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In the decade before the U.S. Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) facility was established in 1992, early modern climate models exhibited large differences in the climate response [...]

On July 21,2020, we lost a good colleague and a dear friend, Michael Mishchenko, a scientist at NASA Goddard Institute for Space Studies. In 1983, Michael graduated from one of [...]

The National Oceanic and Atmospheric Administration (NOAA) Deep Space Climate Observatory (DSCOVR) spacecraft was launched on 11 February 2015 and in June 2015 achieved its orbit at the first Lagrange [...]

A new analytical statistical model describing the structure of broken cloud fields is presented. It depends on two parameters (cell size and occupancy probability) and provides chord distributions of clouds and gaps between them by length, as well as the cloud fraction distribution. This approach is based on the assumption that the structure of a cloud field is determined by a semiregular grid of cells (an abstraction of the atmospheric convective cells), which are filled with cloud with some probability. First, a simple discrete model is introduced, where clouds and gaps can occupy an integer number of cells, and then a continuous analog is developed, allowing for arbitrary cloud and gap sizes. The influence of a finite sample size on the retrieved statistics is also described. DOI: 10.1175/2010JAS3364.1

Cellular statistical models are designed to provide a simple two-parameter characterization of the structure of broken cloud fields described through distributions of cloud fraction and of chord lengths for clouds and clear gaps. In these analytical models cloud fields are assumed to occur on a semiregular grid of cells (which can be vaguely interpreted as atmospheric convective cells). In a simple, discrete cell model, cell size is fixed and each cell can either be completely filled with cloud with some probability or remain empty. Extending the discrete model to a continuous case provides more realism by allowing arbitrary cloud and gap sizes. Here the continuous cellular model is tested by comparing its statistics with those from large-eddy simulations (LES) of marine boundary layer clouds based on case studies from three trade-cumulus field projects. The statistics largely agree with some differences in small sizes approaching the LES model grid spacing. Exponential chord-length distributions follow from the assumption that the probability of any cell being cloudy is constant, appropriate for a given meteorological state (narrow sampling). Relaxing that assumption, and instead allowing this probability to have its own distribution, leads to a power-law distribution of chord lengths, appropriate to a broader sample of meteorological states (diverse sampling). DOI: 10.1175/2010JAS3365.1

A simple and fast algorithm for generating two correlated stochastic two-dimensional (2D) cloud fields is described. The algorithm is illustrated with two broken cumulus cloud fields: cloud optical depth and cloud-top height retrieved from the Moderate Resolution Imaging Spectroradiometer (MODIS). Only two 2D fields are required as an input. The algorithm output is statistical realizations of these two fields with approximately the same correlation and joint distribution functions as the original ones. The major assumption of the algorithm is statistical isotropy of the fields. In contrast to fractals and the Fourier filtering methods frequently used for stochastic cloud modeling, the proposed method is based on spectral models of homogeneous random fields. To retain the same probability density function as the (first) original field, the method of inverse distribution function is used. When the spatial distribution of the first field has been generated, a realization of the correlated second field is simulated using a conditional distribution matrix. This paper serves as a theoretical justification of the publicly available software 'Simulation of a two-component cloud field,' which has been recently released. Although 2D rather than full 3D, the stochastic realizations of two correlated cloud fields that mimic statistics of given fields have proven to be very useful to study 3D radiative transfer features of broken cumulus clouds for a better understanding of shortwave radiation and the interpretation of remote sensing retrievals.

Laser beams emitted from the Geoscience Laser Altimeter System (GLAS), as well as other spaceborne laser instruments, can only penetrate clouds to a limit of a few optical depths. As a result, only optical depths of thinner clouds (< about 3 for GLAS) are retrieved from the reflected lidar signal. This paper presents a comprehensive study of possible retrievals of optical depth of thick clouds using solar background light and treating GLAS as a solar radiometer. To do so one must first calibrate the reflected solar radiation received by the photon-counting detectors of the GLAS 532-nm channel, the primary channel for atmospheric products. Solar background radiation is regarded as a noise to be subtracted in the retrieval process of the lidar products. However, once calibrated, it becomes a signal that can be used in studying the properties of optically thick clouds. In this paper, three calibration methods are presented: (i) calibration with coincident airborne and GLAS observations, (ii) calibration with coincident Geostationary Operational Environmental Satellite (GOES) and GLAS observations of deep convective clouds, and (iii) calibration from first principles using optical depth of thin water clouds over ocean retrieved by GLAS active remote sensing. Results from the three methods agree well with each other. Cloud optical depth (COD) is retrieved from the calibrated solar background signal using a one-channel retrieval. Comparison with COD retrieved from GOES during GLAS overpasses shows that the average difference between the two retrievals is 24%. As an example, the COD values retrieved from GLAS solar background are illustrated for a marine stratocumulus cloud field that is too thick to be penetrated by the GLAS laser. Based on this study, optical depths for thick clouds will be provided as a supplementary product to the existing operational GLAS cloud products in future GLAS data releases.

The Multiangle Imaging Spectroradiometer (MISR) views the earth with nine cameras, ranging from a 70[degrees] zenith angle viewing forward through nadir to 70[degrees] viewing aft. MISR does not have an operational cloud optical depth retrieval algorithm, but previous research has hinted that solar reflection measured in multiple directions might improve cloud optical depth retrievals. This study explores the optical depth information content of MISR's multiple angles using a retrieval simulation approach. Hundreds of realistic boundarylayer cloud fields are generated with large-eddy simulation (LES) models for stratocumulus, small trade cumulus, and land surface-forced fair-weather cumulus. Reflectances in MISR directions are computed with three-dimensional radiative transfer from the LES cloud fields over an ocean surface and averaged to MISR resolution and sampled at MISR 275-m pixel spacing. Neural networks are trained to retrieve the mean and standard deviation of optical depth over different size pixel patches from the mean and standard deviation of simulated MISR reflectances. Various configurations of MISR cameras are input to the retrieval, and the rms retrieval errors are compared. For 5 x 5 pixel patches the already low mean optical depth retrieval error for stratocumulus decreases 41% and 23% (for 25[degrees] and 45[degrees] solar zenith angles, respectively) from using only the nadir camera to using seven MISR cameras. For cumulus, however, the much higher normalized optical depth retrieval error only decreases around 14%. These small improvements suggest that measurements of solar reflection in multiple directions do not contribute substantially to more accurate optical depth retrievals for small cumulus clouds. The 3D statistical retrievals, however, even with only the nadir camera, are much more accurate for small cumulus than standard nadir plane-parallel retrievals; therefore, this approach may be worth pursuing.

The effect of scattering angle on Moderate Resolution Imaging Spectroradiometer (MODIS) retrievals of cloud drop effective radius is studied using ensembles of cumulus clouds with varying sun-satellite scattering geometries. The results are interpreted as shadowing and illumination effects. When 3-D clouds are viewed near the backscatter geometry, well-illuminated cloud surfaces are seen, and the retrievals based on plane-parallel geometry underestimate the effective radius. The reverse is true when the satellite is far from the backscatter position, and the shadowed portions of clouds are observed. The shadowing geometry produces a larger bias than the illuminated geometry. These differences between the shadowed and the illuminated ensembles decrease toward zero as the clouds become shallower. Removing the edge pixels based on 1-km-scale geometry partially reduces biases due to the 3-D effects and surface contamination. Recommendations are provided for reducing the 3-D cloud effects using current satellite retrieval algorithms. Index Terms--Clouds, remote sensing, terrestrial atmosphere.

The interaction of clouds with solar and terrestrial radiation is one of the most important topics of climate research. In recent years it has been recognized that only a full three-dimensional (3D) treatment of this interaction can provide answers to many climate and remote sensing problems, leading to the worldwide development of numerous 3D radiative transfer (RT) codes. The international Intercomparison of 3D Radiation Codes (I3RC), described in this paper, sprung from the natural need to compare the performance of these 3D RT codes used in a variety of current scientific work in the atmospheric sciences. I3RC supports intercomparison and development of both exact and approximate 3D methods in its effort to 1) understand and document the errors/limits of 3D algorithms and their sources; 2) provide 'baseline' cases for future code development for 3D radiation; 3) promote sharing and production of 3D radiative tools; 4) derive guidelines for 3D radiative tool selection; and 5) improve atmospheric science education in 3D RT. Results from the two completed phases of I3RC have been presented in two workshops and are expected to guide improvements in both remote sensing and radiative energy budget calculations in cloudy atmospheres.

By analyzing aircraft measurements of individual drop sizes in clouds, it has been shown in a companion paper that the probability of finding a drop of radius r at a linear scale I decreases as [l.sup.D(r)], where 0 [less than or equal to] D(r) [less than or equal to] 1. This paper shows striking examples of the spatial distribution of large cloud drops using models that simulate the observed power laws. In contrast to currently used models that assume homogeneity and a Poisson distribution of cloud drops, these models illustrate strong drop clustering, especially with larger drops. The degree of clustering is determined by the observed exponents D(r). The strong clustering of large drops arises naturally from the observed power-law statistics. This clustering has vital consequences for rain physics, including how fast rain can form. For radiative transfer theory, clustering of large drops enhances their impact on the cloud optical path. The clustering phenomenon also helps explain why remotely sensed cloud drop size is generally larger than that measured in situ.

A new method for retrieving cloud optical depth from ground-based measurements of zenith radiance in the red (RED) and near-infrared (NIR) spectral regions is introduced. Because zenith radiance does not have a one-to-one relationship with optical depth, it is absolutely impossible to use a monochromatic retrieval. On the other side, algebraic combinations of spectral radiances, such as normalized difference cloud index (NDCI), while largely removing nonuniqueness and the radiative effects of cloud inhomogeneity, can result in poor retrievals due to its insensitivity to cloud fraction. Instead, both RED and NIR radiances as points on the 'RED versus NIR' plane are proposed to be used for retrieval. The proposed retrieval method is applied to Cimel measurements at the Atmospheric Radiation Measurements (ARM) site in Oklahoma. Cimel, a multichannel sun photometer, is a part of the Aerosol Robotic Network (AERONET)--a ground-based network for monitoring aerosol optical properties. The results of retrieval are compared with the ones from microwave radiometer (MWR) and multifilter rotating shadowband radiometer (MFRSR) located next to Cimel at the ARM site. In addition, the performance of the retrieval method is assessed using a fractal model of cloud inhomogeneity and broken cloudiness. The preliminary results look very promising both theoretically and from measurements.

Statistical scale-by-scale analysis, for the first time, has been applied to the aerosol optical thickness (AOT) retrieved from the Multi-Filter Rotating Shadowband Radiometer (MFRSR) network. The MFRSR data were collected in September 2000 from the dense local network operated by the U.S. Department of Energy Atmospheric Radiation Measurement program, located in Oklahoma and Kansas. These data have 20-s temporal resolution. The instrument sites form an irregular grid with the mean distance between neighboring sites about 80 km. It is found that temporal variability of AOT can be separated into two well-established scale-invariant regimes: 1) microscale (0.5-15 km), where fluctuations are governed by 3D turbulence, and 2) intermediate scale (15 100 km), characterized by a transition toward large-scale 2D turbulence. The spatial scaling of AOT was determined by the comparison of retrievals between different instrument sites (distance range 30-400 km). The authors investigate how simultaneous determination of AOT scaling in space and time can provide means to examine the validity of Taylor's frozen turbulence hypothesis. The temporal evolution of AOT scaling exponents during the month appeared to be well correlated with changes in aerosol vertical distribution, while their spatial variability reflects the concavity/convexity of the site topography. Explanations based on dynamical processes in atmospheric convective boundary layer are suggested.

Here, previous work using photon diffusion theory to describe radiative transfer through dense plane-parallel clouds at nonabsorbing wavelengths is extended. The focus is on the scaling of space- and time-domain moments for transmitted light with respect to cloud thickness H and optical depth [tau]; and the new results are as follows: accurate prefactors for asymptotic scaling, preasymptotic correction terms in closed form, 3D effects for internal variability in [tau], and the rms transit time or pathlength. Mean pathlength is [alpha]H for dimensional reasons and, from random-walk theory, we already know that it is also [alpha](1 - g)[tau] for large enough [tau] (g being the asymmetry factor). Here, it is shown that the prefactor is precisely 1/2 and that corrections are significant for (1 - g)[tau] < 10, which includes most actual boundary layer clouds. It is also shown that rms pathlength is not much larger than the mean for transmittance (its prefactor is [square root of 7/20] [approximately equal to 0.59]); this proves that, in sharp contrast with reflection, pathlength distributions are quite narrow in transmission. If the light originates from a steady point source on a cloud boundary, a fuzzy spot is observed on the opposite boundary. This problem is formally mapped to the pulsed source problem, and it is shown that the rms radius of this spot slowly approaches [square root of 2/3H] as [tau] increases; it is also shown that the transmitted spot shape has a flat top and an exponential tail. Because all preasymptotic corrections are computed here, the diffusion results are accurate when compared to Monte Carlo counterparts for [tau] [greater than or equal to] 5, whereas the classic scaling relations apply only for [tau] [greater than or equal to] 70, assuming g = 0.85. The temporal quantities shed light on observed absorption properties and optical lightning waveforms. The spatial quantity controls the three-dimensional radiative smoothing process in transmission, which was recently observed in spectral analyses of time series of zenith radiance at 725 nm. Opportunities in ground-based cloud remote sensing using the new developments are described and illustrated with simulations of 3D solar radiative transfer in realistic models of stratocumulus. Finally, since this analytical diffusion study applies only to weakly variable stratus layers, extensions to more complex cloud systems using anomalous diffusion theory are discussed.

When cloud properties are retrieved from satellite observations, current calculations apply one-dimensional (1D) theory to the three-dimensional (3D) world: they consider only vertical processes and ignore horizontal interactions. This paper proposes a novel approach that estimates 3D effects in cloud optical thickness retrievals. The proposed method combines visible and thermal infrared images to see whether 3D radiative effects make clouds appear asymmetric--that is, whether cloud surfaces tilted toward the sun are systematically brighter than surfaces tilted away from it. The observed asymmetries are then used to estimate 3D effects for 1-km-size pixels as well as 50-km-size areas. Initial results obtained for Moderate-Resolution Imaging Spectroradiometer (MODIS) images reveal that 3D effects cause abundant uncertainties in the 1-km-resolution 1D retrievals. Averaging over 50 km by 50 km areas greatly reduces the errors but does not remove them completely. Conservative estimates show that the mean optical thickness values are biased by more than 10% in 10% of the areas, and the errors in the areas' standard deviation values are more than 10% in about 20% of areas. (Manuscript received 14 May 2001, in final form 7 September 2001)

The independent pixel approximation (IPA), which treats radiative properties of each pixel independently by using standard plane-parallel calculations preserves scale-invariance found in the analyses of the horizontal variability of liquid water in marine stratocumulus clouds. Several studies, however, report a violation of scale-invariance in LANDSAT cloud radiance fields that are much smoother than cloud structure on small scales. This shows a limitation of IPA on small scales: it is unable to simulate the smooth small-scale behavior that is due to the horizontal photon transport. This paper introduces a 'nonlocal' independent pixel approximation (NIPA) that extends the IPA by incorporating empirically the smoothing effects of horizontal interpixel fluxes through a convolution product of the IPA and an approximate Green function for radiative transfer. We also address the inverse problem of cloud optical depth retrieval from satellite data, showing how NIPA can be used to overcome the limitations of current IPA-based methods at small scales.

This is the second of two papers analyzing the internal liquid water content (LWC) structure of marine stratocumulus (Sc) based on observations taken during the First ICCP (International Commission on Cloud Physics) Regional Experiment (FIRE) 1987 and Atlantic Stratocumulus Transition Experiment (ASTEX) 1992 field programs. Part I examined wavenumber spectra and the three-decade scale range (tens of meters to tens of kilometers) over which scale invariance holds; the inability of spectral analysis to distinguish between different random processes was also underscored. This indetermination is removed in this part by applying multifractal analysis techniques to the LWC fields, leading to a characterization of the role of intermittency in marine Sc. Two multiscaling statistics are computed and associated nonincreasing hierarchies of exponents are obtained: structure functions and H(q), singular measures and D(q). The real variable q is the order of a statistical moment (e.g., q = 1.0 yields a mean); D(q) quantifies intermittency, H(q) nonstationarity. Being derived from the slopes of lines on log(statistic) versus log(scale) plots, these exponents are only defined when those lines are reasonably straight and where this happens defines the scale-invariant range. Being nonconstant, the derived H(q) and D(q) indicate multifractality rather than monofractality of LWC fields. Two exponents can serve as first-order measures of nonstationarity and intermittency: [H.sub.1] = H(1) and [C.sub.1] = 1 - D(1). For the ensemble average of all FIRE and all ASTEX data, the authors find the two corresponding points in the ([H.sub.1], [C.sub.1]) plane to be close: (0.28, 0.10) for FIRE and (0.29, 0.08) for ASTEX. This indicates that the dynamics determining the internal structure of marine Sc depend little on the local climatology. In contrast, the scatter of spatial averages for the individual flight around the ensemble average illustrates ergodicity violation. Finally, neither multiplicative cascades (with [H.sub.1] = 0) nor additive Gaussian models such as fractional Brownian motions (with [C.sub.1] = 0) adequately reproduce the LWC fluctuations in marine Sc.

Several studies have uncovered a break in the scaling properties of Landsat cloud scenes at nonabsorbing wavelengths. For scales greater than 200-400 m, the wavenumber spectrum is approximately power law in [k.sup.-5/3], but from there down to the smallest observable scales (50-100 m) follows another [k.sup.-[Beta]] law with [Beta] > 3. This implies very smooth radiance fields. The authors reexamine the empirical evidence for this scale break and explain it using fractal cloud models, Monte Carlo simulations, and a Green function approach to multiple scattering theory. In particular, the authors define the 'radiative smoothing scale' and relate it to the characteristic scale of horizontal photon transport. The scale break was originally thought to occur at a scale commensurate with either the geometrical thickness A. of the cloud, or with the 'transport' mean free path [l.sub.t] = [[(1 - g)[Sigma]].sup.-1], which incorporates the effect of forward scattering ([Sigma] is extinction and g the asymmetry factor of the phase function). The smoothing scale is found to be approximately [square root of [l.sub.t][[Delta].sub.z]] at cloud top; this is the prediction of diffusion theory which applies when [(1 - g).sub.[Tau]] = [[Delta].sub.z]/[l.sub.t] [greater than or equal to] 1 ([Tau] is optical thickness). Since the scale break is a tangible effect of net horizontal radiative fluxes excited by the fluctuations of [Tau], the smoothing scale sets an absolute lower bound on the range where one can neglect these fluxes and use plane-parallel theory locally, even for stratiform clouds. In particular, this constrains the retrieval of cloud properties from remotely sensed data. Finally, the characterization of horizontal photon transport suggests a new lidar technique for joint measurements of optical and geometrical thicknesses at about 0.5-km resolution.

This study investigates the internal structure of marine stratocumulus (Sc) using the spatial fluctuations of liquid water content (LWC) measured along horizontal flights off the coast of southern California during the First ISCCP Regional Experiment (FIRE) in summer of 1987. The results of FIRE 87 data analyses are compared to similar ones for marine Sc probed during the Atlantic Stratocumulus Transition Experiment (ASTEX) in summer 1992 near the Azores. In this first of two parts, the authors use spectral analysis to determine the main scale-invariant regimes, defined by the ranges of scales where wavenumber spectra follow power laws; from there, they discuss stationarity issues. Although crucial for obtaining meaningful spatial statistics (e.g., in climate diagnostics), the importance of establishing stationarity - statistical invariance under translation - is often overlooked. The sequel uses multifractal analysis techniques and addresses intermittency issues. By improving our understanding of both nonstationarity and intermittency in atmospheric data, we are in a better position to formulate successful sampling strategies. Comparing the spectral responses of different instruments to natural LWC variability, the authors find scale breaks (characteristic scales separating two distinct power law regimes) that are spurious, being traceable to well-documented idiosyncrasies of the Johnson - Williams probe and forward scattering spectrometer probes. In data from the King probe, the authors find no such artifacts; all spectra are of the scale-invariant form [k.sup.-[Beta]] with exponents [Beta] in the range 1.1-1.7, depending on the flight. Using the whole FIRE 87 King LWC database, the authors find power-law behavior with [Beta] = 1.36 [+ or -] 0.06 from 20 m to 20 km. From a spectral vantage point, the ASTEX cloud system behaves statistically like a scaled-up version of FIRE 87: a similar exponent [Beta] = 1.43 [+ or -] 0.08 is obtained, but the scaling range is shifted to [60 m, 60 km], possibly due to the 2-3 times greater boundary layer thickness. Finally, the authors reassess the usefulness of spectral analysis: Its main shortcoming is ambiguity: very different looking stochastic processes can yield similar, even identical, spectra. This problem impedes accurate modeling of the LWC data and, ultimately, is why multifractal methods are required. Its main asset is applicability in stationary and nonstationary situations alike and, in conjunction with scaling, it can be used to detect nonstationary behavior in data. Having [Beta] [greater than] 1, LWC fields in marine Sc are nonstationary within the scaling range and stationary only at larger scales. Nonstationarity implies long-range correlations, and we demonstrate the damage these cause when trying to estimate means and standard deviations with limited amounts of LWC data.

This paper presents a simple yet general approach to estimate the uncertainties that arise in satellite retrievals of cloud optical depth when the retrievals use one-dimensional radiative transfer theory for heterogeneous clouds that have variations in all three dimensions. For the first time, preliminary error bounds are set to estimate the uncertainty of cloud optical depth retrievals. These estimates can help us better understand the nature of uncertainties that three-dimensional effects can introduce into retrievals of this important product of the Moderate Resolution Imaging Spectroradiometer instrument. The probability distribution of resulting retrieval errors is examined through theoretical simulations of shortwave cloud reflection for a set of cloud fields that represent the variability of stratocumulus clouds. The results are used to illustrate how retrieval uncertainties change with observable and known parameters, such as solar elevation or cloud brightness. Furthermore, the results indicate that a tendency observed in an earlier study--clouds appearing thicker for oblique sun--is indeed caused by three-dimensional radiative effects.

In the third part of the 'Cellular Statistical Models of Broken Cloud Fields' series the cloud statistics formalism developed in the first two parts is interpreted in terms of the theory of Markov processes. The master matrix introduced in this study is a unifying generalization of both the cloud fraction probability distribution function (PDF) and the Markovian transition probability matrix. To illustrate the new concept, the master matrix is used for computation of the moments of the cloud fraction PDF-in particular, the variance-which until now has not been analytically derived in the framework of the authors' previous work. This paper also serves as a bridge to the proposed future studies of the effects of sampling and averaging on satellite-based cloud masks. [ABSTRACT FROM AUTHOR]

A method for inferring cloud optical depth [tau] is introduced and assessed using simulated surface radiometric measurements produced by a Monte Carlo algorithm acting on fields of broken, single-layer, boundary layer clouds derived from Landsat imagery. The method utilizes a 1D radiative transfer model and time series of zenith radiances and irradiances measured at two wavelengths, [[lambda].sub.1] and [[lambda].sub.2], from a single site with surface albedos [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Assuming that clouds transport radiation in accordance with 1D theory and have spectrally invariant optical properties, inferred optical depths [tau]' are obtained through cloud-base reflectances that are approximated by differencing spectral radiances and estimating upwelling fluxes at cloud base. When initialized with suitable values of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and cloud-base altitude h, this method performs well at all solar zenith angles. Relative mean bias errors for [tau]' are typically less than 5% for these cases. Relative variances for [tau]' for given values of inherent [tau] are almost independent of inherent [tau] and are Though results are shown only for surfaces resembling green vegetation (i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the performance of this method depends little on the values of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, if radiometric data have sufficient signal-to-noise ratios and suitable wavelengths can be found, this method should yield reliable estimates of [tau] for broken clouds above many surface types.

Approximate Entropy and Sample Entropy are two algorithms for determining the regularity of series of data based on the existence of patterns. Despite their similarities, the theoretical ideas behind those techniques are different but usually ignored. This paper aims to be a complete guideline of the theory and application of the algorithms, intended to explain their characteristics in detail to researchers from different fields. While initially developed for physiological applications, both algorithms have been used in other fields such as medicine, telecommunications, economics or Earth sciences. In this paper, we explain the theoretical aspects involving Information Theory and Chaos Theory, provide simple source codes for their computation, and illustrate the techniques with a step by step example of how to use the algorithms properly. This paper is not intended to be an exhaustive review of all previous applications of the algorithms but rather a comprehensive tutorial where no previous knowledge is required to understand the methodology. [ABSTRACT FROM AUTHOR]