g., Hu et al., 1997; Oldak et al., 2002], the wavelet coefficients of near-surface soil moisture were reported to behave as monofractals or to exhibit simple scaling [Hu et al., 1998].

[5] Various researchers [e.g., Kumar, 1999; Crow and Wood, 2002] have proposed and applied downscaling methodologies for near-surface soil moisture fields. For instance, Crow and Wood [2002] used a simple downscaling scheme to evaluate the impact of subgrid variability of near-surface soil moisture in land-surface model predictions of energy fluxes. These researchers generated spatially uncorrelated subgrid-scale near-surface soil moisture fields having variances in accordance to simple scaling relationships. This approach is capable of capturing the subpixel variability of near-surface soil moisture fields. However, it cannot preserve their spatial dependence. Nonetheless, Crow and Wood [2002] reported that significant improvements in predictions of latent heat flux at 32-km resolution could result from the representation of subgrid variability of near-surface soil moisture. On the other hand, Kumar [1999] employed a multiscale state-space-model based downscaling scheme for the near-surface soil moisture fields retrieved from ESTAR during Washita '92. The approach proposed by Kumar relies on the assumption that the spatial variability of near-surface soil moisture fields is strongly determined by the spatial distribution of soil texture classes, and on the existence of a linear relationship between mean-area near-surface soil moisture content and available water content for a given soil texture class. The former assumption has been shown to be appropriate for the near-surface soil moisture fields derived during Washita '92, which correspond to a 9-day drying period following approximately uniform precipitation over the entire study region [Jackson et al., 1995; Oldak et al., 2002]. Nonetheless, the near-surface soil moisture images corresponding to SGP97 show a stronger dependence on the spatial variability of precipitation and a less pronounced dependence on soil texture [Jackson et al., 1999; Oldak et al., 2002]. In section 3.2.4, we additionally provide a brief comparison to the work of Perica and Foufoula-Georgiou [1996a, 1996b], who developed a wavelet-based downscaling scheme for rainfall intensity fields.

[6] In this paper, we introduce a general downscaling framework for long memory 1/f processes and apply it to L band microwave radiobrightness temperature images retrieved from ESTAR. The proposed downscaling methodology may also be applicable to the near-surface soil moisture observations derived from the radiobrightness temperature fields under consideration. The gist of our downscaling scheme is the statistical characterization of scale-invariant properties of the wavelet coefficients or fluctuations from long memory 1/f processes. We test the proposed downscaling framework with the radiobrightness temperature images retrieved during SGP97 by producing realizations of these at 800-m resolution given a mean-area value at approximately 30-km resolution (the near-future expected operational scale), and evaluating how well the realizations reproduce the variability and spatial dependence of the corresponding observations.

[7] This paper is organized as follows. Section 2 formulates the proposed downscaling methodology for long memory 1/f processes in full generality and summarizes its underlying statistical and mathematical foundations. Section 3 statistically characterizes the radiobrightness temperature fields derived from ESTAR during SGP97 as long memory 1/f processes and describes the application of the proposed downscaling scheme to these fields. Section 4 presents the results obtained. Finally, section 5 provides our primary conclusions.

2. Downscaling Framework

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Downscaling Framework
  5. 3. Application to L Band Microwave Radiobrightness Temperature Imagery
  6. 4. Results
  7. 5. Conclusions and Future Work
  8. Acknowledgments
  9. References
  10. Supporting Information

[8] In this section, we present a downscaling framework for long memory 1/f processes (e.g., radiobrightness temperature), which relies on the statistical characterization of scale-invariant properties of the wavelet coefficients or fluctuations from these fields. The underlying statistical and mathematical foundations are briefly summarized and relevant references are provided. In section 2.1, we concisely introduce the key concepts from multiresolution analysis based on orthonormal wavelet bases. In section 2.2, we define statistical self-affinity. Section 2.3 introduces long memory 1/f processes, their statistical properties, and the salient features of their multiresolution analysis based on wavelet transforms. Ultimately, section 2.4 describes the proposed downscaling framework for long memory 1/f processes in full generality.

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 2JN < 2J+1 for some integer J ≥ 0, the coarsest scale approximation or mean level is given at spatial scale 2J ℓ, and it is indexed by j = J. The fluctuations or details about the mean level are given at progressively finer spatial resolutions 2j−1 ℓ, for j = J, …, 1. Specifically, the fluctuations indexed by j, denoted Djf(x, y), represent the new details of f(x, y) that become available as we move from spatial scale 2j ℓ to finer scale 2j−1 ℓ, and they may be associated to spatial resolution 2j−1 ℓ.

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

  • equation image
  • equation image
  • equation image

In (1), n and m index positions or spatial locations within f(x, y), and 〈·, ·〉 denotes the inner product in the L2 norm [Boggess and Narcowich, 2001]. Djf(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, Ψj,n,mk(x, y) for k = 1, 2, 3. The {dj,n,mk} 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 2j−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, Aj−1f(x, y), with a bandpass filter with nominal bandpass range [2−(j+1), 2j] [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 2R × 2R 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.

2.2. Brief Review of Self-Affinity

[14] As noted in the work of Parada and Liang [2003b], the term “self-affine” is a neologism introduced to designate processes also known as “self-similar” [Mandelbrot, 2000]. The former and more recent terminology explicitly recognizes the possibility of anisotropic scaling and it is adopted in this paper. Nonetheless, many of our references adhere to the use of the term “self-similar” instead of “self-affine”.

[15] For ease of notation, the definition of self-affinity is provided for the one-dimensional case. Its extension to two dimensions follows naturally from that for one dimension. Beran [1994] and references therein give a more complete discussion on self-affinity than provided here, as well as references on the subject. Let S(x) denote a one-dimensional stochastic process. S(x) is said to be statistically self-affine in strict sense if for any τ > 0:

  • equation image

where “equation image” denotes equality in distribution, and H is the so-called Hurst or self-affinity exponent [Hurst, 1951; Beran, 1994]. The term fractal is often used to denote self-affine processes [Mandelbrot, 2000]. Wide sense self-affinity, on the other hand, applies only up to second-order moments.

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 2JN < 2J+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, cff(u, v):

  • equation image

In (3), i = equation image, (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 σff2 [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]:

  • equation image

where

  • equation image
  • equation image
  • equation image

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:

  • equation image

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.,

  • equation image

where Cc > 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].

[19] Expressions (5) and (6) evince that long memory 1/f processes are statistically self-affine in wide sense since equation image(Aα, Bβ) ∝ ∣AλBλequation image(α, β) as α, β [RIGHTWARDS ARROW] 0 [Beran, 1994; Percival and Walden, 2000]. In particular, the power spectrum in (5) is self-affine in both α and β, which implies that the autocovariance function is allowed to approach zero at different rates in the x and y directions. The work by Parada and Liang [2003b] provides a more thorough description of long memory 1/f processes than intended here. It cites areas in which such models commonly arise, and discusses a methodology for inference of statistical parameters.

[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):

  • equation image

where ∥ · ∥ denotes distance in the L2 norm [Boggess and Narcowich, 2001]. On the basis of (7), a decomposition of the variance of f(x, y) across scales is given by:

  • equation image

where σj,k2 denotes the variance of the wavelet coefficients associated with Ψj,m,nk (see (1)). Since the wavelet coefficients at scale j are related to a bandpass filter with nominal bandpass range [2−(j+1), 2j] 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):

  • equation image

[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 Ψk(x, y) for k = 1, 2, 3, we make a closure hypothesis and let the variance of the coefficients for each direction scale as:

  • equation image

(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:

  • equation image

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:

  • equation image
  • equation image

where covj,k(uj, vj) represents the autocovariance of the wavelet coefficients {dj,n,mk} (see (1)) at horizontal and vertical lags uj and vj, 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 AJf(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 (CH (x, y)), vertical (CV (x, y)), and diagonal (CD (x, y)) wavelet coefficients at a single scale, denoted ARMAH, ARMAV, and ARMAD, 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 AJf(x, y) denotes the observation for a single such pixel.

image

Figure 1. Schematic of downscaling methodology. Ajf(x, y) denotes a coarse scale approximation for radiobrightness temperature. ξH(x, y), ξV(x, y), and ξD(x, y) connote i.i.d. Gaussian fields used to generate the horizontal, vertical and diagonal wavelet coefficients at scales j = J, J − 1, …, 1. These i.i.d. input fields are such that their variances show a power law increase with increasing scale dictated by ∣λ + γ∣ (see (12a)). ARMAH, ARMAV, and ARMAD represent the ARMA models for the horizontal (CH(x, y)), vertical (CV(x, y)), and diagonal (CD(x, y)) wavelet coefficients, respectively. Realizations are obtained via (1).

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[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 ξH (x, y), ξV (x, y), and ξD (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 AJf(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.

3. Application to L Band Microwave Radiobrightness Temperature Imagery

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Downscaling Framework
  5. 3. Application to L Band Microwave Radiobrightness Temperature Imagery
  6. 4. Results
  7. 5. Conclusions and Future Work
  8. Acknowledgments
  9. References
  10. Supporting Information

[28] In this section, we start by providing a brief description of the study region and data sources from SGP97 that are relevant to this investigation. Then, we proceed to demonstrate that the L band microwave radiobrightness temperature observations retrieved from ESTAR during SGP97 may be statistically characterized as long memory 1/f processes and describe how the downscaling scheme presented in section 2.4 may be applied to these fields.

3.1. Study Region and Data Sources

[29] The SGP97 hydrology experiment is described by Jackson et al. [1999]. The study region is located in a subhumid setting in Oklahoma and extends over approximately 10,000 km2. This location was selected because of its gentle topography, moderate vegetation cover, and remarkable data collection networks for measurement of meteorological and hydrologic variables, such as precipitation, temperature, wind speed, solar radiation, and soil moisture. Typical land conditions for the months of the experiment are a mixture of pasture grasses and senesced or recently harvested winter wheat [Wickel et al., 2001].

[30] Radiobrightness temperature images were retrieved using a L band electronically scanned thinned array radiometer (ESTAR) [Jackson et al., 1995, 1999]. These images were taken at an original footprint of 400 meters for 18 days included in the period from 16 June through 17 July1997. During this time span, three drying periods took place following main rainfall events: 18–20 June, 1–3 July, and 12–14 July [Jackson et al., 1999; Oldak et al., 2002]. Each radiobrightness temperature image was gridded to a final product resolution of 800 meters.

3.2. Statistical Characterization of Radiobrightness Temperature Images

[31] In this section, we show that the downscaling framework developed in section 2 is applicable to the radiobrightness temperature images retrieved from ESTAR during SGP97. Specifically, we focus on the nine 800-m resolution radiobrightness temperature images taken during the three main drying periods of SGP97 described in section 3.1. Because the use of wavelet analysis as applied in this study requires that the horizontal and vertical dimensions of the image to be analyzed be the same, five to nine subregions were selected from each of these images. These subregions were identified based on three criteria: maximizing the dimensions of the subregions, choosing subregions with no missing values, and having at least a section of 32 × 32 nonoverlapping pixels in each subregion (approximately 30-km resolution as in the expected operational scale). Table 1 lists the number of pixels within all subregions chosen for analysis for the nine days of interest.

Table 1. Number of 800-m by 800-m Pixels Within All Subregions in the Images for the Dates Analyzeda
 Subregion
123456789
  • a

    NA denotes not available. Subregions with the same numbers in distinct images do not necessarily correspond to the same geographic location.

18 June522522532412422NANANANA
19 June522532532532532542NANANA
20 June532532532532532532542NANA
1 July532532532562552552552542542
2 July532532532542552552542552552
3 July522532532552552552552542NA
12 July532532532542562552542NANA
13 July542532542542552552552552542
14 July532582472562562552562542NA

[32] The radiobrightness temperature images for all subregions are observed to behave as long memory 1/f processes. They are characterized by slowly vanishing autocorrelation structures at large separation distances in the vertical, the horizontal, or both directions and by periodograms that tend to infinity as the wave numbers approach zero as in (5). Consequently, linear relations apply between the logarithm of the variance of wavelet coefficients for all directions and the logarithm of scale. To illustrate these points for a typical subregion, Figure 2 displays the horizontal and vertical autocorrelation matrices as well as the smoothed periodogram for subregion 1 in the image for 20 June 1997. Figure 3 depicts log-log plots of variance versus scale for the horizontal, vertical, and diagonal wavelet coefficients for the same subregion and date. These statements are in agreement with the findings of previous studies that report the presence of scaling properties in the near-surface soil moisture fields derived from radiobrightness temperature images [Rodriguez-Iturbe et al., 1995; Hu et al., 1997; Oldak et al., 2002; and references therein] or in the corresponding wavelet coefficients [Hu et al., 1998].

image

Figure 2. (top) Horizontal correlation matrix, (middle) vertical correlation matrix, and (bottom) periodogram for subregion 1 in the image for 20 June 1997. For correlation matrices, values along the main diagonal denote correlation at lag zero. Values along diagonals offset by u from the main diagonal denote correlation at lag u.

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image

Figure 3. Scaling relations for the (top) horizontal, (middle) vertical, and (bottom) diagonal wavelet coefficients corresponding to subregion 1 in the image for 20 June 1997. The wavelet coefficients at scale j may be associated to spatial resolution 2j−1 × ℓ, where ℓ = 800 m. The solid black line was obtained by least squares regression. The dashed black lines provide 95% confidence limits for the linear fit.

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[33] The remainder of this section is devoted to the statistical characterization of the wavelet coefficients obtained for each of the subregions in Table 1. Section 3.2.1 justifies our choice of wavelet basis, and provides arguments in favor of the assumptions of stationarity and normality of the wavelet coefficients. Section 3.2.2 describes the autocorrelation structure of the wavelet coefficients and provides optimal models to represent it. Section 3.2.3 outlines two approaches for determining the scaling exponent (λ + γ) for the wavelet coefficients. Lastly, section 3.2.4 summarizes how the downscaling scheme presented in section 2.4 is applied to the radiobrightness temperature observations of interest.

3.2.1. On the Choice of Wavelet Basis, and the Assumptions of Stationarity and Normality of the Resulting Wavelet Coefficients

[34] The Daubechies wavelet basis of order 2 was chosen upon careful consideration of various factors. It was required to select a wavelet basis such that the resulting wavelet coefficients were stationary and weakly correlated, which translate to utilizing a Daubechies wavelet of order R > max(∣λ∣, ∣γ∣)/2 as discussed in section 2.3. On the other hand, higher-order Daubechies filters have longer supports. This increases the number of wavelet coefficients affected by boundary conditions, which must be discarded to prevent introducing bias into the analysis [Percival and Walden, 2000]. Consequently, it was of interest to select a wavelet filter with the smallest possible support such that it yielded wavelet coefficients being approximately stationary and weakly correlated. To accomplish this goal, we carefully examined running means and standard deviations as well as periodograms for the wavelet coefficients at scale 1 (for j = 1 in section 2.1) from Daubechies wavelets of orders 1 (Haar), 2 and 3 for all subregions in Table 1. From these, it was concluded that the wavelet coefficients of the radiobrightness temperature observations of interest obtained with Daubechies filters of orders 2 and 3 allow for the assumption of stationarity to be made with reasonable confidence and appear to capture and display the same patterns. This was not found to hold for the Haar wavelet basis. Thus a Daubechies wavelet of order 2 was chosen for analysis. A more objective justification for this choice is given in section 3.2.3.

[35] Prior to performing any statistical analysis, outliers were identified by applying the Dixon test [Kanji, 1993] to every row and column of the images for wavelet coefficients at scale j = 1 in the horizontal, vertical and diagonal directions, and replaced by local averages over neighboring points. On average, 2.7 percent of the horizontal, 3.5 percent of the vertical, and 3.7 percent of the diagonal wavelet coefficients at scale 1 were identified as outliers for all subregions. The Dixon test assumes that the underlying population is normal. This hypothesis is found to be reasonable at the 0.05 level of significance for the horizontal, vertical, and diagonal wavelet coefficients at scale j = 1 corresponding to 87 percent, 83 percent, and 87 percent of all 68 subregions in Table 1 by applying the Lilliefors test for normality [Conover, 1980]. Figure 4 visually illustrates the goodness of fit of the normality assumption for the horizontal, vertical, and diagonal wavelet coefficients at scale 1 corresponding subregion 1 in the image for 20 June 1997. On the basis of these results, we subsequently assume that the wavelet coefficients for all directions may be taken as normally distributed.

image

Figure 4. Normal probability plots for the wavelet coefficients at scale j = 1 corresponding to subregion 1 in the image for 20 June 1997.

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3.2.2. Autocorrelation Structure of Wavelet Coefficients

[36] As expected, the autocorrelation among wavelet coefficients for a given subregion is much weaker than that of the corresponding radiobrightness temperature image. Figure 5 displays the sample autocorrelation functions in the West-to-East and South-to-North directions for the horizontal, vertical, and diagonal wavelet coefficients at scale j = 1 corresponding to subregion 1 in the image for 20 June 1997. Ninety five percent confidence limits for white noise are also displayed [Shumway and Stoffer, 2001]. If the underlying wavelet coefficients are Gaussian and independent, 95% of all values should fall within the confidence bounds.

image

Figure 5. Sample autocorrelation functions for the (top) horizontal, (middle) vertical, and (bottom) diagonal wavelet coefficients corresponding to scale j = 1 and subregion 1 in the image for 20 June 1997. Dash-dot lines indicate 95% confidence limits for white noise.

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[37] The sample autocorrelation functions for the wavelet coefficients from all subregions are seen to follow the general trends depicted in Figure 5. The autocorrelation functions for the horizontal and vertical wavelet coefficients decay very rapidly, which indicates that low-parameter ARMA models constitute appropriate choices for characterizing them. Furthermore, the diagonal wavelet coefficients for all subregions behave as independent random variables. Owing to ease of design and application, we focus on finite impulse response filters or moving average (MA) filters for describing the spatial dependence of the horizontal and vertical wavelet coefficients. In particular, we consider the following model:

  • equation image

with

  • equation image

In (13), C connotes a stationary field or image of wavelet coefficients, ξ represents a field of i.i.d. Gaussian random variables with mean zero and variance σξ2, and Θ is a moving average filter with horizontal, vertical, and diagonal components denoted as θH, θV, and θD, respectively. The size of Θ is arbitrary such that NH, NV, and ND represent the number of MA filter components in the horizontal, vertical, and diagonal directions, and NMAX = max(NH, NV, ND). The appropriate filter to use for each field of wavelet coefficients at scale j = 1 was found by minimizing the negative logarithm of the Gaussian likelihood function as approximated by Whittle [1953]. The optimal models for the horizontal and vertical wavelet coefficients for all subregions with respect to the Akaike and Schwarz criteria [Harvey, 1989], which apply a penalty for increasing the number of parameters of the MA filter, were comprised of one horizontal and one vertical filter components (i.e., NH = NV = 1 and ND = 0 in (13)). The ranges of values for the horizontal and vertical filter components corresponding to the horizontal wavelet coefficients for all subregions are 0.09 to 0.30 and −0.20 to 0.07, respectively. The analogous ranges for the vertical wavelet coefficients are −0.17 to −0.01 and 0.11 to 0.29, respectively. To illustrate the typical goodness of fit of the MA models, Figure 6 displays the smoothed periodograms and the corresponding spectral density models for the horizontal and vertical wavelet coefficients at scale j = 1 for subregion 1 in the image for 20 June 1997. For details on the derivation for the confidence limits provided for the smoothed periodograms, the reader is referred to Brillinger [2001].

image

Figure 6. (top row) Smoothed periodograms and (bottom row) spectral density models for the (left column) horizontal and (right column) vertical wavelet coefficients at scale j =1 for subregion 1 in the image for 20 June 1997. 95% confidence limits for the smoothed periodogram are given as black mesh surfaces.

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3.2.3. Simple Scaling Relations for Wavelet Coefficients

[38] In this section we outline two alternative procedures for determining λ + γ in (11), which are used in section 4 to validate our downscaling scheme. Expression (11) and the discussion following it indicate that the wavelet coefficients for long memory 1/f processes exhibit scaling of second-order moments. This behavior is also illustrated in Figure 3. Thus it is possible to obtain an estimate of the slope of the scaling relations for the wavelet coefficients corresponding to all subregions by regressing the logarithm of the variance of the wavelet coefficients at all resolutions on the logarithm of scale. Since (11) is a limit property that is valid as the scale gets coarser, the slope of the scaling relation would be most accurately determined from the coefficients at lower resolutions [Flandrin, 1992]. Unfortunately, coarser scales have progressively less wavelet coefficients and thus the variance estimates for them bear higher uncertainties. Thus obtaining the slope of the scaling relations (λ + γ) by applying (11) directly requires large samples and may be subject to considerable bias [Percival and Walden, 2000] as shown in section 4. In particular, Flandrin [1992] showed, in the context of fractional Brownian motion, that the estimates for λ + γ determined from (11) have larger absolute magnitudes than the true parameter values in the presence of discrete finite samples.

[39] Alternatively, it is possible to derive estimates for λ and γ in (11) by maximum likelihood fitting of autoregressive fractionally integrated moving average (ARFIMA) models to the radiobrightness temperature images for all subregions [Beran, 1994; Kokoszka and Taqqu, 1995, 1996a, 1996b, 1999]. This approach has been shown to lead to more accurate estimates of λ and γ for the radiobrightness temperature observations of interest than various others by Parada and Liang [2003b], and it is also used in this study. Other researchers have also shown that estimates obtained by maximum likelihood estimation of ARFIMA model parameters are convergent, consistent, and tend to have less inherent uncertainty than those attained via various heuristic estimation approaches [Beran, 1994; Percival and Walden, 2000; Taqqu et al., 1995]. Parada and Liang [2003b] further describe the ARFIMA models under consideration, the implications that these have on the spatial structure of radiobrightness temperature images, and how these relate to previous findings regarding the derived near-surface soil moisture fields. In particular Parada and Liang [2003b] report the following ranges of parameter values apply for all subregions: −3.44 ≤ λ ≤ −0.26 and −3.36 ≤ γ ≤ 0. Given these values for λ and γ, it is clear that the use of Daubechies wavelets of order R = 2 meets the criterion provided in sections 2.3 and 3.2.1 for rendering the wavelet coefficients corresponding to long memory 1/f processes stationary and weakly correlated.

3.2.4. Downscaling Framework for Radiobrightness Temperature

[40] In this section, we link the statistical characterization of the wavelet coefficients of radiobrightness temperature to the downscaling methodology presented in section 2.4 and outline the application of the latter to the observations corresponding to each of the subregions in Table 1. For validation purposes, we take AJf(x, y) in Figure 1 to be the average radiobrightness temperature observation over a given subregion and produce realizations at 800-m resolution that mimic the spatial structure of the corresponding observations taken during SGP97. We accomplish this by generating fields of wavelet coefficients for the horizontal and vertical directions by means of MA models as described in section 3.2.2. The diagonal wavelet coefficients are modeled as i.i.d. fields based on the characteristics of the data previously discussed. Finally, expression (1) is applied to the synthetically generated images of wavelet coefficients to produce realizations of radiobrightness temperature possessing the desired spatial autocorrelation structure and variability. This exercise is carried out using estimates of λ + γ obtained through the two alternative approaches outlined in section 3.2.3.

[41] The downscaling framework described above is along the lines of the work presented by Perica and Foufoula-Georgiou [1996a, 1996b] for downscaling of rainfall intensity fields. On the basis of the analysis of a large number of storms, these researchers found that the logarithm of the variance of the normalized Haar wavelet coefficients from rainfall intensity measurements follow linear relationships when plotted against the logarithm of scale. Then, they used the linear slopes from these relationships and i.i.d. Gaussian input fields to produce wavelet coefficients at all desired resolutions needed for downscaling. The key distinction between the analysis provided by Perica and Foufoula-Georgiou [1996a, 1996b] for rainfall intensities and that presented in this study for radiobrightness temperature is that rainfall intensities, as characterized by Perica and Foufoula-Georgiou, appear not to be amenable to characterization as 1/f processes. This stems from the fact that, according to Perica and Foufoula-Georgiou, the unnormalized wavelet coefficients from rainfall intensity fields seem not to exhibit scaling of second-order moments.

4. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Downscaling Framework
  5. 3. Application to L Band Microwave Radiobrightness Temperature Imagery
  6. 4. Results
  7. 5. Conclusions and Future Work
  8. Acknowledgments
  9. References
  10. Supporting Information

[42] The framework described in section 2.4 is used to produce alternative realizations of radiobrightness temperature images at 800-m resolution for all subregions in Table 1 given their mean-area values as discussed in section 3.2.4. Since J = 5 for all subregions under consideration (see section 2.1), by using the downscaling framework described in section 2.4 we assume that (11) holds for the wavelet coefficients at scales j = 5, 4, …, 1. Owing to limitations imposed by the extent of the area for which ESTAR data was retrieved during SGP97 and by the use of wavelet transforms (e.g., square subregions and decreasing number of wavelet coefficients with increasing scale), we cannot test whether the simple scaling property for the variance of the wavelet coefficients illustrated in Figure 3 is valid at scales j = 4 and j = 5, respectively. Consequently, we assume that the wavelet coefficients from the radiobrightness temperature observations under consideration exhibit scaling of second-order moments over the scales of interest. The results discussed below clearly show that the assumption made here constitutes a good approximation.

[43] We conduct several simulations to validate the proposed downscaling framework. Specifically, we produce enough (approximately 550) unconditional realizations of the observed radiobrightness temperature image for each subregion in Table 1 at 800-m resolution until the statistics of interest (e.g., the probability density function) converge. In this process, we evaluate whether the realizations generated reproduce the spatial structure of the observations in an ensemble sense. We present the results obtained from performing this exercise for all subregions using the estimates for λ + γ in expression (11) obtained via the two different approaches described in section 3.2.3. In the first scenario, we produce unconditional realizations utilizing the optimal parameters obtained from ARFIMA models [Parada and Liang, 2003b]. In the second scenario, we apply equation (11) directly to obtain estimates of λ + γ by regressing linearly on log-log plots of variance of the wavelet coefficients versus scale for all subregions as illustrated in Figure 3. In this case, we average the slopes obtained by applying (11) to the wavelet coefficients in the horizontal, vertical, and diagonal directions to be consistent with the closure hypothesis made in (10). Parada and Liang [2003b] provide a more detailed inter-comparison of the estimates of λ + γ obtained via the two approaches described here together with those from two additional techniques.

[44] In general, we found that matching the standard deviation of the observed radiobrightness temperature fields can be quite challenging given the relatively small sample sizes available within each subregion of interest for estimation of λ + γ. Specifically, the ensemble average standard deviation estimate over all unconditional realizations for a given subregion is seen to be highly sensitive to the estimates of λ + γ. This is to be expected since ∣λ + γ∣ describes power law rates of growth for the variances of the wavelet coefficients with increasing scale as given in (11) and (12a). We utilize the following objective measures to quantitatively evaluate how our two estimates of λ + γ perform in terms of reproducing the standard deviation of the observations for each subregion in an ensemble sense:

  • equation image
  • equation image
  • equation image

In (14), T represents the total number of realizations generated for a given subregion, σr denotes the standard deviation for a single realization, and σobs connotes the standard deviation of the corresponding observations. The 〈 · 〉 and std( · ) operators denote the ensemble mean and standard deviation over all T realizations, respectively.

[45] Table 2 lists the fraction of all 68 subregions for which the absolute value of (14c), denoted ∣Δσ∣, is within various ranges when we use the two available estimates of λ + γ for each subregion. The average of stdr) in (14b) over all 68 subregions provides a measure of the uncertainty with which the ensemble mean standard deviations of the observations can be reproduced by the use of each estimation technique. This mean uncertainty is equal to 0.15 for simulations that use the estimates obtained via the ARFIMA approach and 0.24 for simulations that use the estimates computed by direct application of (11). As shown in Table 2, the estimates obtained from ARFIMA models yield realizations which, on average, reproduce the standard deviation of the observed fields with errors less than or equal to 5, 10, 15, 20 and 25 percent for 24, 51, 71, 78, and 79 percent of all 68 subregions respectively. By contrast, the standard deviation of the observations is captured with errors within 5, 10, 15, 20 and 25 percent for only 15, 22, 28, 35, and 40 percent of all 68 subregions when the estimates of λ + γ are obtained via direct use of (11). Notice that the standard deviation of the observed fields can be reproduced with errors within 25 percent for 79 and 40 percent of all 68 subregions if the estimates of λ + γ are obtained via ARFIMA models and direct application of (11), respectively. Therefore we conclude that the estimates for λ + γ derived via the ARFIMA paradigm appear to be much more robust even in the presence of relatively small sample sizes within our subregions.

Table 2. Fraction of All 68 Subregions Having Absolute Δσ (∣Δσ∣) Values Less Than Those Specified in the First Column When λ + γ Is Obtained Via Two Estimation Techniquesa
Maximum ∣Δσ∣, %ARFIMAln (wavelet variance)
  • a

    The two estimation techniques are (1) ARFIMA models and (2) direct application of equation (11) (denoted as ln(wavelet variance)).

50.240.15
100.510.22
150.710.28
200.780.35
250.790.40
300.820.46
400.850.59
6010.69
8010.81
10010.85
12510.93
15010.97
20010.99
25011

[46] On the other hand, we found that the autocorrelation structure of the observations for all subregions can be reproduced quite well and with relative ease via our downscaling scheme whether λ + γ is estimated by means of fitting ARFIMA models or by direct application of equation (11). To illustrate this, Figure 7 depicts the ensemble mean sample autocorrelation function estimates and their corresponding uncertainty bounds along the horizontal and vertical directions obtained by estimating λ + γ through the ARFIMA approach for subregion 1 in the image for 20 June 1997. Figure 8 shows the observed radiobrightness temperature field and its periodogram for the same subregion as in Figure 7, a typical unconditional realization of the radiobrightness temperature field and its periodogram, and the ensemble average field and periodogram. Note that the ensemble average radiobrightness temperature for all pixels converges to the mean level of the observations as expected.

image

Figure 7. (top) Sample autocorrelation functions for the observed radiobrightness temperature field, (middle) a sample unconditional realization of it, (bottom) and averaged over various unconditional realizations for subregion 1 in the image for 20 June 1997. One lag unit = 800 m. The mean field radiobrightness temperature used to generate the unconditional 800-m realizations for this subregion has a resolution of 42.4 km = 53 lag units.

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image

Figure 8. (left column) Radiobrightness temperature fields at 800-m resolution and (right column) corresponding periodograms for (top row) the observations, for (middle row) a sample unconditional realization of them, and for (bottom row) the average over various unconditional realization for subregion 1 in the image for 20 June 1997.

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[47] To further verify the goodness of fit to the structure of spatial dependence of the radiobrightness temperature fields of interest, we obtained estimates of the Fourier transform of the autocorrelation function for each subregion by normalizing its periodogram (see (4a)) by the variance of the corresponding observations. The motivation underlying the use of the normalized periodogram instead of the sample autocorrelation functions lies in the simplicity of the approximate confidence bounds for the periodogram as described by Brillinger [2001]. We additionally computed uncertainty bounds for the ensemble mean normalized periodogram for each subregion directly from the unconditional realizations for it. Given the normalized periodogram for each subregion and the corresponding ensemble mean normalized periodogram, we tested whether these two estimates could be considered to be the same with 95% confidence. We rejected this hypothesis if there was no overlap between the 95% confidence bounds for the normalized periodogram obtained from the observations and the corresponding ensemble mean normalized periodogram for at least 5% of all normalized periodogram values. The hypothesis could not be rejected for 80% of all subregions analyzed. Moreover, the minimum, maximum, mean, and median fractions of normalized periodogram values exhibiting overlap in the 95% confidence bounds for the observed and ensemble mean normalized periodograms are 0.79, 0.99, 0.96, and 0.97, respectively, as summarized in Table 3. These results attest that the proposed downscaling scheme can reproduce the autocorrelation structure of the radiobrightness temperature observations of interest quite well.

Table 3. Statistics on the Fractions of Observed and Ensemble Mean Normalized Periodogram Values Exhibiting Overlap in 95% Confidence Bounds for All 68 Subregions Studied
 Fraction of Normalized Periodogram Values
Least0.79
Highest0.99
Mean0.96
Median0.97

5. Conclusions and Future Work

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Downscaling Framework
  5. 3. Application to L Band Microwave Radiobrightness Temperature Imagery
  6. 4. Results
  7. 5. Conclusions and Future Work
  8. Acknowledgments
  9. References
  10. Supporting Information

[48] In this paper, we introduce a general downscaling framework and apply it to the L band microwave radiobrightness temperature images retrieved from ESTAR during SGP97. The proposed downscaling methodology is based on the statistical characterization of scale-invariant properties of the wavelet coefficients or fluctuations from long memory 1/f processes. Thus we first show that the radiobrightness temperature observations of interest can be characterized as long memory 1/f processes. This in turn implies that the wavelet coefficients of radiobrightness temperature must exhibit scaling of second-order moments as given in expressions (11) and (12) [Flandrin, 1992; Tewfik and Kim, 1992; Wornell, 1993; Percival and Walden, 2000]. We also permit for the wavelet coefficients at all scales to manifest spatial dependence in any direction as required by the data by characterizing them as ARMA processes. Moreover, the statistical inference setting within our downscaling scheme allows us to draw a direct link to the ARFIMA time series paradigm. Hence we make use of the ARFIMA machinery to perform robust estimation of the scaling exponent (λ + γ) corresponding to the wavelet coefficients as shown by Parada and Liang [2003b] and outlined in section 3.2.3.

[49] We test the proposed downscaling framework with the radiobrightness temperature images taken during SGP97 by producing unconditional realizations of these at 800-m resolution given a mean-area value at approximately 30-km resolution (the near-future expected operational scale), and evaluating how well the realizations reproduce the statistics of the corresponding observations. As shown in section 4, the proposed downscaling methodology for radiobrightness temperature fields is capable of accurately preserving the variability and overall structure of spatial dependence of the observed radiobrightness temperature fields.

[50] Currently, efforts are being devoted to the application of the proposed downscaling framework to the characterization of subpixel variability of near-surface soil moisture fields. We are also exploring alternatives for estimation of the scaling exponent (λ + γ) of the wavelet coefficients and the ARMA models that describe their spatial dependence in terms of physical descriptors such as readily available measurements or outputs from land-surface models.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Downscaling Framework
  5. 3. Application to L Band Microwave Radiobrightness Temperature Imagery
  6. 4. Results
  7. 5. Conclusions and Future Work
  8. Acknowledgments
  9. References
  10. Supporting Information

[51] The authors are thankful to David Brillinger from the Statistics Department at the University of California, Berkeley, and to the anonymous reviewers for their helpful comments and suggestions. This work is supported by NASA under grant NAG5-10673 to the University of California, Berkeley. Partial funding for the first author provided by the Berkeley Atmospheric Sciences Center is also greatly appreciated.

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  2. Abstract
  3. 1. Introduction
  4. 2. Downscaling Framework
  5. 3. Application to L Band Microwave Radiobrightness Temperature Imagery
  6. 4. Results
  7. 5. Conclusions and Future Work
  8. Acknowledgments
  9. References
  10. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Downscaling Framework
  5. 3. Application to L Band Microwave Radiobrightness Temperature Imagery
  6. 4. Results
  7. 5. Conclusions and Future Work
  8. Acknowledgments
  9. References
  10. Supporting Information

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