968 research outputs found

    Progress toward characterization of the atmospheric boundary layer over northern Alabama using observations by a vertically pointing, S-band profiling radar during VORTEX-Southeast

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    During spring 2016 and spring 2017, a vertically pointing, S-band FMCW radar (UMass FMCW) was deployed in northern Alabama under the auspices of the Verification of the Origins of Rotation in Tornadoes Experiment (VORTEX) – Southeast. In total, ~14 weeks’ worth of data were collected, in conditions ranging from quiescent clear skies to severe thunderstorms. The principal objective of these deployments was to characterize the boundary layer evolution near the VORTEX-Southeast domain. In this paper, we describe intermediate results in service of this objective. Specifically, we describe updates to the UMass FMCW system, document its deployments for VORTEX-Southeast, and apply three automated algorithms: (1) an dealiasing algorithm to the Doppler velocities, (2) a fuzzy logic scatterer classification scheme to separate precipitation from non-precipitation observations, (3) a bright band / melting layer identification algorithm for stratiform precipitation, and (4) an extended Kalman filter-based convective boundary layer depth (mixing height) measurement algorithm for non-precipitation observations. Results from the latter two applications are qualitatively verified against retrieved soundings from a collocated thermodynamic profiling system.Peer ReviewedPostprint (author's final draft

    Planetary boundary layer height variability over Athens, Greece, based on the synergy of Raman lidar and radiosonde data: application of the Kalman filter and other techniques (2011-2016)

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    The temporal evolution of the Planetary Boundary Layer height over Athens, Greece for a 5-year period (2011-2016) is presented. Using the EOLE Raman lidar system, the range-corrected lidar signals were selected around 12:00 UTC and 00:00 UTC for a total of 332 cases (165 days and 167 nights). The Kalman filter and other techniques were used to determine PBL height. The mean PBL height was found to be around 1617±324 m (12:00 UTC) and 892±130 m (00:00 UTC).Peer ReviewedPostprint (published version

    Application and testing of the extended-Kalman-filtering technique for determining the planetary boundary-layer height over Athens, Greece

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    The final publication is available at Springer via http://dx.doi.org/10.1007/s10546-020-00514-zWe investigate the temporal evolution of the planetary boundary-layer (PBL) height over the basin of Athens, Greece, during a 6-year period (2011–2016), using data from a Raman lidar system. The range-corrected lidar signals are selected around local noon (1200 UTC) and midnight (0000 UTC), for a total of 332 cases: 165 days and 167 nights. In this dataset, the extended-Kalman filtering technique is applied and tested for the determination of the PBL height. Several well-established techniques for the PBL height estimation based on lidar data are also tested for a total of 35 cases. The lidar-derived PBL heights are compared to those derived from radiosonde data. The mean PBL height over Athens is found to be 1617¿±¿324 m at 1200 UTC and 892¿±¿130 m at 0000 UTC for the period examined, while the mean PBL-height growth rate is found to be 170¿±¿64 m h-1 and 90¿±¿17 m h-1 during daytime and night-time, respectively.The research leading to these results has received additional funding from the European Union 7th Framework Program (FP7/2011-2015) and Horizon 2020/2015-2021 Research and Innovation program (ACTRIS) under grant agreements nos 262254, 654109, and 739530, as well as from Spanish National Science Foundation and FEDER funds PGC2018-094132-B-I00. CommSensLab-UPC is a María-de-Maeztu Excellence Unit, MDM-2016-0600, funded by the Agencia Estatal de Investigación, Spain.Peer ReviewedPostprint (author's final draft

    Lidar and S-band radar profiling of the atmosphere : adaptive processing for boundary-layer monitoring, optical-parameter error estimation, and application cases

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    This Ph.D. thesis addresses remote sensing of the atmosphere by means of lidar and S-band clear-air weather radar, and related data signal processing. Active remote sensing by means of these instruments offers unprecedented capabilities of spatial and temporal resolutions for vertical atmospheric profiling and the retrieval of key optical and physical atmospheric products in an increasing environmental regulatory framework. The first goal is this Ph.D. concerns the estimation of error bounds in the inversion of the profile of the atmospheric backscatter coefficient from elastic lidar signals (i.e., without wavelength shift in reception when interacting with atmospheric scatterers) by means of the two-component inversion algorithm (the so-called Klett-Fernald-Sasano¿s algorithm). This objective departs from previous works at the Remote Sensing Lab. (RSLab) of the Universitat Politècnica de Catalunya (UPC) and derives first-order error-propagated bounds (approximate) and total-increment bounds (exact). As distinctive feature in the state of the art, the error bounds merge into a single body both systematic (i.e., user-calibration inputs) and random error sources (finite signal-to-noise ratio, SNR) yielding an explicit mathematical form. The second goal, central to this Ph.D., tackles retrieval of the Atmospheric Boundary Layer Height (ABLH) from elastic lidar and S-band Frequency-Modulated Continuous-Wave (FMCW) radar observations by using adaptive techniques based on the Extended Kalman Filter (EKF). The filter is based on morphological modelling of the Mixing-Layer-to-Free-Troposphere transition and continuous estimation of the noise covariance information. In the lidar-EKF realization the proposed technique is shown to outperform classic ABLH estimators such as those based on derivative techniques, thresholded decision, or the variance centroid method. The EKF formulation is applied to both ceilometer and UPC lidar records in high- and low-SNR scenes. The lidar-EKF approach is re-formulated and successfully extended to S-band radar scenes (Bragg¿s scattering) in presence of interferent noise sources (Rayleigh scattering from e.g., insects and birds). In this context, the FMCW feature enables the range-resolved capability. EKF-lidar and EKF-radar ABLH estimates are cross-examined from field campaign results. Finally, the third goal deals with exploitation of the existing UPC lidar station: In a first introductory part, a modified algorithm for enhancing the dynamic range of elastic lidar channels by ¿gluing¿ analog and photon-counting data records is formulated. In a second part, two case examples (including application of the gluing algorithm) are presented to illustrate the capabilities of the UPC lidar in networked atmospheric observation of two recent volcano eruption events as part of the EARLINET (European Aerosol Research Lidar Network). The latter is part of GALION (Global Atmospheric Watch Atmospheric Lidar Observation Network)-GEOSS (Global Earth Observation System of Systems) framework.La tesis doctoral aborda la teledetecció atmosfèrica amb tècniques lidar i radar (banda S) i llur tractament del senyal. La teledetecció activa amb aquests instruments ofereix resolucions espacials i temporals sense precedents en la perfilometria vertical de l'atmosfera i recuperació de productes de dades òptics i físics atmosfèrics en un marc de creixent regulació mediambiental. El primer objectiu d'aquesta tesi concerneix l'estimació de cotes d'error en la inversió del perfil del coeficient de retrodispersió atmosfèrica a partir de senyals lidar de tipus elàstic (és a dir, sense desplaçament de la longitud d'ona en recepció al interactuar amb els dispersors atmosfèrics) mitjançant l'algorisme d'inversió de dues components de Klett-Fernald-Sasano. Aquest objectiu parteix de treballs previs en el Remote Sensing Lab. (RSLab) de la Universitat Politècnica de Catalunya (UPC) i permet obtenir cotes de primer ordre (aproximades) basades en propagació d'errors i cotes (exactes) basades en el increment total de l'error. Característica diferencial en front l'estat de l'art és l'assimilació d'errors sistemàtics (per exemple, entrades de cal.libració d'usuari) i aleatoris (relació senyal-soroll, SNR, finita) en forma matemàtica explícita. El segon objectiu, central de la tesis, aborda l'estimació de l'altura de la capa límit atmosfèrica (ABLH) a partir de senyal lidar elàstics i d'observacions radar en banda S (ona continua amb modulació en freqüència, FMCW) utilitzant tècniques adaptatives basades en filtrat estès de Kalman (EKF). El filtre es basa en modelat morfològic de la transició atmosfèrica entre la capa de mescla i la troposfera lliure i en l'estimació continua de la informació de covariança del soroll. En el prototipus lidar-EKF la tècnica proposada millora clarament les tècniques clàssiques d'estimació de la ABLH como són les basades en mètodes derivatius, decisió de llindar, o el mètode de la variança-centroide. La formulació EKF s'aplica tant a mesures procedents de ceilòmetres lidar como de la pròpia estació lidar UPC en escenes d'alta i baixa SNR. Addicionalment, l'enfoc lidar-EKF es reformula i s'estén amb èxit a escenes radar en banda S (dispersió Bragg) en presència de fonts de soroll interferent (dispersió Rayleigh de, per exemple, insectes i ocells). En aquest context, la característica FMCW permet la capacitat de resolució en distància. L'estimació de la ABLH amb els prototipus lidar-EKF i radar-EKF s'intercompara en campanyes de mesura. Finalment, el tercer objectiu atén a l'explotació de l'estació lidar UPC existent: En una primera part introductòria, es formula un algorisme modificat de "gluing" per a la millora del marge dinàmic de canals lidar elàstics mitjançant combinació (o "enganxat") de senyals lidar adquirits analògicament i amb foto-comptatge. En una segona part, es presenten dos exemples (incloent l'aplicació de l'algorisme de "gluing") que il.lustren les capacitats del lidar de la UPC en l'observació atmosfèrica de dos recents erupcions volcàniques des de la xarxa d'observació EARLINET (European Aerosol Research Lidar Network). Aquesta última és part de GALION (Global Atmospheric Watch Atmospheric Lidar Observation Network)-GEOSS (Global Earth Observation System of Systems)

    Bayesian approach to ionospheric imaging with Gaussian Markov random field priors

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    Ionosphere is the partly ionised layer of Earth's atmosphere caused by solar radiation and particle precipitation. The ionisation can start from 60 km and extend up to 1000 km altitude. Often the interest in ionosphere is in the quantity and distribution of the free electrons. The electron density is related to the ionospheric refractive index and thus sufficiently high densities affect the electromagnetic waves propagating in the ionised medium. This is the reason for HF radio signals being able to reflect from the ionosphere allowing broadcast over the horizon, but also an error source in satellite positioning systems. The ionospheric electron density can be studied e.g. with specific radars and satellite in situ measurements. These instruments can provide very precise observations, however, typically only in the vicinity of the instrument. To make observations in regional and global scales, due to the volume of the domain and price of the aforementioned instruments, indirect satellite measurements and imaging methods are required. Mathematically ionospheric imaging suffers from two main complications. First, due to very sparse and limited measurement geometry between satellites and receivers, it is an ill-posed inverse problem. The measurements do not have enough information to reconstruct the electron density and thus additional information is required in some form. Second, to obtain sufficient resolution, the resulting numerical model can become computationally infeasible. In this thesis, the Bayesian statistical background for the ionospheric imaging is presented. The Bayesian approach provides a natural way to account for different sources of information with corresponding uncertainties and to update the estimated ionospheric state as new information becomes available. Most importantly, the Gaussian Markov Random Field (GMRF) priors are introduced for the application of ionospheric imaging. The GMRF approach makes the Bayesian approach computationally feasible by sparse prior precision matrices. The Bayesian method is indeed practicable and many of the widely used methods in ionospheric imaging revert back to the Bayesian approach. Unfortunately, the approach cannot escape the inherent lack of information provided by the measurement set-up, and similarly to other approaches, it is highly dependent on the additional subjective information required to solve the problem. It is here shown that the use of GMRF provides a genuine improvement for the task as this subjective information can be understood and described probabilistically in a meaningful and physically interpretative way while keeping the computational costs low.Ionosfääri on noin 60–1000 kilometrin korkeudella sijaitseva ilmakehän kerros, jossa kaasuatomien ja -molekyylien elektroneja on päässyt irtoamaan auringon säteilyn ja auringosta peräisin olevien nopeiden hiukkasten vaikutuksesta. Näin syntyneillä ioneilla ja vapailla elektroneilla on sähkö- ja magneettikenttien kanssa vuorovaikuttava sähkövaraus. Ionosfäärillä on siksi merkittävä rooli radioliikenteessä. Se voi mahdollistaa horisontin yli tapahtuvat pitkät radiolähetykset heijastamalla lähetetyn sähkömagneettisen signaalin takaisin maata kohti. Toisaalta ionosfääri vaikuttaa myös sen läpäiseviin korkeampitaajuuksisiin signaaleihin. Esimerkiksi satelliittipaikannuksessa ionosfäärin vaikutus on parhaassakin tapauksessa otettava huomioon, mutta huonoimmassa se voi estää paikannuksen täysin. Näkyvin ja tunnetuin ionosfääriin liittyvä ilmiö lienee revontulet. Yksi keskeisistä suureista ionosfäärin tutkimuksessa on vapaiden elektronien määrä kuutiometrin tilavuudessa. Käytännössä elektronitiheyden mittaaminen on mahdollista mm. tutkilla, kuten Norjan, Suomen ja Ruotsin alueilla sijaitsevalla EISCAT-tutkajärjestelmällä, sekä raketti- tai satelliittimittauksilla. Mittaukset voivat olla hyvinkin tarkkoja, mutta tietoa saadaan ainoastaan tutkakeilan suunnassa tai mittalaitteen läheisyydestä. Näillä menetelmillä ionosfäärin tutkiminen laajemmalla alueella on siten vaikeaa ja kallista. Olemassa olevat paikannussatelliitit ja vastaanotinverkot mahdollistavat ionosfäärin elektronitiheyden mittaamisen alueellisessa, ja jopa globaalissa mittakaavassa, ensisijaisen käyttötarkoituksensa sivutuotteena. Satelliittimittausten ajallinen ja paikallinen kattavuus on hyvä, ja kaiken aikaa kasvava, mutta esimerkiksi tarkkoihin tutkamittauksiin verrattuna yksittäisten mittausten tuottama informaatio on huomattavasti vähäisempää. Tässä väitöstyössä kehitettiin tietokoneohjelmisto ionosfäärin elektronitiheyden kolmiulotteiseen kuvantamiseen. Menetelmä perustuu matemaattisten käänteisongelmien teoriaan ja muistuttaa lääketieteessä käytettyjä viipalekuvausmenetelmiä. Satelliittimittausten puutteellisesta informaatiosta johtuen työssä on keskitytty etenkin siihen, miten ratkaisun löytymistä voidaan auttaa tilastollisesti esitetyllä fysikaalisella ennakkotiedolla. Erityisesti työssä sovellettiin gaussisiin Markovin satunnaiskenttiin perustuvaa uutta korrelaatiopriori-menetelmää. Menetelmä vähentää merkittävästi tietokonelaskennassa käytettävän muistin tarvetta, mikä lyhentää laskenta-aikaa ja mahdollistaa korkeamman kuvantamisresoluution

    Bayesian approach to ionospheric imaging with Gaussian Markov random field priors

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    Ionosfääri on noin 60–1000 kilometrin korkeudella sijaitseva ilmakehän kerros, jossa kaasuatomien ja -molekyylien elektroneja on päässyt irtoamaan auringon säteilyn ja auringosta peräisin olevien nopeiden hiukkasten vaikutuksesta. Näin syntyneillä ioneilla ja vapailla elektroneilla on sähkö- ja magneettikenttien kanssa vuorovaikuttava sähkövaraus. Ionosfäärillä on siksi merkittävä rooli radioliikenteessä. Se voi mahdollistaa horisontin yli tapahtuvat pitkät radiolähetykset heijastamalla lähetetyn sähkömagneettisen signaalin takaisin maata kohti. Toisaalta ionosfääri vaikuttaa myös sen läpäiseviin korkeampitaajuuksisiin signaaleihin. Esimerkiksi satelliittipaikannuksessa ionosfäärin vaikutus on parhaassakin tapauksessa otettava huomioon, mutta huonoimmassa se voi estää paikannuksen täysin. Näkyvin ja tunnetuin ionosfääriin liittyvä ilmiö lienee revontulet. Yksi keskeisistä suureista ionosfäärin tutkimuksessa on vapaiden elektronien määrä kuutiometrin tilavuudessa. Käytännössä elektronitiheyden mittaaminen on mahdollista mm. tutkilla, kuten Norjan, Suomen ja Ruotsin alueilla sijaitsevalla EISCAT-tutkajärjestelmällä, sekä raketti- tai satelliittimittauksilla. Mittaukset voivat olla hyvinkin tarkkoja, mutta tietoa saadaan ainoastaan tutkakeilan suunnassa tai mittalaitteen läheisyydestä. Näillä menetelmillä ionosfäärin tutkiminen laajemmalla alueella on siten vaikeaa ja kallista. Olemassa olevat paikannussatelliitit ja vastaanotinverkot mahdollistavat ionosfäärin elektronitiheyden mittaamisen alueellisessa, ja jopa globaalissa mittakaavassa, ensisijaisen käyttötarkoituksensa sivutuotteena. Satelliittimittausten ajallinen ja paikallinen kattavuus on hyvä, ja kaiken aikaa kasvava, mutta esimerkiksi tarkkoihin tutkamittauksiin verrattuna yksittäisten mittausten tuottama informaatio on huomattavasti vähäisempää. Tässä väitöstyössä kehitettiin tietokoneohjelmisto ionosfäärin elektronitiheyden kolmiulotteiseen kuvantamiseen. Menetelmä perustuu matemaattisten käänteisongelmien teoriaan ja muistuttaa lääketieteessä käytettyjä viipalekuvausmenetelmiä. Satelliittimittausten puutteellisesta informaatiosta johtuen työssä on keskitytty etenkin siihen, miten ratkaisun löytymistä voidaan auttaa tilastollisesti esitetyllä fysikaalisella ennakkotiedolla. Erityisesti työssä sovellettiin gaussisiin Markovin satunnaiskenttiin perustuvaa uutta korrelaatiopriori-menetelmää. Menetelmä vähentää merkittävästi tietokonelaskennassa käytettävän muistin tarvetta, mikä lyhentää laskenta-aikaa ja mahdollistaa korkeamman kuvantamisresoluution.Ionosphere is the partly ionised layer of Earth's atmosphere caused by solar radiation and particle precipitation. The ionisation can start from 60 km and extend up to 1000 km altitude. Often the interest in ionosphere is in the quantity and distribution of the free electrons. The electron density is related to the ionospheric refractive index and thus sufficiently high densities affect the electromagnetic waves propagating in the ionised medium. This is the reason for HF radio signals being able to reflect from the ionosphere allowing broadcast over the horizon, but also an error source in satellite positioning systems. The ionospheric electron density can be studied e.g. with specific radars and satellite in situ measurements. These instruments can provide very precise observations, however, typically only in the vicinity of the instrument. To make observations in regional and global scales, due to the volume of the domain and price of the aforementioned instruments, indirect satellite measurements and imaging methods are required. Mathematically ionospheric imaging suffers from two main complications. First, due to very sparse and limited measurement geometry between satellites and receivers, it is an ill-posed inverse problem. The measurements do not have enough information to reconstruct the electron density and thus additional information is required in some form. Second, to obtain sufficient resolution, the resulting numerical model can become computationally infeasible. In this thesis, the Bayesian statistical background for the ionospheric imaging is presented. The Bayesian approach provides a natural way to account for different sources of information with corresponding uncertainties and to update the estimated ionospheric state as new information becomes available. Most importantly, the Gaussian Markov Random Field (GMRF) priors are introduced for the application of ionospheric imaging. The GMRF approach makes the Bayesian approach computationally feasible by sparse prior precision matrices. The Bayesian method is indeed practicable and many of the widely used methods in ionospheric imaging revert back to the Bayesian approach. Unfortunately, the approach cannot escape the inherent lack of information provided by the measurement set-up, and similarly to other approaches, it is highly dependent on the additional subjective information required to solve the problem. It is here shown that the use of GMRF provides a genuine improvement for the task as this subjective information can be understood and described probabilistically in a meaningful and physically interpretative way while keeping the computational costs low

    The ESPAS e-infrastructure

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    ESPAS provides an e-Infrastructure to support access to a wide range of archived observations and model derived data for the near-Earth space environment, extending from the Earth's middle atmosphere up to the outer radiation belts. To this end, ESPAS will serve as a central access hub for researchers who wish to exploit multi-instrument multipoint data for scientific discovery, model development and validation, and data assimilation, among others. Observation based and model enhanced scientific understanding of the physical state of the Earth's space environment and its evolution is critical to advancing space weather and space climate studies, two very active branches of current scientific research. ESPAS offers an interoperable data infrastructure that enables users to find, access, and exploit near-Earth space environment observations from ground-based and spaceborne instruments and data from relevant models, obtained from distributed repositories. In order to facilitate efficient user queries ESPAS allows a highly flexible workflow scheme to select and request the desired data sets. ESPAS has the strategic goal of making Europe a leading player in the efficient use and dissemination of near-Earth space environment information offered by institutions, laboratories and research teams in Europe and worldwide, that are active in collecting, processing and distributing scientific data. Therefore, ESPAS is committed to support and foster new data providers who wish to promote the easy use of their data and models by the research community via a central access framework. ESPAS is open to all potential users interested in near-Earth space environment data, including those who are active in basic scientific research, technical or operational development and commercial applications

    COMBAT SYSTEMS Volume 1. Sensor Elements Part I. Sensor Functional Characteristics

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    This document includes: CHAPTER 1. SIGNATURES, OBSERVABLES, & PROPAGATORS. CHAPTER 2. PROPAGATION OF ELECTROMAGNETIC RADIATION. I. – FUNDAMENTAL EFFECTS. CHAPTER 3. PROPAGATION OF ELECTROMAGNETIC RADIATION. II. – WEATHER EFFECTS. CHAPTER 4. PROPAGATION OF ELECTROMAGNETIC RADIATION. III. – REFRACTIVE EFFECTS. CHAPTER 5. PROPAGATION OF ELECTROMAGNETIC RADIATION IV. – OTHER ATMOSPHERIC AND UNDERWATER EFFECTS. CHAPTER 6. PROPAGATION OF ACOUSTIC RADIATION. CHAPTER 7. NUCLEAR RADIATION: ITS ORIGIN AND PROPAGATION. CHAPTER 8. RADIOMETRY, PHOTOMETRY, & RADIOMETRIC ANALYSIS. CHAPTER 9. SENSOR FUNCTIONS. CHAPTER 10. SEARCH. CHAPTER 11. DETECTION. CHAPTER 12. ESTIMATION. CHAPTER 13. MODULATION AND DEMODULATION. CHAPTER 14. IMAGING AND IMAGE-BASED PERCEPTION. CHAPTER 15. TRACKING. APPENDIX A. UNITS, PHYSICAL CONSTANTS, AND USEFUL CONVERSION FACTORS. APPENDIX B. FINITE DIFFERENCE AND FINITE ELEMENT TECHNIQUES. APPENDIX C. PROBABILITY AND STATISTICS. INDEX TO VOLUME 1. Note by author: Note: Boldface entries in the table of contents are not yet completed

    Opportunistic rain rate estimation from measurements of satellite downlink attenuation: A survey

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    Recent years have witnessed a growing interest in techniques and systems for rainfall surveillance on regional scale, with increasingly stringent requirements in terms of the following: (i) accuracy of rainfall rate measurements, (ii) adequate density of sensors over the territory, (iii) space‐time continuity and completeness of data and (iv) capability to elaborate rainfall maps in near real time. The devices deployed to monitor the precipitation fields are traditionally networks of rain gauges distributed throughout the territory, along with weather radars and satellite remote sensors operating in the optical or infrared band, none of which, however, are suitable for full compliance to all of the requirements cited above. More recently, a different approach to rain rate estimation techniques has been proposed and investigated, based on the measurement of the attenuation induced by rain on signals of pre‐existing radio networks either in terrestrial links, e.g., the backhaul connections in cellular networks, or in satellite‐to‐earth links and, among the latter, notably those between geostationary broadcast satellites and domestic subscriber terminals in the Ku and Ka bands. Knowledge of the above rain‐induced attenuation permits the retrieval of the corresponding rain intensity provided that a number of meteorological and geometric parameters are known and ultimately permits estimating the rain rate locally at the receiver site. In this survey paper, we specifically focus on such a type of “opportunistic” systems for rain field monitoring, which appear very promising in view of the wide diffusion over the territory of low‐cost domestic terminals for the reception of satellite signals, prospectively allowing for a considerable geographical capillarity in the distribution of sensors, at least in more densely populated areas. The purpose of the paper is to present a broad albeit synthetic overview of the numerous issues inherent in the above rain monitoring approach, along with a number of solutions and algorithms proposed in the literature in recent years, and ultimately to provide an exhaustive account of the current state of the art. Initially, the main relevant aspects of the satellite link are reviewed, including those related to satellite dynamics, frequency bands, signal formats, propagation channel and radio link geometry, all of which have a role in rainfall rate estimation algorithms. We discuss the impact of all these factors on rain estimation accuracy while also highlighting the substantial differences inherent in this approach in comparison with traditional rain monitoring techniques. We also review the basic formulas relating rain rate intensity to a variation of the received signal level or of the signal‐to-noise ratio. Furthermore, we present a comprehensive literature survey of the main research issues for the aforementioned scenario and provide a brief outline of the algorithms proposed for their solution, highlighting their points of strength and weakness. The paper includes an extensive list of bibliographic references from which the material presented herein was taken
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