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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. D22, 8862, doi:10.1029/2003JD003567, 2003

A stochastic modeling approach for characterizing the spatial structure of L band radiobrightness temperature imagery Laura M. Parada and Xu Liang Department of Civil and Environmental Engineering, University of California, Berkeley, California, USA Received 6 March 2003; revised 29 July 2003; accepted 13 August 2003; published 13 November 2003.

[1] This study focuses on the statistical characterization of the spatial structure of L band

microwave radiobrightness temperature fields retrieved during the Southern Great Plains hydrology experiment of 1997 (SGP97). It is found that the radiobrightness temperature observations of interest can be considered nonstationary scaling processes that exhibit persistence or long memory. This implies that the spatial dependence of observations decays very slowly at large separation distances such that models with exponentially decaying autocorrelation (e.g., autoregressive moving average models or exponential models commonly used in geostatistics) are not appropriate. It is further shown that the radiobrightness temperature fields retrieved during SGP97 exhibit distinct scaling behaviors along the horizontal (west to east) and vertical (south to north) directions. A two-dimensional implementation of the fractionally integrated moving average (FIMA) time series model is shown capable of capturing the spatial autocovariance structure of the observations. The results presented evince that the FIMA paradigm allows for robust estimation of distinct scaling exponents along the horizontal and vertical directions both in stationary and nonstationary situations. Comparisons to alternative heuristic methods for determining the scaling exponent(s) further demonstrate that FIMA models yield estimates with superior accuracy. Additionally, within the FIMA framework it is possible to jointly and accurately model the spatial dependence of radiobrightness temperature for INDEX TERMS: 1866 Hydrology: Soil observations separated by short and long distances. moisture; 3250 Mathematical Geophysics: Fractals and multifractals; 3322 Meteorology and Atmospheric Dynamics: Land/atmosphere interactions; 3360 Meteorology and Atmospheric Dynamics: Remote sensing; KEYWORDS: brightness temperature, spatial correlation and variability, fractional ARIMA model Citation: Parada, L. M., and X. Liang, A stochastic modeling approach for characterizing the spatial structure of L band radiobrightness temperature imagery, J. Geophys. Res., 108(D22), 8862, doi:10.1029/2003JD003567, 2003.

1. Introduction [2] Radiobrightness temperature observations retrieved from electronically scanned thinned array radiometer (ESTAR) are of pivotal importance for validation of radiation transfer models, as well as to derive fields of volumetric soil moisture content for the top 5 cm of soil [Jackson et al., 1995]. Soil moisture plays a key role in processes occurring at a wide range of resolutions: from the characterization of capillary forces exerted on water at soil pores to the partitioning of incoming radiation into latent, sensible, and ground heat fluxes needed for evaluation of water and energy budgets from the catchment to the global scales. [3] The National Aeronautics and Space Administration (NASA) and the U.S. Department of Agriculture (USDA) have jointly sponsored several large-scale measurement campaigns such as Washita ’92 [Jackson et al., 1995] and the Southern Great Plains hydrology experiment of 1997 (SGP97) [Jackson et al., 1999], which have been the focus of significant efforts to evaluate the feasibility of large area mapping of soil moisture. During Washita ’92 and SGP97, Copyright 2003 by the American Geophysical Union. 0148-0227/03/2003JD003567

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L band radiobrightness temperature observations (1.4 GHz) were retrieved for the Little Washita watershed in Oklahoma and surrounding areas by flying an ESTAR on board an airplane over the study regions. These radiobrightness temperature images were utilized to derive observations of volumetric near-surface soil moisture content at final resolutions of 200 m and 800 m, respectively [Jackson et al., 1995, 1999]. [4] The statistical characterization of the radiobrightness temperature images retrieved with ESTAR and of the derived near-surface soil moisture fields is of main interest for a variety of purposes. For instance, (1) it may yield insight on the spatial structure and dynamics of soil moisture and pinpoint dominant aspects of the process that must be considered and reproduced by the parameterizations in land-surface models. (2) It may allow for inference of radiobrightness temperature and near-surface soil moisture states at unobserved locations. (3) It is required for process simulation, which may be needed, for example, for the evaluation of the predictive uncertainty of radiation transfer and land-surface models within a Monte Carlo framework. (4) It is of key relevance for assimilation of radiobrightness temperature or the derived near-surface soil moisture into

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PARADA AND LIANG: VARIABILITY OF BRIGHTNESS TEMPERATURE

land-surface models and hence to improve predictions of soil moisture states [e.g., Walker and Houser, 2001; Reichle et al., 2002; Crow and Wood, 2003; Parada and Liang, 2003a]. Data assimilation schemes merge model predictions of given states with observations of these, if available, to yield better estimates and attempt to improve the predictive capabilities of the model. In doing this, statistical characterizations for the model output and the observations must be specified for the construction of a framework in which to perform statistical inference and assimilation. [5] Several researchers have reported that the near-surface soil moisture images derived from ESTAR or other L band remote sensing instruments possess scaling or fractal properties. Such invariance of normalized statistical properties with the resolution of the observation has been observed over scales ranging from 800 m to within 10 km [e.g., Rodriguez-Iturbe et al., 1995; Hu et al., 1997, 1998; Cosh and Brutsaert, 1999; Crow et al., 2000; Oldak et al., 2002]. These findings imply the possibility of characterizing subgrid variability of radiobrightness temperature and nearsurface soil moisture within a statistical framework. Specifically, previous work has focused on determining the Hurst exponent [Hurst, 1951; Beran, 1994], H, and on evaluating how estimates for this parameter evolve when computed for statistical moments of orders 2 and higher [e.g., Hu et al., 1997, 1998; Oldak et al., 2002]. In most studies mentioned thus far, the Hurst exponent for near-surface soil moisture was determined by aggregation of these fields into progressively coarser resolutions by averaging over neighboring pixels. The statistical moments of the resulting near-surface soil moisture images at various resolutions were then computed and the Hurst exponent was determined from plotting the logarithm of these moments versus the logarithm of scale (see section 2.3 for more details). Significant efforts have also been devoted to trying to provide insight as to what physical attributes may be used to infer the value of H [e.g., Rodriguez-Iturbe et al., 1995; Peters-Lidard et al., 2001; Oldak et al., 2002]. [6] For certain ranges of the Hurst exponent, fractal fields possess autocorrelation functions that decay very slowly with increasing lag such that observations separated by large distances may still display nonnegligible dependence or long memory (see section 2). This implies that standard time series (e.g., autoregressive moving average or ARMA processes) and geostatisical (e.g., exponential) models, which possess exponentially decaying spatial dependence, are not appropriate for modeling scaling processes that exhibit long memory [Beran, 1994]. Moreover, scaling long memory fields become nonstationary beyond certain ranges of the Hurst or scaling exponent as discussed in section 2. In this context, the work by Cosh and Brutsaert [1999] shows that the derived near-surface soil moisture fields obtained during Washita ’92 are (1) nonstationary for distances beyond 5 km; (2) that the assumption of stationarity may be valid if the study region is restricted to lie within an area with uniform soil type or within ranges of 5 km; and (3) that within a stationary region the autocorrelation range is approximately 1 km. [7] In this paper, we present an alternative approach for modeling the spatial structure of L band microwave radiobrightness temperature fields by extending the autoregressive fractionally integrated moving average (ARFIMA) time

series paradigm to two dimensions. ARFIMA models are well suited for situations in which stationarity may not be assumed. Furthermore, ARFIMA models provide a robust and unified framework for modeling long memory, i.e., slowly (hyperbolically) decaying dependence of observations separated by large distances, as well as short-distance autocorrelation. The ARFIMA models presented in this paper also permit for distinct scaling exponents to be estimated along the horizontal (west to east) and vertical (south to north) directions of scaling fields with superior accuracy than various alternative heuristic approaches. It is worth stressing that the work presented here is concerned solely with the characterization of spatial autocovariance structure. No attempts are made to capture statistical moments of order higher than two. Hence the framework presented relies only on the assumption of simple scaling of the variance or wide sense self-affinity as defined in section 2. While validation of the proposed methodology is performed with the radiobrightness temperature observations retrieved from ESTAR during SGP97, the methodology presented may also be applicable to the derived near-surface soil moisture images. [8] This paper is organized as follows. Section 2 briefly introduces the statistical concepts and terminology employed in the paper. Section 3 statistically characterizes the spatial dependence of radiobrightness temperature fields. Section 4 presents the results obtained from applying the ARFIMA paradigm to model radiobrightness temperature. Finally, section 5 provides our primary conclusions.

2. Theoretical Foundation [9] The statistical principles and terminology relevant to the work presented in this paper are introduced in this section. Self-affinity is briefly defined in section 2.1. Long-memory 1/f processes are introduced in section 2.2. Heuristic techniques for estimation of the scaling exponent are described in section 2.3. Autoregressive fractionally integrated moving average (ARFIMA) models for long memory 1/f processes are described in section 2.4, which also provides some background on inference of optimal parameters for these models. 2.1. Self-affine Processes [10] The term self-affine is a neologism introduced to designate processes that are also known as self-similar [Mandelbrot, 2000], and it is used throughout this paper. Nonetheless, many of our references adhere to the use of the term self-similar instead of self-affine. A thorough treatment of statistically self-affine processes is provided by Beran [1994] and references therein. [11] The definition of self-affinity is given here for the one-dimensional case for ease of notation. Its extension to two dimensions follows naturally from the one provided here. Let S(x) denote a one-dimensional stochastic process. S(x) is statistically self-affine in strict sense if for any t > 0: D

S ð xÞ ¼ tH S ð txÞ; D

ð1Þ

where ‘¼’ denotes equality in distribution, and H 0 is the so-called Hurst exponent. Specifically, S(x) is statistically identical to its transform given a contraction or expansion in


x, namely S(tx), followed by change in intensity by the factor tH. Since the rescaling ratios of x and S in (1) are not identical, the transformation from S(x) to tHS(tx) constitutes an affinity, rather than a similitude [Mandelbrot, 2000], as the latter term would imply equality in rescaling ratios. Hence the term self-affine explicitly recognizes that scale invariance may result from anisotropic transformations, that is, by rescaling with different factors along different directions. The term fractal is commonly used to denote self-affine processes [Mandelbrot, 2000]. Wide sense or second-order self-affinity, on the other hand, applies only up to the second-order moment or autocovariance function. S(x) in (1) is formally nonstationary unless H = 0 (in which case it is a constant), but possesses stationary increments, S(x) = {S(x) S(x 1)}, for 0 H < 1 [Beran, 1994]. On the other hand, when H 1 the increments of S(x) are no longer stationary. [12] The first comprehensive model for self-affine processes having 0 H < 1 and Gaussian increments is due to Mandelbrot and van Ness [1968], who designate S(x) as fractional Brownian motion (fBm) and its stationary increments, S(x), as fractional Gaussian noise (fGn). For 0.5 < H < 1, fractional Gaussian noise exhibits persistence or long memory, whereas for 0 < H < 0.5 it is antipersistent. Further details on this topic are given in Appendix A. In sections 2.4, we introduce alternative models for long memory processes that can accommodate nonstationarity. 2.2. Long Memory 1//f Processes [13] Percival and Walden [2000] and references therein provide a thorough description of long memory 1/f processes. Let f (x, y) denote a two-dimensional stochastic process or image consisting of N N pixels and having autocovariance function cff (u, v), where u and v index lags in the x and y directions, respectively. The power spectrum of f (x, y) is defined as the Fourier transform (denoted by ^) of the autocovariance function, and it is connoted as c^ff (a, b): _ c^ff ða; bÞ ¼

X

X

cff ðu; vÞ expði2pðua þ v bÞÞ: ð2Þ

1u1 1v1

pffiffiffiffiffiffiffi In (2), i = 1 and (a, b) is referred to as the wave number. It can be shown that the integral of the power spectrum equals the variance of f (x, y), which we denote as sff2 [Brillinger, 2001]. Thus the power spectrum yields a decomposition of variance with wave number that makes evident what fraction of sff2 can be associated to each (a, b) and its corresponding spatial scale (a1 ‘, b1 ‘), where ‘ represents the spatial resolution at which f (x, y) is sampled. Let ‘’ and superscript ‘T’ denote complex conjugation and matrix transpose, respectively. An estimate of the power spectrum is provided by the periodogram [Brillinger, 2001]: _ _ 1 c_^ff ða; bÞ ¼ 2 f ð x; yÞ m ^ f ð x; yÞ m ^T ; N

ð3Þ

^ denotes where a = a/N and b = b/N for a, b = 1, . . ., N, and m the sample mean of f (x, y). Since the periodogram is even and has period N [Brillinger, 2001], it suffices to display it for a 0.5 and b 0.5. An improved estimate of the power spectrum termed smoothed periodogram can be obtained by

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applying a moving average filter to the periodogram [Brillinger, 2001]. [14] A long memory 1/f process is such that its power spectrum goes to infinity as the wave number approaches zero [Beran, 1994; Percival and Walden, 2000] as: c^ff ða; bÞ / C jajl jbjg

as a; b ! 0;

ð4Þ

where C > 0, l 0, and g 0. This in turn implies that the autocovariance function of a long memory 1/f process approaches zero at a hyperbolic rate as the separation distance increases, i.e.: cff ðu; vÞ / Cc jujh jvjV

as u; v ! 1;

ð5Þ

where Cc > 0 and h= l 1, V = g 1 for 1 < l < 0 and 1 < g < 0 [Percival and Walden, 2000]. Thus observations separated by large distances may still display nonnegligible dependence or exhibit long memory. In contrast, standard time series models, such as autoregressive moving average (ARMA) processes [Brillinger, 2001; Shumway and Stoffer, 2001], and models commonly used in geostatistics (e.g., exponential models) have autocovariance functions that approach zero much faster (exponentially) than long memory 1/f processes as the separation distance increases [Beran, 1994]. [15] Expressions (4) and (5) evince that long memory 1/f processes are statistically self-affine in a wide sense since c^ff (Aa, Bb) / jAjl jBjgc^ff (a, b) as a, b ! 0. Throughout this paper, we use the general term scaling exponent to designate l, g, and H, and we reserve use of the term Hurst exponent to refer exclusively to H as in section 2.1. Details on how long memory 1/f processes relate to fractional Brownian motion and fractional Gaussian noise and on the correspondence between l, g, and H for these cases are provided in Appendix A. [16] The power spectrum in (4) permits for different scaling behaviors in a and b, which implies that the autocovariance functions in the x and y directions are allowed to approach zero at different rates as the separation distance increases. Moreover, for 1 < l < 0 and 1 < g < 0, a long memory 1/f process has finite variance and is said to be stationary. Whereas for l 1 and/or g 1, the power spectrum increases to infinity so fast that it becomes nonintegrable. This implies that the variance of the process becomes infinite. Hence second- and higher-order moments are not defined and the process is said to be nonstationary [Beran, 1994; Kokoszka and Taqqu, 1995, 1996a, 1996b, 1999; Percival and Walden, 2000]. Note that while secondorder moments are not formally defined for long memory 1/f processes with infinite variance, given a finite sample it is always possible to compute the sample estimate of the autocovariance and the periodogram. These sample estimates help us evaluate the validity of (4) and (5) as well as the overall structure of spatial dependence. [17] Self-affine processes with finite and infinite variance find applications in characterizing a wide range of natural and artificial phenomena. They were first reported in the context of characterizing river flows by Hurst [1951] and are known to be pervasive in hydrologic data [Mandelbrot and Van Ness, 1968; Mandelbrot and Wallis, 1968;

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Montanari et al., 1997]. Long memory 1/f processes with infinite variance are particularly common in hydrology since they permit modeling of persistence (e.g., long periods of droughts and floods) and high variability (e.g., the presence of either extremely high or low floods) simultaneously [Taqqu, 1987]. In this context, it is expected that the Hurst exponent, H, is greater than or equal to 0.5 since we expect hydrologic data to display persistence or long memory rather than antipersistence [Mandelbrot and Wallis, 1968; Beran, 1994; Percival and Walden, 2000]. These expectations are in agreement with the values for the scaling exponents for the near-surface soil moisture data obtained from ESTAR during Washita ‘92 reported by RodriguezIturbe et al. [1995], which fall in the [0.75, 0.98] range. Conversely, Hu et al. [1997] and Cosh and Brutsaert [1999] report values of H outside of the range given. This appears to be due partly to the fact that these studies characterize near-surface soil moisture as fractional Brownian motion, namely S(x) as defined in section 2.1 and further described in Appendix A, rather than that as fractional Gaussian noise, S(x), to determine the value of the Hurst exponent [Beran, 1994, pp. 50– 53, 92– 94]. We conjecture that if a characterization as fractional Gaussian noise had been adopted in these studies, their values for H might have fallen within the range reported by Rodriguez-Iturbe et al. [1995]. [18] Because of the slow rate of decay in the autocorrelation function of long memory 1/f processes, their spatial structure is such that the observations tend to occur in clusters of predominantly high or low values. Thus, if one views small data sets, deterministic trends may appear to be present on the data as the process evolves from a cluster of high values to a cluster of low values [Beran, 1994]. Furthermore, if one zooms enough into a cluster of high or low observations and focuses on the autocorrelation within this smaller portion of the data, it may appear that the spatial dependence of process decays exponentially such that standard time series models may be valid. The autoregressive fractionally integrated moving average (ARFIMA) models for long memory 1/f processes discussed in section 2.4 allow for characterizing the observations as a whole rather than being restricted to modeling fractions of the data that look stationary and appear to have exponentially decaying autocovariance functions separately. 2.3. Heuristic Approaches for Estimation of the Scaling Exponent [19] Expressions (4) and (5) suggest two different approaches for estimation of l and g. Taking the logarithm of both sides of (4), we see that a log-log plot of the power spectrum versus wave number should increase to infinity linearly in a and b with slopes l and g, respectively, as a, b ! 0. In practice, it is difficult to determine the range of values of a and b for which (4) holds and large biases may be introduced by considering values outside of this range [Beran, 1994]. Furthermore, empirical studies show that even when large samples are available the uncertainty of the estimates obtained by this approach may be quite large [Taqqu et al., 1995]. Alternatively, it is possible to estimate h and V from the linear relationships that arise as we take the logarithm of both sides of (5). However, this approach suffers from the same pitfalls as that previously described.

[20] A third approach is based on computing the variance of the self-affine process at successively coarser aggregation levels. The variance is initially computed at the finest scale available. Then, neighboring pixels are averaged repeatedly to obtain fields of the process at successively coarser resolutions and the variance for each of these is estimated. For a long memory process, a log-log plot of variance versus pixel area for each resolution should yield a linear relationship with slope equal to (h + V)/2 (or 2H 2 for fGn as shown in Appendix A) at coarse scales [Beran, 1994]. This approach has been used by various researchers to compute the Hurst exponent of the near-surface soil moisture images derived from radiobrightness temperature fields obtained with ESTAR during Washita ‘92 [e.g., RodriguezIturbe et al., 1995; Hu et al., 1997, 1998; Oldak et al., 2002] and SGP97 [e.g., Oldak et al., 2002]. This technique is based on a limit property and is valid as the level of aggregation increases [Beran, 1994]. Thus it suffers from the same limitations as the two approaches previously described. Moreover, this approach only allows for the computation of a single Hurst or scaling exponent. Hence it assumes that the rate of decay of the autocorrelation function at large separation distances is the same for the horizontal and vertical directions. [21] It is also possible to obtain estimates of l + g employing orthonormal wavelet transforms. More precisely, for a long memory 1/f process, a log-log plot of the variance of the wavelet detail coefficients versus scale shows a linear increase with a slope approximately equal to (l + g) at coarse scales [Wornell, 1993; Parada and Liang, 2003b]. Hu et al. [1998] reported scaling holds for the variance of the wavelet coefficients for the near-surface soil moisture images derived from ESTAR during Washita ’92. 2.4. ARFIMA Models for 1//f Processes [22] In this section, we introduce autoregressive fractionally integrated moving average (ARFIMA) models. For a more thorough coverage on this topic, the reader is referred to Granger and Joyeux [1980], Hosking [1981], Beran [1994], and references therein. ARFIMA models are of interest since they provide a parametric framework for modeling long memory 1/f processes. [23] For completeness, let us define the procedure for applying a linear time invariant filter in two dimensions. Given a two-dimensional filter A with 2Ni + 1 rows and 2Nj + 1 columns such that A(i, j) denotes the component in its ith row and jth column, and a discrete two-dimensional process f (x, y), the operation filtering f (x, y) with A is given by: A f ð x; yÞ

Nj Ni X X

A i þ Ni þ 1; j þ Nj þ 1 f ð x þ j; y iÞ;

i¼Ni j¼Nj

ð6Þ

where ‘*’ denotes filtering or convolution. Let us further define the back shift operators in the, x, Bx, and, y, By , directions as follows: Bux f ð x; yÞ f ð x u; yÞ; and

ð7aÞ

Bvy f ð x; yÞ f ð x; y vÞ;

ð7bÞ

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for integers u, v. Given two dimensional filters and , f (x, y) is said to follow an autoregressive fractionally integrated moving average model if the following holds: d ð1 Bx Þdx 1 By y ð f ð x; yÞ mÞ ¼ eð x; yÞ:

ð8Þ

In the above, m denotes the mean of f (x, y). is an autoregressive (AR) filter since it reflects the dependence of f (x, y) on itself. e(x, y) is a field of zero-mean independent and identically distributed (i.i.d.) variates usually taken to be Gaussian with variance se2. is a moving average (MA) filter acting on the i.i.d. field. Finally, dx and dy are real numbers termed differencing parameters since they act on differences or increments of f (x, y) in the x and y directions, respectively. Certain conditions must be met by the components of , and to ensure stationarity and invertibility of such filters, respectively [Shumway and Stoffer, 2001]. Since the Fourier transform turns filtering into point-wise multiplication, the power spectrum for the model in (8) is given by:

c^ff ða; bÞ ¼

j1 eia j

2dx

2d 2 1 eib y ^ j^j

2

s2e :

ð9Þ

[24] The differencing components of (8) and (9) allow for modeling of long memory with l = 2dx and g = 2dy, such that dx > 0 and dy > 0 correspond to l < 0 and g < 0 in equation (4), respectively [Percival and Walden, 2000]. On the other hand, pure ARMA models (models with dx = dy = 0) have exponentially decaying autocorrelation functions and thus are good for modeling the linear dependence of observations separated by short distances (recall that long memory is a limit property defined as the separation distance gets large). Hence the model in equations (8) and (9) allows for the unified characterization of the dependence between observations separated by either large or short distances. From the spectral point of view, differencing allows for modeling of the behavior of the power spectrum as it diverges to infinity, namely as the wave number approaches zero or the separation distance increases. On the other hand, the ARMA component in equation (9) allows for modeling the behavior of the power spectrum at higher frequencies (or short separation distances), where stationarity may be assumed. Note that the model in equations (8) and (9) can easily capture nonstationary long memory 1/f processes by simply allowing dx 0.5 and dy 0.5 since in this case l 1 and g 1, respectively. This corresponds to the increments of equation (1) being nonstationary or H > 1 as discussed in Appendix A. The intuition behind letting dx 0.5 and dy 0.5 is that upon taking higher-order differences of a nonstationary 1/f long memory process, we may indeed obtain stationary fluctuations [Percival and Walden, 2000]. [25] Parameter estimation for equation (9) can be performed by constrained minimization of the following objective measure, J: J ðqÞ ¼

XX a

b

(

) c_^ff ða; bÞ ln c^ff ða; bÞ þ _ c^ff ða; bÞ

ð10Þ

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

Date Analyzed

1

2

3

4

5

6

7

8

9

18 June 19 June 20 June 1 July 2 July 3 July 12 July 13 July 14 July

522 522 532 532 532 522 532 542 532

522 532 532 532 532 532 532 532 582

532 532 532 532 532 532 532 542 472

412 532 532 562 542 552 542 542 562

422 532 532 552 552 552 562 552 562

NA 542 532 552 552 552 552 552 552

NA NA 542 552 542 552 542 552 562

NA NA NA 542 552 542 NA 552 542

NA NA NA 542 552 NA NA 542 NA

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

where q denotes all parameters in equation (9), namely the components of and , se2, dx, and dy . The cost function in equation (10) is known as Whittle’s approximation to the negative of the Gaussian log likelihood function [Whittle, 1953; Beran, 1994]. It has been shown to be asymptotically consistent and convergent for maximum likelihood estimation of model parameters for Gaussian and non-Gaussian short memory (e.g., ARMA) as well as finite or infinite variance long memory (e.g., ARFIMA) processes. For a thorough coverage on the consistency and properties of equation (10) the reader is referred to the work of Kokoszka and Taqqu [1995, 1996a, 1996b, 1999] and references therein. [26] Empirical studies attest that estimates obtained by maximum likelihood estimation of ARFIMA model parameters through minimization of equation (10) are convergent, consistent (i.e., the uncertainty of the estimates decreases with increasing sample size), and tend to have less inherent uncertainty than those from various heuristic estimation approaches [Beran, 1994; Taqqu et al., 1995; Percival and Walden, 2000].

3. Statistical Characterization of Radiobrightness Temperature Images [27] This work focuses on nine radiobrightness temperature images retrieved during three drying periods that took place following main rainfall events during SGP97. The time spans for these periods are: 18– 20 June, 1– 3 July, and 12– 14 July [Jackson et al., 1999; Oldak et al., 2002]. The radiobrightness temperature images of interest were subdivided for analysis into 5 to 9 square subregions to facilitate the application of spectral techniques needed for this study (fast Fourier transforms) as well as for further work on downscaling (discrete wavelet transforms) by the same authors [Parada and Liang, 2003b]. These subregions were selected based on three criteria: (1) maximizing the size of each subregion, (2) choosing areas with no missing values, and (3) having at least 32 32 nonoverlapping pixels in each subregion. Table 1 lists the sizes for all subregions under consideration, which range from 412 to 582 800-meter pixels. [28] Prior to performing any analysis, outliers were removed from the radiobrightness temperature field corresponding to each subregion to prevent them from introducing bias into the results. Outliers were initially identified by visual inspection of the data as extremely

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Figure 1. (top) Horizontal correlation matrix, (middle) vertical correlation matrix, and (bottom) periodogram for subregion 3 in the image for 19 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. See color version of this figure in the HTML.

bright or dark spots in the images that were not coherent with the spatial structure of the surrounding observations. If the observations visually identified as abnormal were found to be greater than a local mean plus 1.5 times the standard deviation of the image, they were replaced by their local means. On average, 0.49% of the observations for each subregion were identified as outliers. For all subregions, the fraction of observations considered to be outliers was less than or equal to 3.15%.

[29] Figures 1 and 2 illustrate the general patterns and contrasts discerned by analyzing the spatial structure of all subregions in Table 1. Figure 1 shows the autocorrelation matrices for a subregion having horizontal and vertical autocorrelation that decays at comparably slow rates in both directions. On the other hand, Figure 2 displays the autocorrelation matrices for a subregion having noticeably more rapidly decaying vertical than horizontal autocorrelation. In both cases, the periodograms tend to infinity near


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Figure 2. As in Figure 1, but for subregion 9 in the image for 1 July 1997. See color version of this figure in the HTML.

the origin. The following assertions summarize the findings exposed in Figures 1 and 2. (1) All subregions are characterized by slowly (nonexponentially) decaying autocorrelation at large lags in either the horizontal (west to east) direction, the vertical (south to north) direction, or both directions. (2) The rate of decay of the autocorrelation function for a given subregion is seen to be distinct in the horizontal and vertical directions either to a small

extent or, in most cases, to a quite noticeable degree. (3) Since none of the subareas studied displays exponentially decaying spatial dependence in both the horizontal and vertical spatial directions, all of them are characterized by periodograms that diverge to infinity as the wave number approaches zero. [30] The characteristics of the radiobrightness temperature images discussed above suggest that these may be

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modeled as long memory 1/f processes. The work by several researchers who report that the near-surface soil moisture images derived from radiobrightness temperature observations obtained during Washita ’92, SGP97, and other experiments [e.g., Rodriguez-Iturbe et al., 1995; Hu et al., 1997, 1998; Cosh and Brutsaert, 1999; Crow et al., 2000; Oldak et al., 2002] behave as fractals supports this claim. The observation that the autocorrelation of radiobrightness temperature observations may decay at markedly distinct rates in the horizontal and vertical directions highlight the importance of adopting a model flexible enough to capture and portray such a property. In particular, the ARFIMA paradigm given in equations (8) and (9) is capable of modeling unequal rates of decay of the autocorrelation in the horizontal and vertical directions by assigning a distinct differencing parameter to each of them. As discussed in section 2.4, this model has the additional appeal of allowing for an accurate and simultaneous representation of short and long-range spatial dependence.

4. Statistical Modeling of Radiobrightness Temperature Images [31] This section presents the results obtained from the novel application of ARFIMA models for characterizing the spatial structure of the radiobrightness temperature imagery for all subregions in Table 1. Section 4.1 provides details on the specific form of the ARFIMA models used for this purpose. Section 4.2 presents the optimal or maximumlikelihood models for all subregions, and section 4.3 validates these models and inter-compares them to alternative heuristic approaches. 4.1. Fractionally Integrated Moving Average (FIMA) Models [32] In this section, we describe the application of the ARFIMA paradigm given in equations (8) and (9) to characterize the spatial structure of the radiobrightness temperature observations corresponding to all subregions of interest. In doing this, we restrict our attention to models having no autoregressive (AR) component termed fractionally integrated moving average (FIMA) models. Letting = 1 in equations (8) and (9) we obtain: d ð1 Bx Þdx 1 By y ð f ð x; yÞ mÞ ¼ eð x; yÞ; 2d 2dy ^2 2 c^ff ða; bÞ ¼ 1 eia x 1 eib se :

and simplifies the analysis and model selection procedure. Specifically, the MA filters used in this study take the following form: 2

0 6 6 . 6 .. 6 6 6 6 0 6 6 6 6 6q ¼ 6 h;Nh 6 6 6 0 6 6 6 . 6 .. 6 6 6 4 0

cðu; v ¼ 0Þ ¼

FIMA models yield a representation for the exponentially decaying autocorrelation used to model dependence at short separation distances in terms of a finite number of terms, or as a pure moving average (MA) model as opposed to an ARMA model. Models having different from one (AR or ARMA models) have autocorrelation functions for short lags that are expressed in terms of infinite sums. We further restrict our attention to moving average filters having only horizontal and vertical components. This causes the short-lag horizontal and vertical autocorrelation functions to depend on filter components in their corresponding directions only

0

qv;Nv

0

..

.

.. .

.. .

.. .

. ..

0

qv;1

0

qh;1

1

qh;1

0

qv;1

0

.

.. .

.. .

.. .

..

0

qv;Nv

0

..

.

0

3

7 .. 7 . 7 7 7 7 0 7 7 7 7 7 qh;Nh 7 7; 7 7 0 7 7 7 .. 7 . 7 7 7 7 0 5

ð13Þ

where qh and qv represent the MA filter components in the horizontal and vertical directions, respectively. [33] The results exposed in sections 4.2 and 4.3 demonstrate that the simplifying assumptions made regarding the form of the ARFIMA model used do not compromise its ability to capture the spatial structure of the radiobrightness temperature images of interest. These simplifications lead us to having to optimize Nh + Nv moving average filter components for each subregion in addition to se2, dx, and dy to fully specify our model. In all cases, optimization was performed by minimizing (10) with the parameters constrained to proper ranges to guarantee model invertibility [Shumway and Stoffer, 2001]. The Newton-Raphson algorithm was utilized for this purpose with several starting points selected to attempt to minimize the chances of convergence to local minima. The appropriate number of MA filter components to use for each subregion was selected by considering the following: (1) the marginal reductions of the cost function achieved by increasing the number of filter components; and (2) the visual goodness of fit to the periodogram of the observations near the origin. [34] For a two-dimensional pure MA model (the model obtained by letting dx = dy = 0 in (11) and (12)) having as in (13), the autocovariance functions in the horizontal and vertical directions can be shown to be:

ð11Þ

ð12Þ

8 > < s2 e

> :

NhP juj j¼ðNh jujÞ

qh; j qh; jþjuj for juj Nh ; and

ð14Þ

0 otherwise

In (14), we let qu = qu and q0 = 1. The MA autocovariance in the vertical direction is the same as that in (14) with u replaced by v and subscript h replaced by subscript v. The MA autocovariance functions provide insight on how the model in (11) and (12) is representing the spatial dependence at short lags. It indicates how the dependence of nearby observations would be characterized if it were possible to isolate the influence of long memory from the model, e.g., if we were to zoom into a stationary cluster of predominantly high or low observations such that long memory became nonobservable to us.


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Table 2. Optimal Differencing Parameters (dx, dy) in FIMA Models Within All Subregions in the Images for the Dates Analyzed Date Analyzed 18 June 19 June 20 June 1 July 2 July 3 July 12 July 13 July 14 July

Subregion 1 1.42, 0.17, 0.17, 0.22, 0.18, 0.13, 1.06, 1.18, 0.41,

0.30 1.48 1.49 1.53 1.48 1.56 1.12a 0.84 1.65a

2 0.17, 1.36, 1.34, 0.28, 0.43, 0.38, 0.43, 0.38, 0.32,

3 1.47 0.19 0.17 1.43 1.29 1.33 1.43 1.49 1.46

1.46, 1.00, 1.37, 1.63, 1.54, 1.55, 1.47, 1.53, 1.27,

4 0.12 0.82 0.36 0.23 0.32 0.21 0.39 0.29 0.45

1.42, 1.49, 1.59, 1.13, 1.46, 1.39, 1.40, 1.30, 1.07,

0.13 0.36 0.32 0.49a 0.48 0.44 0.30a 0.44a 0.66

5 0.73, 1.65, 1.72, 1.29, 1.22, 1.15, 0.95, 1.41, 0.69,

1.02 0.11 0.03 0.22a 0.36a 0.31a 0.92 0.30 1.11

6 NA 1.24, 1.48, 1.63, 1.57, 1.55, 0.77, 0.18, 0.22,

0.32a 0.22 0.42 0.40 0.37 1.13a 1.68+ 1.54

7 NA NA 0.18, 1.43, 1.17, 1.55, 1.49, 1.42, 1.42,

1.59 0.30 0.43 0.27a 0.15 0.28 0.16

8 NA NA NA 1.33, 1.14, 0.98, NA 1.44, 0.35,

9

0.16 0a 0.48 0.33 1.52

NA NA NA 1.39, 0.25 1.18, 0.23a NA NA 0.34, 1.56 NA

a Subregions for which these parameters resulted in image simulations having ensemble standard deviations with errors >25% with respect to the observations.

4.2. Analysis of Optimal Fractionally Integrated Moving Average (FIMA) Models [35] Table 2 shows the optimal or maximum likelihood estimates of dx and dy (see (11) and (12)) corresponding to each subregion in Table 1. Figure 3 exposes the high degree of anisotropy in scaling behavior captured by the optimal differencing parameters. Specifically, notice that only 2 (i.e., 3%) out of all 68 subregions studied have dx and dy within ±10% from their average, which is denoted as hdi = (dx + dy)/2. The results of Table 2 and Figure 3 illustrate the ability of FIMA models to capture and portray the different scaling behaviors observed along the horizontal and vertical directions for all subregions. To exemplify this further we consider the two subregions that have autocorrelation functions as given in Figures 1 and 2, respectively. Figures 4 and 5 compare the periodograms for the observations of these subregions to the fitted models (see equation (12)). In the first case, as depicted in Figures 1 and 4, dx = 1.00 and dy = 0.82. In the second case, displayed in Figures 2 and 5, dx = 1.39 and dy = 0.25. In these two instances, we see the clear and expected correspondence between increased (decreased) values of the differencing parameters, slower (faster) rates of decay of the autocorrelation matrices (see equation (5)), and faster (slower) rates of divergence to infinity in the smoothed periodograms and corresponding spectral density models (see equations (4) and (12)). [36] Since dx > 0 and/or dy > 0, we have that the brightness temperature fields of all subregions exhibit persistence or long memory as expected (see the discussion in section 2.2). Moreover, the reported ranges of differencing parameters are consistent with the behavior of the periodograms for the observations, which tend to infinity at the origin (this corresponds to dx > 0 and/or dy > 0 in equation (4)) rather than to zero (as would be the case if dx < 0 and dx < 0). A startling feature of the observations is that the differencing parameters for either or both the horizontal and vertical directions are beyond the stationary range, i.e., they exceed 0.5. This indicates that radiobrightness temperature images possess very high inherent variability since in this case the variance of the fitted FIMA models is formally infinite as discussed in section 2.2. [37] A final remark must be made regarding the differences in maximum likelihood differencing parameters even for subregions corresponding to observations in the same day. The results in Table 2 show that the differencing parameters are highly location-dependent. Establishing the causes of this variability constitutes a challenging research

topic on its own and falls beyond the scope of this paper. Nonetheless, work by other researchers may shed some light into this subject. Studies thus far suggest that the spatial structure of near-surface soil moisture fields is highly dependent on the soil and vegetation characteristics as well as on the spatial variability of precipitation. For instance, the results by Cosh and Brutsaert [1999] indicate that the near-surface soil moisture observations for the Washita ‘92 hydrology experiment may be considered stationary if one limits the study region to lie within an area having homogeneous soil type. This suggests that the clusters of predominantly high or low observations that are typical of scaling processes may coincide with areas of homogeneous soil properties. Further support for this hypothesis is found in the work by Rodriguez-Iturbe et al. [1995], who found that the porosity of the soil in the Washita ’92 study region also exhibits scaling properties, and by Peters-Lidard et al. [2001], who report scaling of field capacity, wilting point, and residual soil moisture for the same region. On the other hand, the near-surface soil moisture observations taken

Figure 3. Fraction of all 68 subregions having (top) dx and (bottom) dy within given deviations from their mean, which is denoted as hdi = (dx + dy)/2.

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Figure 4. Smoothed periodogram for the (top) observations and (bottom) spectral density model for subregion 3 in the image for 19 June 1997. Approximate 95% confidence limits for the smoothed periodogram are given as black mesh surfaces [Brillinger, 2001]. See color version of this figure in the HTML. during SGP97 show a strong response to rainfall variability and rather weak dependence on soil type given the much more heterogeneous spatial distribution of precipitation in SGP97 than in Washita ‘92 [Jackson et al., 1999; Oldak et al., 2002]. The work by Vinnikov et al. [1996] on the spatial structure of soil moisture fields also evinces a very strong dependence of the spatial distribution of soil moisture on the spatial structure of precipitation. Studies by Famiglietti et al. [1999] and Oldak et al. [2002] additionally report a marked dependence of the variance of soil moisture fields on the mean soil moisture content. Last, the distribution of vegetative cover has also been demonstrated to play a key role in the near surface soil moisture dynamics during SGP97 [Famiglietti et al., 1999]. Moreover, Peters-Lidard et al. [2001] report scaling properties hold for the leaf area index (LAI) of the Washita ’92 study region. [38] Table 3 shows the average number of horizontal and vertical moving average filter components (see equation (13)) for all subregions in the observations for a given day. Note that the MA autocorrelation functions for the horizon-

tal and vertical directions given in equation (14) are distinct from zero up to lag units equal to Nh and Nv , respectively (one lag unit equals 800 m). The results provided here have the same order of magnitude as those of Cosh and Brutsaert [1999], who report that the near-surface soil moisture images derived from radiobrightness temperature observations taken with ESTAR during Washita ’92 can be considered stationary within distances of approximately 5 km and modeled with exponential semivariograms having ranges on the order of 1 km over which near-surface soil moisture was spatially correlated. 4.3. Validation of Optimal FIMA Models and Comparison to Heuristic Approaches [39] In this section, we further validate the optimal FIMA models and establish comparisons to two of the heuristic approaches discussed in section 2.3. To accomplish this, we use the methodology presented by Parada and Liang [2003b], who employ the optimal FIMA models presented in section 4.2 coupled with a radiobrightness temperature

23 - 11

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Figure 5. As in Figure 4, but for subregion 9 in the image for 1 July1997. See color version of this figure in the HTML. downscaling and image simulation framework, to produce alternative realizations of the radiobrightness temperature fields for all subregions. In particular, we produce enough (approximately 550) unconditional realizations that mimic the autocovariance structure of the observed radiobrightness temperature field for each subregion so that the ensemble statistics of the process (e.g., the probability density function) converge. We present the results obtained from performing this exercise using the differencing parameters (dx and dy) obtained via four different approaches. In the first scenario, we produce realizations utilizing the optimal parameters obtained from FIMA models and reported in Table 2. In the second scenario, we fit FIMA models having the same number of horizontal and vertical MA filter components (see equation (13)) as those for which results are given in Tables 2 and 3, but a single differencing parameter (dx = dy). In the third case, we obtain estimates of dx and dy by linear regression on log-log plots of the periodograms for all subregions versus wave number as discussed in section 2.3. In doing this, we consider only the lowest seven frequencies (a, b = 1,. . ., 7 in equation (3)) to be consistent with the fact that long memory is a limit

property (see equation (4)). As noted in section 2.3, the choice on the cutoff or highest frequency considered for regression is subjective and may introduce bias into the results. Nonetheless, we have attempted to make a proper selection and found that including higher frequencies could result in significant overestimation of the standard deviation of the observed radiobrightness temperature images by the simulated fields. Finally, we use the estimates of dx + dy obtained by linear regression on log-log plots of the variance of the wavelet detail coefficients versus scale as outlined in section 2.3 and discussed further by Parada and Liang [2003b]. [40] Parada and Liang [2003b] found that while the autocorrelation structure of radiobrightness temperature Table 3. Optimal Number of Horizontal and Vertical MA Filter Components (Nh, Nv) Employed in FIMA Models Averaged Over All Subregions for a Given Date 18 June 19 June 20 June 1 July 2 July 3 July 12 July 13 July 14 July Nh Nv

3 4

3 4

3 4

4 4

3 3

3 3

2 2

3 3

2 2

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observations could be reproduced with relative ease, it proved much harder to mimic the standard deviation of these fields. This stems from a high sensitivity to the estimates of the differencing parameters (dx and dy), and the biases that may be introduced by the use of heuristic estimation techniques as discussed in section 2.3. Hence we focus on evaluating how well the realizations from the four different estimation approaches for dx and dy described above reproduce the standard deviation of the radiobrightness temperature observations for all subregions in an ensemble sense via the following objective measure: s ¼

hsr i 1 100: sobs

ð15Þ

In (15), hsri and sobs denote the estimates for the average standard deviation over many realizations and the standard deviation for the observations, respectively. For an evaluation on the goodness of fit of the autocorrelation structure of the realizations the reader is referred to Parada and Liang [2003b]. [41] Figure 6 illustrates how the differencing parameters derived from all four estimation techniques under consideration compare to one another. Clearly, the differencing parameters derived via linear regression on log-log plots of periodograms versus wave number are biased low with respect to the optimal estimates reported in Table 2. The opposite holds for the estimates obtained via linear regression on log-log plots of the variance of wavelet coefficients versus scale. On the other hand, the estimates obtained via optimal fitting of FIMA models with dx = dy seem to

Table 4. Fraction of All 68 Subregions Having Ensemble Average Standard Deviation Values, hsri, Less Than or Equal to or Greater Than Those for the Corresponding Observed Fields, sobs, When the Differencing Parameters Are Obtained Via Four Estimation Techniquesa FIMA: dx 6¼ dy FIMA: dx = dy ln(periodogram) ln(wavelet variance)

hsri sobs

hsri > sobs

0.57 0.53 0.91 0.26

0.43 0.47 0.09 0.74

a The four estimation techniques are (1) FIMA models with dx 6¼ dy, (2) FIMA models with dx = dy, (3) linear regression on log-log plots of periodograms versus scale (denoted ln(periodogram)), and (4) linear regression on log-log plots of wavelet variance versus scale (denoted ln(wavelet variance)).

fluctuate more or less randomly about those reported in Table 2. [42] Table 4 evaluates the ensemble bias incurred by the realizations in mimicking the standard deviation of the observations from all subregions when using the four different approaches to estimate the differencing parameters. These results clearly confirm the information depicted in Figure 6 and discussed above. Note that FIMA models with equal or distinct differencing parameters are capable of yielding realizations with unbiased ensemble average standard deviations, and that this does not apply for the two other heuristic estimation approaches considered. [43] Table 5 provides a quantitative measure based on (15) on how well the realizations that use the four different estimation approaches for the differencing parameters can reproduce the standard deviations of the observations. Notice that the estimates from FIMA models with dx 6¼ dy yield realizations which, on average, reproduce the standard deviation of the observed fields with errors less than or equal 5, 10, 15, 20 and 25% for 24, 51, 71, 78, and 79% of all 68 subregions respectively. Subregions for which the standard deviations of the observations could not be reproduced within reasonable error may require diagonal MA filter components, AR components, or simply larger samples to achieve a better fit. By contrasting these results to the ones from the realizations that make use of the other Table 5. Fraction of All 68 Subregions Having Absolute s (jsj) Values Less Than Those Specified in the First Column When the Differencing Parameters Are Obtained Via Four Estimation Techniques As in Table 4a

Figure 6. Comparison of differencing parameters for all 68 subregions obtained via four estimation techniques: (1) FIMA models with dx 6¼ dy; (2) FIMA models with dx = dy; (3) linear regression on log-log plots of periodograms versus wave number (denoted ln(periodogram)); and (4) linear regression on log-log plots of wavelet variance versus scale (denoted ln(wavelet variance)). The differencing parameters obtained via techniques 2, 3, and 4 are given in the ordinate and plotted against the corresponding estimates from technique 1 given in the abscissa.

Max. jsj, %

FIMA: dx 6¼ dy

FIMA: dx = dy

ln (periodogram)

ln (wavelet variance)

5 10 15 20 25 30 40 60 80 100 125 150 200 250

0.24 0.51 0.71 0.78 0.79 0.82 0.85 1 1 1 1 1 1 1

0.09 0.25 0.29 0.37 0.46 0.52 0.66 0.88 0.94 0.98 1 1 1 1

0.01 0.07 0.12 0.18 0.22 0.26 0.32 0.63 0.98 1 1 1 1 1

0.15 0.22 0.28 0.35 0.40 0.46 0.59 0.69 0.81 0.85 0.93 0.97 0.99 1

a Columns 2 (FIMA: dx 6¼ dy) and 5 (ln(wavelet variance)) are as in the work of Parada and Liang [2003b].


estimation techniques described, we conclude that the data at hand seems to be better characterized by the use of distinct differencing parameters along the horizontal and vertical directions. This verifies the assertion made in section 3 regarding the observed different rates of decay of the autocorrelation functions at large separation distances along the horizontal and vertical directions. Additionally, the results in Tables 4 and 5 demonstrate that FIMA models with dx 6¼ dy appear to provide not only unbiased but also robust estimates of scaling or differencing parameters. In contrast, FIMA models with dx = dy are unbiased, but seem to incur a significant loss in accuracy or robustness arising from the assumption of equal differencing parameters along the horizontal and vertical directions. Moreover, the estimates of the scaling exponents obtained by linear regression on loglog plots of periodograms versus wave number and variance of wavelet detail coefficients versus scale are not only biased as previously shown, they are also significantly less accurate than those corresponding to FIMA models with dx 6¼ dy .

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addition, it relates self-affine processes to long memory 1/f processes so that comparisons between the present study and earlier work on the characterization of near-surface soil moisture can be easily established. In doing this, we restrict the Hurst exponent such that 0 H < 1, namely the range over which fractional Brownian motion and fractional Gaussian noise are defined. For a more thorough coverage on this material, the reader may refer to Beran [1994], Mandelbrot [2000], and Percival and Walden [2000]. [47] Consider the following zero-mean self-affine process as in expression (1) of the main text: D

S ð xÞ ¼ tH S ð txÞ;

ðA1Þ

Consider also the first-order difference of S(x) or increment process S(x) {S(x) S(x 1)}. It can be shown that S(x) is also self-affine [Wornell, 1993; Beran, 1994; Percival and Walden, 2000] such that D

0

S ð xÞ ¼ tH S ð txÞ;

ðA2aÞ

H 0 ¼ H 1:

ðA2bÞ

5. Conclusions and Future Work [44] The analysis performed on the radiobrightness temperature observations derived from ESTAR during SGP97 suggest that these may be modeled as nonstationary long memory 1/f processes having distinct scaling properties along the horizontal (west to east) and vertical (south to north) directions. Moreover, the scaling properties of the radiobrightness temperature fields studied seem to vary significantly within the SGP97 study region. The validation results presented evince that FIMA models, as extended to two dimensions in this study, provide a suitable and robust framework for modeling stationary and nonstationary scaling long memory processes such as the radiobrightness temperature fields under consideration. As a consequence, they provide increased modeling flexibility since the need to subdivide a study region into stationary components is eliminated. The models also constitute a unified alternative for the joint modeling of the spatial dependence of observations separated by short distances by means of their MA components, as well as for observations that are far apart by means of their scaling or differencing components. Furthermore, the proposed FIMA models are sensitive enough to capture the distinct scaling properties along the horizontal and vertical directions observed to exist in the radiobrightness temperature fields obtained during SGP97, while preventing the introduction of biases inherent to other heuristic and less robust estimation techniques. [45] Currently, efforts are being devoted to the application of the FIMA framework for characterizing the spatial structure of the near-surface soil moisture images derived from the radiobrightness temperature fields analyzed in this study. We are also exploring alternatives that allow for more than one scaling or differencing parameter to describe the rate of divergence of the power spectrum to infinity within different ranges of wave numbers along each spatial direction.

Appendix A: fBm, fGn, and Their Representation as 1//f Processes [46] This appendix provides further details on self-affine processes as introduced in section 2.1 of the main text. In

Notice that log-log plots of variance versus pixel area must have slopes of 2H for S(x) and 2H 2 for S(x), respectively. To see this, let us rearrange and compute the variances in equations (A1) and (A2): h 2 i ¼ t2H var½S ð xÞ; var½S ð txÞ ¼ var tH S ð xÞ E tH S ð xÞ ðA3Þ h 2 i var½S ð txÞ ¼ var tH1 S ð xÞ E tH1 S ð xÞ ¼ t2H2 var½S ð xÞ;

ðA4Þ

where E[] and var[] denote the expectation and variance operators, respectively. If we now take the log on both sides of (A3) and (A4), respectively, the following linear relationships arise: logðvar½S ð txÞÞ ¼ 2H logðtÞ þ logðvar½S ð xÞÞ;

ðA5Þ

logðvar½S ð txÞÞ ¼ ð2H 2Þ logðtÞ þ logðvar½S ð xÞÞ: ðA6Þ

Moreover, both S(x) and S(x) can be characterized as 1/f processes having power spectra of the following form: c^S ðaÞ ¼ C jajl as a ! 0:

ðA7Þ

For S(x), l = 2H 1 whereas for S(x), l = 2H + 1, [Mandelbrot, 2000; Percival and Walden, 2000]. As stated in section 2.2 of the main text, a 1/f process exhibits long memory if l < 0. Furthermore, a long memory 1/f process has finite variance and hence is stationary for 1 < l < 0. For l 1, the variance of the long memory process becomes infinite, and thus the process is nonstationary [Beran, 1994; Percival and Walden, 2000]. This clearly indicates that S(x) with 0 H < 1 as in equation (A1) is

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nonstationary. On the other hand, the increment process S(x) is stationary for 0 H < 1. For H 1, both S(x) and S(x) are nonstationary. [48] The first comprehensive model for self-affine processes as in equations (A1) and (A2) with 0 H < 1, and S(x) being Gaussian is due to Mandelbrot and van Ness [1968], who designate S(x) as fractional Brownian motion and its stationary increments, S(x), as fractional Gaussian noise. Two widely known instances of fractional Gaussian noise include white noise with H = 0.5 and perfectly correlated sequences with H = 1. Hence from equation (A6), log-log plots of variance versus resolution should yield slopes of 1 and 0 for white noise and perfectly correlated sequences, respectively. [49] Acknowledgments. The authors are thankful to the anonymous reviewers for their insightful comments. 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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X. Liang and L. M. Parada, Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720, USA. ([email protected])