Probability Distribution Of Intensity Research Articles

Maximum Likelihood Deconvolution of Beamforming Images with Signal-Dependent Speckle Fluctuations

Ocean Acoustic Waveguide Remote Sensing (OAWRS) typically utilizes large-aperture linear arrays combined with coherent beamforming to estimate the spatial distribution of acoustic scattering echoes. The conventional maximum likelihood deconvolution (DCV) method uses a likelihood model that is inaccurate in the presence of multiple adjacent targets with significant intensity differences. In this study, we propose a deconvolution algorithm based on a modified likelihood model of beamformed intensities (M-DCV) for estimation of the spatial intensity distribution. The simulated annealing iterative scheme is used to obtain the maximum likelihood estimation. An approximate expression based on the generalized negative binomial (GNB) distribution is introduced to calculate the conditional probability distribution of the beamformed intensity. The deconvolution algorithm is further simplified with an approximate likelihood model (AM-DCV) that can reduce the computational complexity for each iteration. We employ a direct deconvolution method based on the Fourier transform to enhance the initial solution, thereby reducing the number of iterations required for convergence. The M-DCV and AM-DCV algorithms are validated using synthetic and experimental data, demonstrating a maximum improvement of 73% in angular resolution and a sidelobe suppression of 15 dB. Experimental examples demonstrate that the imaging performance of the deconvolution algorithm based on a linear small-aperture array consisting of 16 array elements is comparable to that obtained through conventional beamforming using a linear large-aperture array consisting of 96 array elements. The proposed algorithm is applicable for Ocean Acoustic Waveguide Remote Sensing (OAWRS) and other sensing applications using linear arrays.

Exploring the non-Gaussianity of the cosmic infrared background and its weak gravitational lensing

ABSTRACT Gravitational lensing deflects the paths of photons, altering the statistics of cosmic backgrounds and distorting their information content. We take the cosmic infrared background (CIB), which provides plentiful information about galaxy formation and evolution, as an example to probe the effect of lensing on non-Gaussian statistics. Using the Websky simulations, we first quantify the non-Gaussianity of the CIB, revealing additional detail on top of its well-measured power spectrum. To achieve this, we use needlet-like multipole-band filters to calculate the variance and higher-point correlations. Using our simulations, we show the two-, three- and four-point spectra, and compare our calculated power spectra and bispectra to Planck values. We then lens the CIB, shell-by-shell with corresponding convergence maps, to capture the broad redshift extent of both the CIB and its lensing convergence. The lensing of the CIB changes the three- and four-point functions by a few tens of per cent at large scales, unlike with the power spectrum, which changes by less than two per cent. We expand our analyses to encompass the full intensity probability distribution functions (PDFs) involving all n-point correlations as a function of scale. In particular, we use the relative entropy between lensed and unlensed PDFs to create a spectrum of templates that can allow estimation of lensing. The underlying CIB model is missing the important role of star bursting, which we test by adding a stochastic lognormal term to the intensity distributions. The novel aspects of our filtering and lensing pipeline should prove useful for any radiant background, including line intensity maps.

FractalRG: Advanced fractal region growing using Gaussian mixture models for left atrium segmentation

This paper presents an advanced region growing method for precise left atrium (LA) segmentation and estimation of atrial wall thickness in CT/MRI scans. The method leverages a Gaussian mixture model (GMM) and fractal dimension (FD) analysis in a three-step procedure to enhance segmentation accuracy. The first step employs GMM for seed initialization based on the probability distribution of image intensities. The second step utilizes fractal-based texture analysis to capture image self-similarity and texture complexity. An enhanced approach for generating 3D fractal maps is proposed, providing valuable texture information for region growing. In the last step, fractal-guided 3D region growing is applied for segmentation. This process expands seed points iteratively by adding neighboring voxels meeting specific similarity criteria. GMM estimations and fractal maps are used to restrict the region growing process, reducing the search space for global segmentation and enhancing computational efficiency. Experiments on a dataset of 10 CT scans with 3,947 images resulted in a Dice score of 0.85, demonstrating superiority over traditional techniques. In a dataset of 30 MRI scans with 3,600 images, the proposed method achieved a competitive Dice score of 0.89±0.02, comparable to Deep Learning-based models. These results highlight the effectiveness of our approach in accurately delineating the LA region across diverse imaging modalities.

Intense Tropical Cyclones in the Western North Pacific Under Global Warming: A Dynamical Downscaling Approach

AbstractThis study aims to assess the global warming's impact on intense tropical cyclones (TCs) over the western North Pacific (WNP) through dynamical downscaling. 379 and 179 TCs reaching Category 1 in the High‐Resolution Atmospheric Model (HiRAM) are downscaled for use in the Weather Research and Forecasting (WRF) model at 5‐km horizontal resolution in the current climate (1979–2015) and for use in the Representative Concentration Pathways 8.5 (RCP8.5) in the future climate (2074–2100) scenarios, respectively. Inclusion of the downscaling simulations by WRF helps better reproduce the probability distribution of the TC's lifetime maximum intensity (LMI). In the warmer climate, the LMI of very intense TCs in WNP are projected to be stronger. Such an increase in intensity is statistically significant, and can be primarily explained by enhanced intensification rate. Meanwhile, TCs among the top 5% in LMI can reach higher intensities which cannot be attained in the current climate. After downscaling, the probability of WNP TCs reaching Category 4–5 increases by 6.5 percentage points in the late 21st century. This increase is 1.7 percentage points higher than the increase projected exclusively by HiRAM. Moreover, for TCs among the top 5% in LMI, a 233‐km and 300‐km westward shift of LMI locations is identified in the late 21st century for simulations with and without the downscaling approach, respectively. Both results suggest that very intense TCs would pose a higher threat to the WNP lands under global warming, as they become substantially stronger, and as their LMI locations migrate toward the coast.

Late gadolinium enhancement entropy as a new measure of myocardial tissue heterogeneity for prediction of adverse cardiac events in patients with hypertrophic cardiomyopathy

ObjectivesEntropy is a new late gadolinium enhanced (LGE) cardiac magnetic resonance (CMR)–derived parameter that is independent of signal intensity thresholds. Entropy can be used to measure myocardial tissue heterogeneity by comparing full pixel points of tissue images. This study investigated the incremental prognostic value of left ventricular (LV) entropy in patients with hypertrophic cardiomyopathy (HCM).MethodsThis study enrolled 337 participants with HCM who underwent 3.0-T CMR. The LV entropy was obtained by calculating the probability distribution of the LV myocardial pixel signal intensities of the LGE sequence. Patients who underwent CMR imaging were followed up for endpoints. The primary endpoint was defined as readmission to the hospital owing to heart failure. The secondary endpoint was the composite of the primary endpoint, sudden cardiac death and non-cardiovascular death.ResultsDuring the median follow-up of 24 months ± 13 (standard deviation), 43 patients who reached the primary and secondary endpoints had a higher entropy (6.20 ± 0.45, p < 0.001). The patients with increased entropy (≥ 5.587) had a higher risk of the primary and secondary endpoints, compared with HCM patients with low entropy (p < 0.001 for both). In addition, Cox analysis showed that LV entropy provided significant prognostic value for predicting both primary and secondary endpoints (HR: 1.291 and 1.273, all p < 0.001). Addition of LV entropy to the multivariable model improved model performance and risk reclassification (p < 0.05).ConclusionLV entropy assessed by CMR was an independent predictor of primary and secondary endpoints. LV entropy assessment contributes to improved risk stratification in patients with HCM.Critical relevance statementMyocardial heterogeneity reflected by entropy the derived parameter of LGE has prognostic value for adverse events in HCM. The measurement of LV entropy helped to identify patients with HCM who were at risk for heart failure and sudden cardiac death.Key points• Left ventricular entropy can reflect myocardial heterogeneity in HCM patients.• Left ventricular entropy was significantly higher in HCM patients who reached endpoint events.• Left ventricular entropy helps to predict the occurrence of heart failure and death in HCM patients.Graphical

Detection of Meditation-Induced HRV Dynamics Using Averaging Technique-Based Oversampled Feature Set and Machine Learning Classifiers

In this paper, we propose a textural information-based analysis of scalogram image obtained from continuous wavelet transform of heart rate variability (HRV) signal to study its dynamics during meditation. In addition to features from scalogram image, visibility graph-based complexity measures and multiscale permutation entropies (MPEs) from HRV signal are used to elucidate the modulation in autonomic activity of heart during meditative and non-meditative state. Significant changes in the probability distribution of pixel intensities of scalogram image and undiminished permutation entropy at higher scales are observed during the meditative state. From the extracted features, we have selected the top-ranked features based on ReliefF algorithm and minimum threshold weight of importance set at 0.04. Considering the small sample size (12 subjects) of meditation dataset, we have employed a novel data augmentation technique based on averaging of feature sets to overcome the issue of overfitting. To examine the efficacy of the proposed technique, both the non-augmented and augmented data are applied to four different types of classifiers, namely k-nearest neighbor, support vector machine (SVM), logistic regression and random forest classifiers. Experimental results demonstrate that performance of classifiers in distinguishing meditative and pre-meditative state are much superior with the augmented data as compared to that with regular non-augmented data. Out of these classifiers, radial basis function (RBF)-based SVM results in the best performance with an average accuracy of 96.67%, sensitivity of 95.83% and specificity of 97.5%.

400 Gb/s physical random number generation based on deformed square self-chaotic lasers

A circular-sided square microcavity laser etched a central hole has achieved chaos operation with a bandwidth of 20.8 GHz without external optical feedback or injection, in which the intensity probability distribution of a chaotic signal with a two-peak pattern was observed. Based on the self-chaotic microlaser, physical random numbers at 400 Gb/s were generated by extracting the four least significant bits without other complex post-processing methods. The solitary chaos laser and minimal post-processing have predicted a simpler and low-cost on-chip random number generator in the future.

Multi-hazard joint probability distribution model for wind speed, wind direction and rain intensity

Multiple disasters such as strong wind and torrential rain pose great threats to civil infrastructures. However, most existing studies ignored the dependence structure between them, as well as the effect of wind direction. From the dimension of the engineering sector, this paper introduces the vine copula to model the joint probability distribution (JPD) of wind speed, wind direction and rain intensity based on the field data in Yangjiang, China during 1971–2020. First, the profiles of wind and rain in the studied area are statistically analyzed, and the original rainfall amounts are converted into short-term rain intensity. Then, the marginal distributions of individual variables and their pairwise dependence structures are built, followed by the development of the trivariate joint distribution model. The results show that the constructed vine copula-based model can well characterize the dependence structure between wind speed, wind direction and rain intensity. Meanwhile, the JPD characteristics of wind speed and rain intensity show significant variations depending on wind direction, thus the effect of wind direction cannot be neglected. The proposed JPD model will be conducive for reasonable and precise performance assessment of structures subjected to multiple hazards of wind and rain actions.

Machine learning analysis of instabilities in noise-like pulse lasers.

Neural networks have been recently shown to be highly effective in predicting time-domain properties of optical fiber instabilities based only on analyzing spectral intensity profiles. Specifically, from only spectral intensity data, a suitably trained neural network can predict temporal soliton characteristics in supercontinuum generation, as well as the presence of temporal peaks in modulation instability satisfying rogue wave criteria. Here, we extend these previous studies of machine learning prediction for single-pass fiber propagation instabilities to the more complex case of noise-like pulse dynamics in a dissipative soliton laser. Using numerical simulations of highly chaotic behaviour in a noise-like pulse laser operating around 1550 nm, we generate large ensembles of spectral and temporal data for different regimes of operation, from relatively narrowband laser spectra of 70 nm bandwidth at the -20 dB level, to broadband supercontinuum spectra spanning 200 nm at the -20 dB level and with dispersive wave and long wavelength Raman extension spanning from 1150-1700 nm. Using supervised learning techniques, a trained neural network is shown to be able to accurately correlate spectral intensity profiles with time-domain intensity peaks and to reproduce the associated temporal intensity probability distributions.

Entropy as a Measure of Myocardial Tissue Heterogeneity in Patients With Ventricular Arrhythmias

ObjectivesThe authors investigated the incremental prognostic value of entropy, a novel measure of myocardial tissue heterogeneity by cardiac magnetic resonance (CMR) imaging in patients presenting with ventricular arrhythmias (VAs). BackgroundCMR can characterize myocardial areas serving as arrhythmogenic substrate. MethodsConsecutive patients undergoing CMR imaging for VAs were followed for major adverse cardiac events (MACEs) defined by all-cause death, incident VAs requiring therapy, or heart failure hospitalization. Entropy was derived from the probability distribution of pixel signal intensities of the left ventricular (LV) myocardium. ResultsA total of 583 patients (age 54 ± 15 years, female 39%, left ventricular ejection fraction [LVEF] 54 ± 13%) were followed for a median of 4.4 years and experienced 141 MACEs. Entropy showed strong unadjusted association with MACE (HR: 1.88; 95% CI: 1.63-2.17; P < 0.001). In a multivariable model including LVEF, QRS duration, late gadolinium enhancement, and presenting arrhythmia, entropy maintained independent association with MACE (HR: 1.61; 95% CI: 1.32-1.96; P < 0.001). Entropy was further significantly associated with MACE in patients without myocardial scar (HR: 2.43; 95% CI: 1.55-3.82; P < 0.001) and in those presenting with nonsustained VAs (HR: 2.16; 95% CI: 1.43-3.25; P < 0.001). Addition of LV entropy to the baseline multivariable model significantly improved model performance (C-statistic improvement: 0.725 to 0.754; P = 0.003) and risk reclassification. ConclusionsIn patients with VAs, CMR-assessed LV entropy was independently associated with MACE and provided incremental prognostic value, on top of LVEF and late gadolinium enhancement. LV entropy assessment may help risk stratification in patients with absence of myocardial scar or with nonsustained VAs.

Nonlinear dynamics of a self-mixing thin-slice solid-state laser subjected to Doppler-shifted optical feedback.

Chaotic oscillations of a linearly polarized single longitudinal-mode thin-slice Nd:GdVO_{4} laser placed in a self-mixing laser Doppler velocity scheme were dynamically characterized in terms of the intensity probability distribution, joint time-frequency analysis, and short-term Fourier transformation of temporal evolutions, and the degree of disorder in the amplitude and phase of the long-term temporal evolutions. The transition from chaotic relaxation oscillations (ROs) to chaotic spiking oscillations (SOs) was explored via the chaotic itinerancy (CI) regime by increasing the feedback ratio toward the laser from a rotating scattering object. The intensity probability distribution was found to change from an exponential decay in the RO regime to an inverse power law in the SO regime, which manifests itself in self-organized critical behavior, while stochastic subharmonic frequency locking among the two periodicities of RO and SO takes place in the CI regime featuring quantum-noise (spontaneous-emission)-induced order in the amplitude and phase of the spiking oscillations. All of the experimental results were reproduced by numerical simulations of a model equation of a single-mode self-mixing solid-state laser subjected to Doppler-shifted optical feedback from a rotating scattering object.

Intensity of Respiratory Cortical Arousals Is a Distinct Pathophysiologic Feature and Is Associated with Disease Severity in Obstructive Sleep Apnea Patients.

Background: We investigated whether the number, duration and intensity of respiratory arousals (RA) on C3-electroencephalographic (EEG) recordings correlate with polysomnography (PSG)-related disease severity in obstructive sleep apnea (OSA) patients. We also investigated if every patient might have an individual RA microstructure pattern, independent from OSA-severity. Methods: PSG recordings of 20 OSA patients (9 female; age 27–80 years) were analyzed retrospectively. Correlation coefficients were calculated between RA microstructure (duration, EEG-intensity) and RA number and respiratory disturbance index (RDI), oxygen desaturation index (ODI) and arousal index (AI). Intraclass correlations (ICC) for both RA duration and intensity were calculated. Sleep stage-specific and apnea- and hypopnea-specific analyses were also done. The probability distributions of duration and intensity were plotted, interpolated with a kernel which fits the distribution. A Bayesian posterior distribution analysis and pair-wise comparisons of each patient with all other 19 patients were performed. Results: Of the analyzed 2600 RA, strong positive correlations were found between average RA intensity and both RDI and AI. The number of PSG-recorded RA was strongly positively correlated with RDI. Significant correlations between average RA intensity in REM, NREM2 and NREM3 sleep stages and total ODI were identified. No sleep stage-specific correlations of arousal microstructure with age, sex, RDI or AI were identified. Although between-subjects ICC values were <0.25, within-subject ICC values were all >0.7 (all p < 0.05). While apnea-related RA duration did not differ from hypopnea-related RA duration, RA intensity was significantly higher (p = 0.00135) in hypopneas than in apneas. A clear individual pattern of arousal duration for each patient was made distinct. For arousal intensity, a Gaussian distribution was identified in most patients. The Bayesian statistics regarding the arousal microstructure showed significant differences between each pair of patients. Conclusions: Each individual patient with OSA might have an individual pattern of RA intensity and duration indicating a distinct individual pathophysiological feature. Arousal intensity was significantly higher in hypopneic than in apneic events and may be related causally to the diminished (compared to apneas) respiratory distress associated with hypopneas. RA intensity in REM, NREM2 and NREM3 strongly correlated with ODI.

Mixture Density Networks-Based Knock Simulator

This paper proposes a statistical simulator for the engine knock based on the Mixture Density Network (MDN) and the accept-reject method. The proposed simulator can generate the random knock intensity signal corresponding to the input signal. The generated knock intensity has a consistent probability distribution with the real engine. Firstly, the statistical analysis is conducted with the experimental data. From the analysis results, some important assumptions on the statistical properties of the knock intensity are made. Regarding the knock intensity as a random variable on the discrete-time index, it is independent and identically distributed if the input of the engine is identical. The probability distribution of the knock intensity under identical input can be approximated by the Gaussian Mixture Model(GMM). The parameter of the GMM is a function of the input. Based on these assumptions, two sub-problems for establishing the statistical simulator are formulated: One is to approximate the function from input to the parameters of the knock intensity distribution with an absolutely continuous function; The other one is to design a random number generator that outputs the random data consistent with the given distribution. The MDN is applied to approximate the probability density of the knock intensity and the accept-reject algorithm is used for the random number generator design. The proposed method is evaluated in experimental data-based validation.

Failure analysis of a transmission line considering the joint probability distribution of wind speed and rain intensity

Traditionally, only the intensity measure of wind speed is used to perform the failure analysis of transmission lines (TLs). However, during typhoon landing, both the wind and rainfall effects on the failure probability of TLs should be considered. To fill the gap of rainfall amount data measured with long sampling periods, this study presents a conversion approach of rain intensities with various time resolutions based on the meteorological data of 1336 stations. Then, the 12-hour rain intensity samples recorded over 60 years are converted into 10-min rain intensity samples, followed by the establishment of the marginal probability distributions of yearly extreme wind speed and rain intensity. Furthermore, according to the theory of Copula function, the joint probability distribution of wind speed and rain intensity is proposed using the Clayton copula which has the best goodness of fit. Finally, a case study is performed and the failure probabilities of a TL under combined wind and rain loads are calculated. The failure probability of the employed TL under combined wind and rain loads has reached 6.19%, and the value is only 2.88% if ignoring the rainfall effect, demonstrating that rainfall has significant influence on the structural safety of a TL.

Speckle characteristics of simulated deep Fresnel region under shadowing effect

After the three-dimensional self-affine fractal random surface simulation, we use the optical scattering theory to calculate the deep Fresnel region speckle (DFRS) under consideration of the more strict shadowing effect. The evolution of DFRS with the scattering distance and the intensity probability distribution are studied. It is found that the morphology of the scatterer has an antisymmetric relationship with the intensity distribution of DFRS, and the effect of micro-lenses on the scattering surface causes the intensity probability distribution of DFRS to deviate from the Gaussian speckle in the high light intensity area.

Probability Distribution of Pixel Intensities of EBT3 Films and its Application in the Correction of Uncertainty Budget.

Background and Aim:Modern radiotherapy modalities, such as Intensity-Modulated Radiotherapy and Volumetric Modulated Arc Therapy involve complex dose delivery. The dose delivery is complex as it involves beam modulation, hence, manual dose calculations for these techniques are not possible. Film dosimetry is commonly used method of dose verification for these modalities because of the advantages associated with it. The quantification of uncertainty associated with a film dosimetry system under clinical use becomes important for accurate dosimetry. The spread in the distribution of the pixel values (PV) of the irradiated film contributes to the uncertainty. The probability distribution (PD) of the PV was studied for the clinical photon beam energies of 6, 10, and 15 MV.Methods and Materials:Gafchromic EBT3 film and EPSON 10000XL flatbed scanner were used for this purpose and using the resulting PD, the uncertainty budgets for these energies in the red, green and blue color channels were estimated.Results:The PV of exposed films for the energies studied follows t-distribution, the sum of the squares of the deviation of the measured data from the fitted value was of the order of 10−7, this indicates the goodness of fit. The “t” value corrected combined standard uncertainty (CSU) at 1σ confidence level for exposed film and dose measurement at 200 cGy were 1.42%, 1.48%, and 1.63% and 1.99%, 3.23%, and 5.08% for 6, 10, and 15 MV energies, respectively, in the red colour channel.Conclusion:In the case of the limited number of measurements of a quantity, the SU values must be corrected using the “t” value to get the correct CSU.

Repeatability enhancing method for one-shot LIBS analysis via spectral intensity correction based on probability distribution

A correction model is established between the spectrum of one-shot measurement and the averaged spectrum of multiple measurements based on spectral intensity probability distribution. Only one measured spectrum is needed during the test stage.

Temporally downscaling precipitation intensity factors for Köppen climate regions in the United States

Model inputs for prediction of runoff and soil erosion commonly require precipitation intensity information. Intensity is often estimated if precipitation data with high temporal resolution are unavailable. However, when intensity is time-averaged for fixed measurement intervals, estimates become increasingly underestimated with longer intervals due to the assumption that event durations begin and end at specified measurement intervals. In this study, adjustment factors were determined for downscaling the temporal resolution of intensity values derived from selected resolutions within the range of 10 to 1,440 min for Köppen-Geiger climate regions in the United States. In this case, monthly mean maximum 30 min intensity (MX.5P) was downscaled, which is a parameter used to generate stochastic meteorological inputs for models that include the Rangeland Hydrology and Erosion Model (RHEM) and the Water Erosion Prediction Project model (WEPP). The adjustment factors were given by regressions of reference MX.5P values derived from data with 5 min resolution against MX.5P values derived from data with lower temporal resolutions (≥10 min). In addition to using a slope coefficient for intensity in the regression equation, permutations of the equation included use of an elevation coefficient and constants, resulting in four total permutations. For the 143 stations and 17 climate regions analyzed, the four regression equations had roughly equal performance, and all gave statistically significant results. Regressions for adjusting hourly data using only an intensity coefficient in the equation had standard error of the estimate ranging from 1.01 to 2.96 mm h^–1 with an average of 2.04 mm h^–1. When downscaling daily values, the error range was 2.50 to 10.20 mm h^–1 with an average of 5.63 mm h^–1. Average time-to-peak intensity probability distributions for each climate region were also determined. Finally, a stochastic weather generator, CLIGEN, was used to test the effectiveness of applying the climate-based factors as an alternative to using subhourly data.

Excess Power, Energy, and Intensity of Stochastic Fields in Quasi-Static and Dynamic Environments

The excess power, energy and intensity of a random electromagnetic field above a high threshold level are characterized based on a Slepian--Kac model for upcrossings. For quasi-static fields, the probability distribution of the excess intensity in its regression approximation evolves from $\chi^2_3$ to $\chi^2_2$ when the threshold level increases. The excursion area associated with excess energy exhibits a chi-cubed ($\chi^3_2$) distribution above asymptotically high thresholds, where excursions are parabolic. For dynamic fields, the dependence of the electrical and environmental modulations of the excess power on the hybrid modulation index and threshold level are established. The normalized effective power relative to the quasi-static power increases non-monotonically when this index increases. The mean and standard deviation of the dynamic excess power are obtained in closed form and validated by Monte Carlo simulation.

Stochastic modelling of sandstorms affecting the optimal operation and cleaning scheduling of air coolers in concentrated solar power plants

The operation performance of air-coolers in concentrated solar power plants decays due to particulate deposition on heat transfer surfaces. The deposition process can be seen as a stochastic phenomenon. A modelling approach is proposed to capture the uncertainty and the effect of extreme events, such as sandstorms, affecting the performance of plants located in dry places through dust or sand deposition on the air coolers. A case study of a concentrated solar power plant located in Dubai is analysed. Sandstorms generate acute and drastic fouling of the air coolers, and this is modelled as a stochastic process using historical aerosol dispersion data. Ten scenarios are generated by sampling the probability distribution of sandstorms occurrence and intensity. The optimal operation (cleaning schedule and airflow profiles) of the air coolers is established using Benders decomposition to solve the resulting large-scale mixed integer non-linear programming problem. The results of the stochastic scenarios demonstrate that substantial savings of $ 0.6 M − $ 2.7 M per year are achieved by the optimal operation. Cost is minimized by a combined reactive and proactive cleaning policy which accounts for the frequency, intensity and seasonal variability of sandstorms, in addition to the variability on local radiation and weather conditions.