VOLCANIC ASH CLOUD DETECTION FROM SPACE: …...RST Ash Detection The RST technique computes two...
1
RST Ash Detection RST Ash Detection The RST technique computes two local variation indexes in order to detect volcanic ash plumes [ Pergola, 2004], defined as: The terms T 3 (x,y,t), T 11 (x,y,t) and T 12 (x,y,t), represent the signal (i.e. brightness temperatures) measured at around 3.5µm, 11µm and 12µm, while the terms µ T11-T12 , µ T3-T11 and σ T11-T12 , σ T3-T11 , respectively represent the temporal mean and the temporal standard deviation of the signal differences T 11 -T 12 and T 3 -T 11 calculated processing long-term time series of homogeneous satellite records. The local variation index ⊗ T11-T12 (x,y,t) takes into account the different behavior of volcanic ash cloud at spectral lengths of 11µm and 12µm in comparison with meteorological ones, due to acid component and ash particles. The local variation index ⊗ T3-T11 (x,y,t) instead, takes into account the potential of the spectral difference T 3 -T 11 in contributing to ash particle identification and discrimination. RST approach, considering relative rather than absolute signal variations, is intrinsically protected from site or atmospheric/environmental/observational effects [ Filizzola, 2007]. Alessandro Piscini (1) , Francesco Marchese (2) , Luca Merucci (1) , Nicola Pergola (2,3) , Stefano Corradini (1) , Valerio Tramutoli (3,2) (1) Istituto Nazionale di Geofisica e Vulcanologia, via di Vigna Murata, 605, 00143 Roma, Italy (2) Institute of Methodologies for Environmental Analysis (IMAA), National Research Council, c/da S.Loja- 85050 Tito Scalo (PZ), Italy (3) University of Basilicata, Department of Engineering and Physics of the Environment, via dell’Ateneo Lucano 10, 85100 Potenza, Italy Abstract Abstract: Volcanic eruptions can inject large amounts (Tg) of gas and particles into the troposphere and, sometimes, into the stratosphere. Besides the main gases (H2O, CO2 , SO2 and HCl), volcanic clouds contain a mix of silicate ash particles in the size range 0.1μm to mm or larger. Interest in the ash presence detection is high in particular because it represents a serious hazard for air traffic. Particles with dimension of several millimeters can damage the aircraft structure (windows, wings, ailerons), while particles less than 10 μm may be extremely dangerous for the jet engines and are undetectable by the pilots during night or in low visibility conditions. Satellite data are useful for measuring volcanic clouds because of the large vertical range of these emissions and their likely large horizontal spread. Moreover, since volcanoes are globally distributed and inherently dangerous, satellite measurements offer a practical and safe platform from which to make observations. Two different techniques used to detect volcanic clouds from satellite data are considered here for a preliminary comparison, with possible implications on quantitative retrievals of plume parameters. In particular, the Robust Satellite Techniques (RST) approach and a water vapour corrected version of the Brightness Temperature Difference (BTD) procedure, have been compared. The RST approach is based on the multi-temporal analysis of historical, long-term satellite records, devoted to a former characterization of the measured signal, in terms of expected value and natural variability and a further identification of signal anomalies by an automatic, unsupervised change detection step. The BTD method is based on the difference between the brightness temperature measured in two channels centered around 11 and 12 μm. To take into account the atmospheric water vapour differential absorption in the 11–12 μm spectral range that tends to reduce (and in some cases completely mask) the BTD signal, a water vapor correction procedure, based on measured or synthetic atmospheric profiles, has been applied. Results independently achieved by both methods during recent Mt. Etna eruptions are presented, compared and discussed also in terms of further implications for quantitative retrievals of plume parameters. TEST Cases and Data TEST Cases and Data Mt. Etna is located in the eastern part of Sicily (Italy) and is one the major degassing volcanoes in the world [ Allard, 1991]. Its generally quiescent state is periodically interrupted by eruptive periods when significant ash emissions reach areas surrounding the volcano causing problems to the population of the region [Andronico, 2005] and to air traffic (Catania and Reggio Calabria airports are nearby). Three Mt. Etna eruptions events have been considered as test cases. The first event, the 28 October 2002, is part of the 2002 eruption that began on the evening of 26 October with explosive activity in the NE and S volcano flanks and strombolian activity in the summital area. The explosive activity increased during the following days reaching a paroxysm during 28-31 October, with a fairly sustained eruptive column that rose up to 7 km a.s.l.. Ash fallout caused many problems to the surrounding population and infrastructures, including the closure of the Catania Fontanarossa (27 October) and the Reggio Calabria (31 October) airports. Prevailing winds directed the ash cloud towards the African coast, causing a light fallout in Libya 600 km from Mt. Etna) during the subsequent days [ Andronico, 2008]. The second event, the 24 November 2006, is part of the 2006 eruption. The ash emission started at about 03:00 UTC and ended at about 17:00 UTC; this took place at the SE crater located in the southern flank of Mt. Etna and produced the largest volume of ash in the entire September-December eruptive period. Third one, occured the 6 December 2006, was the erupting activity continued at the South-East Crater of Etna (SEC) involving a number of explosive and effusive vents on and around the SEC cone. AVHRR AVHRR We analyzed different multispectral images related to the eruptions above described recorded by Advanced Very High Resolution Radiometer (AVHRR) on board the Polar Orbiting Environmental Satellites (POES). The AVHRR is a cross-track scanning system with five spectral bands from Visible (VIS) to Thermal InfraRed (TIR) having a resolution of 1.1 km at nadir and a frequency of earth scans twice per day (0230 and 1430 local solar time). The instrument has the two TIR bands used for the volcanic ash detection retrieval (11 and 12 μm central wavelength). All images have been preprocessed in ENVI-IDL environment. They were cutted on specific area, georeferenced and calibrated in Albedo (ch1 and ch2 ) and Brightness Temperature (ch3, ch4, ch5). Ash Detection by using the BTD procedure Ash Detection by using the BTD procedure The ash detection is based on Brightness Temperature Difference (BTD) algorithm by using 11 and 12 μm TIR channels [ Prata, 1989]. BTD= T b (11μm)-T b (12μm) BTD < 0 volcanic ash BTD > 0 meteo clouds Atmospheric Water Vapour Correction (WVC) Atmospheric Water Vapour Correction (WVC) The radiative effect of the atmosphere counteracts the ash radiative effect, so the water vapour absorption can reduce the BTD signal of ash. In case of high columnar content water vapour can completely hide the ash plume effect on the BTD. [ Corradini et al., 2008]. The water vapour contribution on BTD has been estimated by simulating the radiance at sensor by means of a radiative transfer model (MODTRAN) and local atmospheric profiles (P,T,H) taken from European Centre for Medium-Range Weather Forecasts [ECMWF] archive, for each case. Then the BTD and BTD WVC images were generated. Finally, the pixels both in the BTD and in the BTD-WVC image, that satisfied the detection algorithm have been considered for the ash cloud area evaluation. References References P. Allard, J. Carbonelle, D. Dajlevic, J. Le Bronec, P. Morel, M.C. Robe, J.M. Maurenas, R.F. Pierret, D. Martins, J.C. Sabroux, and P. Zettwoog, “Eruptive and diffuse emission of CO2 from Mount Etna,“ Nature 351, 387 (1991) [doi:10.1038/351387a0]. D. Andronico, S. Branca, S. Calvari, M. Burton, T. Caltabiano, R. A. Corsaro, P. Del Carlo, G. Garfi, L. Lodato, L. Miraglia, F. Mur, M. Neri, E. Pecora, M. Pompilio, G. Salerno, and L. Spampinato, “A multi-disciplinary study of the 2002-03 Etna eruption: insights into a complex plumbing system,” Bull. Volcanol. 67 314-330 (2005) [doi:10.1007/s00445-004-0372-8]. D. Andronico, S. Scollo, A. Cristaldi, S. Caruso, The 2002–03 Etna explosive activity: tephra dispersal and features of the deposit, J. Geophys. Res., vol. 180, pp. 123–134, 2008. S. Corradini, C. Spinetti, E. Carboni, C. Tirelli, M.F. Buongiorno, S. Pugnaghi and G. Gangale, Mt. Etna tropospheric ash retrieval and sensitivity analysis using Moderate Resolution Imaging Spectroradiometer measurements, Journal of Applied Remote Sensing, Vol. 2, 023550, 2008. ECMWF, European Centre for Medium-Range Weather Forecasts http://www.ecmwf.int/ Filizzola, C., Lacava, T., Marchese, F., Pergola, N., Scaffidi, I., Tramutoli, V., “Assessing RAT (Robust AVHRR Technique) performances for volcanic ash cloud detection and monitoring in near real-time: The 2002 eruption of Mt. Etna (Italy)”, Remote Sensing Environment, 107, 3, 440-454, 2007. Pergola, N., Tramutoli, V., Marchese, F., Scaffidi, I., Lacava, T., “Improving volcanic ash cloud detection by a robust satellite technique”, Remote Sensing of Environment, 90, pp.1-22, 2004. Prata A. J., “Infrared radiative transfer calculations for volcanic ash clouds”, Geophys. Res. Lett., Vol. 16, No. 11, pp. 1293-1296, 1989. Results Results The figures depict ash cloud detected for BTD technique (left, green), BTD water vapour corrected (center, green) RST approach (right, red). The upper, middle and lower rows refer to 28 October 2006, 24 November 2006 and 6 December 2006 images, respectively. The table shows the ash cloud retrieval obtained for the three events in terms of number of pixels and area in km 2 . Conclusions Conclusions • Refined ash detection methods, like the BTD with WVC and the RST approach may significantly improve ash detection capability and sensitivity obtained by the basic BTD method, up to a 75% for the November 24 test case. • The results, although qualitative, show a good agreement between the areas detected by BTD with WVC and RST procedures, confirming RST capability to account for water vapour content without requiring any ancillary information. • In particular, the December 6 2006 case shows that refined ash detection methods, like the BTD with WVC and the RST approach are necessary to identify ash pixels. • Further investigations, possibly with the availability of independent data and observations, are required in order to better assess differences and similarities of these two methods. • Moreover, more test cases should be analyzed over a number of geographic areas and at different observational conditions (e.g. night-time). Istituto Nazionale di Geofisica e Vulcanologia Ash cloud detected before the Ash cloud detected before the Water Vapour correction: Water Vapour correction: Ash cloud detected after Ash cloud detected after Water Vapour correction: Water Vapour correction: 24 Nov 2006 11:20 UTC 6 Dec 2006 12:37 UTC 28 Oct 2002 13:14 UTC VOLCANIC ASH CLOUD DETECTION FROM SPACE: A PRELIMINARY COMPARISON BETWEEN RST APPROACH AND WATER VAPOUR CORRECTED BTD PROCEDURE European Geosciences Union General Assembly 2010 Vienna, Austria, 02 – 07 May 2010 - Session NH2.2 Hall XY board number XY577 European Geosciences Union General Assembly 2010 Vienna, Austria, 02 – 07 May 2010 - Session NH2.2 Hall XY board number XY577 for further information please write to: [email protected] and [email protected] for further information please write to: [email protected] and [email protected] Date Hour (UTC) Satellite N° ash pixels detected Ash Cloud area (km2) Water Vapour N° ash pixels detected Ash Cloud area (km2) n. ash pixels RST Area ash RST (km2) before water vapour correction corr. (°C) after water vapour correction 28/10/2002 13:14 noaa16 7954 9624 1.1 10.985 13291 9881 11525 24/11/2006 11:20 noaa18 495 617 1.3 1911 2368 1383 1613 06/12/2006 12:37 noaa18 0 0 0.9 240 297 291 339 ) , ( )] , ( )) , , ( ) , , ( [( ) , , ( 12 11 12 11 12 11 12 11 y x y x t y x T t y x T t y x T T T T T T - - - - - = ⊗ σ μ ) , ( )] , ( )) , , ( ) , , ( [( ) , , ( 11 3 11 3 11 3 11 3 y x y x t y x T t y x T t y x T T T T T T - - - - - = ⊗ σ μ x y t Multi-temporal analysis of satellite records in the space time domain following RST scheme
Transcript of VOLCANIC ASH CLOUD DETECTION FROM SPACE: …...RST Ash Detection The RST technique computes two...
![Page 1: VOLCANIC ASH CLOUD DETECTION FROM SPACE: …...RST Ash Detection The RST technique computes two local variation indexes in order to detect volcanic ash plumes [Pergola, 2004], defined](https://reader030.fdocument.org/reader030/viewer/2022040600/5e88fcc6a79ac85418189436/html5/thumbnails/1.jpg)
RST Ash DetectionRST Ash DetectionThe RST technique computes two local variation indexes in order to detect volcanic ash plumes [Pergola, 2004], defined as:
The terms T3(x,y,t), T11(x,y,t) and T12(x,y,t), represent the signal (i.e. brightness temperatures) measured at around 3.5µm, 11µm and 12µm, while the terms µT11-T12, µT3-T11 and σT11-T12, σT3-T11, respectively represent the temporal mean and the temporal standard deviation of the signal differences T11-T12 and T3-T11 calculated processing long-term time series of homogeneous satellite records. The local variation index ⊗T11-T12(x,y,t) takes into account the different behavior of volcanic ash cloud at spectral lengths of 11µm and 12µm in comparison with meteorological ones, due to acid component and ash particles. The local variation index ⊗T3-T11(x,y,t) instead, takes into account the potential of the spectral difference T3-T11 in contributing to ash particle identification and discrimination. RST approach, considering relative rather than absolute signal variations, is intrinsically protected from site or atmospheric/environmental/observational effects [Filizzola, 2007].
Alessandro Piscini (1), Francesco Marchese (2), Luca Merucci (1), Nicola Pergola (2,3), Stefano Corradini (1), Valerio Tramutoli (3,2) (1) Istituto Nazionale di Geofisica e Vulcanologia, via di Vigna Murata, 605, 00143 Roma, Italy
(2) Institute of Methodologies for Environmental Analysis (IMAA), National Research Council, c/da S.Loja- 85050 Tito Scalo (PZ), Italy(3) University of Basilicata, Department of Engineering and Physics of the Environment, via dell’Ateneo Lucano 10, 85100 Potenza, Italy
AbstractAbstract: Volcanic eruptions can inject large amounts (Tg) of gas and particles into the troposphere and, sometimes, into the stratosphere. Besides the main gases (H2O, CO2 , SO2 and HCl), volcanic clouds contain a mix of silicate ash particles in the size range 0.1μm to mm or larger. Interest in the ash presence detection is high in particular because it represents a serious hazard for air traffic. Particles with dimension of several millimeters can damage the aircraft structure (windows, wings, ailerons), while particles less than 10 μm may be extremely dangerous for the jet engines and are undetectable by the pilots during night or in low visibility conditions. Satellite data are useful for measuring volcanic clouds because of the large vertical range of these emissions and their likely large horizontal spread. Moreover, since volcanoes are globally distributed and inherently dangerous, satellite measurements offer a practical and safe platform from which to make observations. Two different techniques used to detect volcanic clouds from satellite data are considered here for a preliminary comparison, with possible implications on quantitative retrievals of plume parameters. In particular, the Robust Satellite Techniques (RST) approach and a water vapour corrected version of the Brightness Temperature Difference (BTD) procedure, have been compared.The RST approach is based on the multi-temporal analysis of historical, long-term satellite records, devoted to a former characterization of the measured signal, in terms of expected value and natural variability and a further identification of signal anomalies by an automatic, unsupervised change detection step. The BTD method is based on the difference between the brightness temperature measured in two channels centered around 11 and 12 μm. To take into account the atmospheric water vapour differential absorption in the 11–12 μm spectral range that tends to reduce (and in some cases completely mask) the BTD signal, a water vapor correction procedure, based on measured or synthetic atmospheric profiles, has been applied. Results independently achieved by both methods during recent Mt. Etna eruptions are presented, compared and discussed also in terms of further implications for quantitative retrievals of plume parameters.
TEST Cases and DataTEST Cases and DataMt. Etna is located in the eastern part of Sicily (Italy) and is one the major degassing volcanoes in the world [Allard, 1991]. Its generally quiescent state is periodically interrupted by eruptive periods when significant ash emissions reach areas surrounding the volcano causing problems to the population of the region [Andronico, 2005] and to air traffic (Catania and Reggio Calabria airports are nearby).Three Mt. Etna eruptions events have been considered as test cases.The first event, the 28 October 2002, is part of the 2002 eruption that began on the evening of 26 October with explosive activity in the NE and S volcano flanks and strombolian activity in the summital area. The explosive activity increased during the following days reaching a paroxysm during 28-31 October, with a fairly sustained eruptive column that rose up to 7 km a.s.l.. Ash fallout caused many problems to the surrounding population and infrastructures, including the closure of the Catania Fontanarossa (27 October) and the Reggio Calabria (31 October) airports. Prevailing winds directed the ash cloud towards the African coast, causing a light fallout in Libya 600 km from Mt. Etna) during the subsequent days [Andronico, 2008].The second event, the 24 November 2006, is part of the 2006 eruption. The ash emission started at about 03:00 UTC and ended at about 17:00 UTC; this took place at the SE crater located in the southern flank of Mt. Etna and produced the largest volume of ash in the entire September-December eruptive period.Third one, occured the 6 December 2006, was the erupting activity continued at the South-East Crater of Etna (SEC) involving a number of explosive and effusive vents on and around the SEC cone.
AVHRRAVHRRWe analyzed different multispectral images related to the eruptions above described recorded by Advanced Very High Resolution Radiometer (AVHRR) on board the Polar Orbiting Environmental Satellites (POES).The AVHRR is a cross-track scanning system with five spectral bands from Visible (VIS) to Thermal InfraRed (TIR) having a resolution of 1.1 km at nadir and a frequency of earth scans twice per day (0230 and 1430 local solar time). The instrument has the two TIR bands used for the volcanic ash detection retrieval (11 and 12 μm central wavelength).All images have been preprocessed in ENVI-IDL environment. They were cutted on specific area, georeferenced and calibrated in Albedo (ch1 and ch2 ) and Brightness Temperature (ch3, ch4, ch5).
Ash Detection by using the BTD procedureAsh Detection by using the BTD procedureThe ash detection is based on Brightness Temperature Difference (BTD) algorithm by using 11 and 12 μm TIR channels [Prata, 1989].
BTD= Tb(11μm)-Tb(12μm)
BTD < 0 volcanic ash BTD > 0 meteo clouds
Atmospheric Water Vapour Correction (WVC)Atmospheric Water Vapour Correction (WVC)The radiative effect of the atmosphere counteracts the ash radiative effect, so the water vapour absorption can reduce the BTD signal of ash. In case of high columnar content water vapour can completely hide the ash plume effect on the BTD. [Corradini et al., 2008]. The water vapour contribution on BTD has been estimated by simulating the radiance at sensor by means of a radiative transfer model (MODTRAN) and local atmospheric profiles (P,T,H) taken from European Centre for Medium-Range Weather Forecasts [ECMWF] archive, for each case. Then the BTD and BTD WVC images were generated. Finally, the pixels both in the BTD and in the BTD-WVC image, that satisfied the detection algorithm have been considered for the ash cloud area evaluation.
ReferencesReferencesP. Allard, J. Carbonelle, D. Dajlevic, J. Le Bronec, P. Morel, M.C. Robe, J.M. Maurenas, R.F. Pierret, D. Martins, J.C. Sabroux, and P. Zettwoog, “Eruptive and diffuse emission of CO2 from Mount Etna,“ Nature 351, 387 (1991) [doi:10.1038/351387a0].
D. Andronico, S. Branca, S. Calvari, M. Burton, T. Caltabiano, R. A. Corsaro, P. Del Carlo, G. Garfi, L. Lodato, L. Miraglia, F. Mur, M. Neri, E. Pecora, M. Pompilio, G. Salerno, and L. Spampinato, “A multi-disciplinary study of the 2002-03 Etna eruption: insights into a complex plumbing system,” Bull. Volcanol. 67 314-330 (2005) [doi:10.1007/s00445-004-0372-8].
D. Andronico, S. Scollo, A. Cristaldi, S. Caruso, The 2002–03 Etna explosive activity: tephra dispersal and features of the deposit, J. Geophys. Res., vol. 180, pp. 123–134, 2008.
S. Corradini, C. Spinetti, E. Carboni, C. Tirelli, M.F. Buongiorno, S. Pugnaghi and G. Gangale, Mt. Etna tropospheric ash retrieval and sensitivity analysis using Moderate Resolution Imaging Spectroradiometer measurements, Journal of Applied Remote Sensing, Vol. 2, 023550, 2008.
ECMWF, European Centre for Medium-Range Weather Forecasts http://www.ecmwf.int/
Filizzola, C., Lacava, T., Marchese, F., Pergola, N., Scaffidi, I., Tramutoli, V., “Assessing RAT (Robust AVHRR Technique) performances for volcanic ash cloud detection and monitoring in near real-time: The 2002 eruption of Mt. Etna (Italy)”, Remote Sensing Environment, 107, 3, 440-454, 2007.
Pergola, N., Tramutoli, V., Marchese, F., Scaffidi, I., Lacava, T., “Improving volcanic ash cloud detection by a robust satellite technique”, Remote Sensing of Environment, 90, pp.1-22, 2004.
Prata A. J., “Infrared radiative transfer calculations for volcanic ash clouds”, Geophys. Res. Lett., Vol. 16, No. 11, pp. 1293-1296, 1989.
ResultsResultsThe figures depict ash cloud detected for BTD technique (left, green), BTD water vapour corrected (center, green) RST approach (right, red). The upper, middle and lower rows refer to 28 October 2006, 24 November 2006 and 6 December 2006 images, respectively.
The table shows the ash cloud retrieval obtained for the three events in terms of number of pixels and area in km2.
ConclusionsConclusions• Refined ash detection methods, like the BTD with WVC and the RST approach may significantly improve ash detection capability and sensitivity obtained by the basic BTD method, up to a 75% for the November 24 test case.
• The results, although qualitative, show a good agreement between the areas detected by BTD with WVC and RST procedures, confirming RST capability to account for water vapour content without requiring any ancillary information.
• In particular, the December 6 2006 case shows that refined ash detection methods, like the BTD with WVC and the RST approach are necessary to identify ash pixels.
• Further investigations, possibly with the availability of independent data and observations, are required in order to better assess differences and similarities of these two methods.
• Moreover, more test cases should be analyzed over a number of geographic areas and at different observational conditions (e.g. night-time).
Istituto Nazionale diGeofisica e Vulcanologia
Ash cloud detected before the Ash cloud detected before the Water Vapour correction:Water Vapour correction:
Ash cloud detected after Ash cloud detected after Water Vapour correction:Water Vapour correction:
24 Nov 2006 11:20 UTC
6 Dec 2006 12:37 UTC
28 Oct 2002 13:14 UTC
VOLCANIC ASH CLOUD DETECTION FROM SPACE: A PRELIMINARY COMPARISON BETWEEN RST APPROACH AND WATER VAPOUR CORRECTED BTD PROCEDURE
European Geosciences Union General Assembly 2010 Vienna, Austria, 02 – 07 May 2010 - Session NH2.2 Hall XY board number XY577 European Geosciences Union General Assembly 2010 Vienna, Austria, 02 – 07 May 2010 - Session NH2.2 Hall XY board number XY577 for further information please write to: [email protected] and [email protected] further information please write to: [email protected] and [email protected]
Date Hour (UTC) Satellite N° ash pixels detected Ash Cloud area (km2) Water Vapour N° ash pixels detected Ash Cloud area (km2) n. ash pixels RST Area ash RST (km2)before water vapour correction corr. (°C) after water vapour correction
28/10/2002 13:14 noaa16 7954 9624 1.1 10.985 13291 9881 1152524/11/2006 11:20 noaa18 495 617 1.3 1911 2368 1383 161306/12/2006 12:37 noaa18 0 0 0.9 240 297 291 339
),()],()),,(),,([(),,(
1211
121112111211 yx
yxtyxTtyxTtyxTT
TTTT
−
−−
−−=⊗σ
µ),(
)],()),,(),,([(),,(113
113113113 yx
yxtyxTtyxTtyxTT
TTTT
−
−−
−−=⊗σ
µ
x
y
tMulti-temporal analysis of satellite records in the space time domain following RST scheme
![Pédagogie - ash-jpp.pagesperso-orange.frash-jpp.pagesperso-orange.fr/pdf/La Pédagogie.pdf · 3 les [doctrines pédagogiques 13 ],[14 sont de grands ensembles théoriques, complexes,](https://static.fdocument.org/doc/165x107/5c6722a909d3f22d638b5c6b/pedagogie-ash-jpppagesperso-pedagogiepdf-3-les-doctrines-pedagogiques.jpg)
![ALGEBRAIC AND EUCLIDEAN LATTICES: OPTIMAL LATTICE ...ALGEBRAIC AND EUCLIDEAN LATTICES: OPTIMAL LATTICE REDUCTION AND BEYOND 5 Regev [11], which proved two di˛erent results. The ˙rst](https://static.fdocument.org/doc/165x107/612f8c891ecc5158694384c9/algebraic-and-euclidean-lattices-optimal-lattice-algebraic-and-euclidean-lattices.jpg)