Ward's Hierarchical Clustering Method: Clustering Criterion and Agglomerative Algorithm

The Ward error sum of squares hierarchical clustering method has been very widely used since its first description by Ward in a 1963 publication. It has also been generalized in various ways. However there are different interpretations in the literature and there are different implementations of the Ward agglomerative algorithm in commonly used software systems, including differing expressions of the agglomerative criterion. Our survey work and case studies will be useful for all those involved in developing software for data analysis using Ward's hierarchical clustering method.Comment: 20 pages, 21 citations, 4 figure

Mumford dendrograms and discrete p-adic symmetries

In this article, we present an effective encoding of dendrograms by embedding them into the Bruhat-Tits trees associated to $p$-adic number fields. As an application, we show how strings over a finite alphabet can be encoded in cyclotomic extensions of $\mathbb{Q}_p$ and discuss $p$-adic DNA encoding. The application leads to fast $p$-adic agglomerative hierarchic algorithms similar to the ones recently used e.g. by A. Khrennikov and others. From the viewpoint of $p$-adic geometry, to encode a dendrogram $X$ in a $p$-adic field $K$ means to fix a set $S$ of $K$-rational punctures on the $p$-adic projective line $\mathbb{P}^1$. To $\mathbb{P}^1\setminus S$ is associated in a natural way a subtree inside the Bruhat-Tits tree which recovers $X$, a method first used by F. Kato in 1999 in the classification of discrete subgroups of $\textrm{PGL}_2(K)$. Next, we show how the $p$-adic moduli space $\mathfrak{M}_{0,n}$ of $\mathbb{P}^1$ with $n$ punctures can be applied to the study of time series of dendrograms and those symmetries arising from hyperbolic actions on $\mathbb{P}^1$. In this way, we can associate to certain classes of dynamical systems a Mumford curve, i.e. a $p$-adic algebraic curve with totally degenerate reduction modulo $p$. Finally, we indicate some of our results in the study of general discrete actions on $\mathbb{P}^1$, and their relation to $p$-adic Hurwitz spaces.Comment: 14 pages, 6 figure

First Odin sub-mm retrievals in the tropical upper troposphere: ice cloud properties

International audienceMore accurate global measurements of the amount of ice in thicker clouds are needed to validate atmospheric models and sub-mm radiometry can be an important component in this respect. A cloud ice retrieval scheme for the first such instrument in space, Odin-SMR, is presented here. Several advantages of sub-mm observations are shown, such as low influence of particle shape and orientation, and a high dynamic range of the retrievals. In the case of Odin-SMR, only cloud ice above ?12.5 km can be measured. The present retrieval scheme gives a detection threshold of about 4 g/m2 above 12.5 km and does not saturate even for thickest observed clouds (>500 g/m2). The main retrieval uncertainties are the assumed particle size distribution and cloud inhomogeneity effects. The overall retrieval accuracy is estimated to be ~75%. The retrieval error is judged to have large random components and to be significantly lower than this value for averaged results, but high fixed errors can not be excluded. However, a firm lower value can always be provided. Initial results are found to be consistent with similar Aura MLS retrievals, but show important differences to corresponding data from atmospheric models. This first retrieval algorithm is limited to lowermost Odin-SMR tangent altitudes, and further development should improve the detection threshold and the vertical resolution. It should also be possible to decrease the retrieval uncertainty associated with cloud inhomogeneities by detailed analysis of other data sets

Nitric acid in the stratosphere based on Odin observations from 2001 to 2009 – Part 2: High-altitude polar enhancements

The wintertime abundance of nitric acid (HNO<sub>3</sub>) in the polar upper stratosphere displays a strong inter-annual variability, and is known to be strongly influenced by energetic particle precipitation (EPP), primarily by protons during solar proton events (SPEs), but also by precipitating auroral or relativistic electrons. We analyse a multi-year record (August 2001 to April 2009) of middle atmospheric HNO<sub>3</sub> measurements by the Sub-Millimeter Radiometer instrument aboard the Odin satellite, with a focus on the polar upper stratosphere. SMR observations show clear evidence of two different types of polar high-altitude HNO<sub>3</sub> enhancements linked to EPP. In the first type, referred to as direct enhancements by analogy with the EPP/NO<sub>x</sub> direct effect, enhanced HNO<sub>3</sub> mixing ratios are observed for a short period (1 week) after a SPE, upwards of a level typically in the mid-stratosphere. In a second type, referred to as indirect enhancements by analogy with the EPP/NO<sub>x</sub> indirect effect, the descent of mesospheric air triggers a stronger and longer-lasting enhancement. Each of the three major SPEs that occurred during the Northern Hemisphere autumn or winter, in November 2001, October–November 2003 and January 2005, are observed to lead to both direct and indirect HNO<sub>3</sub> enhancements. On the other hand, indirect enhancements occur recurrently in winter, are stronger in the Southern Hemisphere, and are influenced by EPP at higher altitudes

Comparison of satellite limb-sounding humidity climatologies of the uppermost tropical troposphere

International audienceHumidity climatologies of the tropical uppermost troposphere from satellite limb emission measurements have been compared. Four instruments are considered; UARS-MLS, Odin-SMR, and Aura-MLS operating in the microwave region, and MIPAS in the IR region. A reference for the comparison is obtained by MOZAIC in-situ measurements. The upper tropospheric humidity products were compared on basis of their empirical probability density functions and seasonally averaged horizontal fields at two altitude layers, 12 and 15 km. The probability density functions of the microwave datasets were found to be in very good agreement with each other, and are also consistent with MOZAIC. The average seasonal humidities differ with less than 10%RHi between the instruments, indicating that stated measurement accuracies of 20?30% are conservative estimates. The systematic uncertainty in Odin-SMR data due to cloud correction was also independently estimated to be 10%RHi. MIPAS humidity profiles were found to suffer from cloud contamination, with only 30% of the measurements reaching into the upper troposphere, but under clear-sky conditions there is a good agreement between MIPAS, Odin-SMR and Aura-MLS. Odin-SMR and the two MLS datasets can be treated as independent, being based on different underlying spectroscopy and technology. The good agreement between the microwave limb-sounders, and MOZAIC, is therefore an important step towards understanding the upper tropospheric humidity. The found accuracy of 10%RHi is approaching the level required to validate climate modelling of the upper troposphere humidity. The comparison of microwave and IR also stresses that microwave limb-sounding is necessary for a complete view of the upper troposphere

A study of polar ozone depletion based on sequential assimilation of satellite data from the ENVISAT/MIPAS and Odin/SMR instruments

International audienceThe objective of this study is to demonstrate how polar ozone depletion can be mapped and quantified by assimilating ozone data from satellites into the wind driven transport model DIAMOND, (Dynamical Isentropic Assimilation Model for OdiN Data). By assimilating a large set of satellite data into a transport model, ozone fields can be built up that are less noisy than the individual satellite ozone profiles. The transported fields can subsequently be compared to later sets of incoming satellite data so that the rates and geographical distribution of ozone depletion can be determined. By tracing the amounts of solar irradiation received by different air parcels in a transport model it is furthermore possible to study the photolytic reactions that destroy ozone. In this study, destruction of ozone that took place in the Antarctic winter of 2003 and in the Arctic winter of 2002/2003 have been examined by assimilating ozone data from the ENVISAT/MIPAS and Odin/SMR satellite-instruments. Large scale depletion of ozone was observed in the Antarctic polar vortex of 2003 when sunlight returned after the polar night. By mid October ENVISAT/MIPAS data indicate vortex ozone depletion in the ranges 80?100% and 70?90% on the 425 and 475 K potential temperature levels respectively while the Odin/SMR data indicates depletion in the ranges 70?90% and 50?70%. The discrepancy between the two instruments has been attributed to systematic errors in the Odin/SMR data. Assimilated fields of ENVISAT/MIPAS data indicate ozone depletion in the range 10?20% on the 475 K potential temperature level, (~19 km altitude), in the central regions of the 2002/2003 Arctic polar vortex. Assimilated fields of Odin/SMR data on the other hand indicate ozone depletion in the range 20?30%

Comparison of satellite limb-sounding humidity climatologies of the uppermost tropical troposphere

International audienceHumidity climatologies of the tropical uppermost troposphere from satellite limb emission measurements have been compared. Four instruments are considered; UARS-MLS, Odin-SMR, and Aura-MLS operating in the microwave region, and MIPAS in the infrared region. A reference for the comparison is obtained by MOZAIC in-situ measurements. The upper tropospheric humidity products were compared on basis of their empirical probability density functions and seasonally averaged horizontal fields at two altitude layers, 12 and 15 km. The probability density functions of the microwave datasets were found to be in very good agreement with each other, and were also consistent with MOZAIC. The average seasonal humidities differ with less than 10%RHi between the instruments, indicating that stated measurement accuracies of 20?30% are conservative estimates. The systematic uncertainty in Odin-SMR data due to cloud correction was also independently estimated to be 10%RHi. MIPAS humidity profiles were found to suffer from cloud contamination, with only 30% of the measurements reaching into the upper troposphere, but under clear-sky conditions there is a good agreement between MIPAS, Odin-SMR and Aura-MLS. Odin-SMR and the two MLS datasets can be treated as independent, being based on different underlying spectroscopy and technology. The good agreement between the microwave limb-sounders, and MOZAIC, is therefore an important step towards understanding the upper tropospheric humidity. The found accuracy of 10%RHi is approaching the level required to validate climate modelling of the upper troposphere humidity. The comparison of microwave and infrared also stresses that microwave limb-sounding is necessary for a complete view of the upper troposphere

Observations of the mesospheric semi-annual oscillation (MSAO) in water vapour by Odin/SMR

International audienceMesospheric water vapour measurements taken by the SMR instrument onboard the Odin satellite between 2002 and 2006 have been analysed with focus on the mesospheric semi-annual circulation in the tropical and subtropical region. This analysis provides the first complete picture of mesospheric SAO in water vapour, covering altitudes above 80 km where the only previous study based on UARS/HALOE data was limited. Our analysis shows a clear semi-annual variation in the water vapour distribution in the entire altitude range between 65 km and 100 km in the equatorial area. Maxima occur near the equinoxes below 75 km and around the solstices above 80 km. The phase reversal occurs in the small layer in-between, consistent with the downward propagation of the mesospheric SAO in the zonal wind in this altitude range. The SAO amplitude exhibits a double peak structure, with maxima at about 75 km and 81 km. The observed amplitudes show higher values than the UARS/HALOE amplitudes. The upper peak amplitude remains relatively constant with latitude. The lower peak amplitude decreases towards higher latitudes, but recovers in the Southern Hemisphere subtropics. On the other hand, the annual variation is much more prominent in the northern hemispheric subtropics. Furthermore, higher volume mixing ratios during summer and lower values during winter are observed in the Northern Hemisphere subtropics, as compared to the corresponding latitude range in the Southern Hemisphere

Comparison between the first Odin-SMR, Aura MLS and CloudSat retrievals of cloud ice mass in the upper tropical troposphere

International audienceEmerging microwave satellite techniques are expected to provide improved global measurements of cloud ice mass. CloudSat, Aura MLS and Odin-SMR fall into this category and first cloud ice retrievals from these instruments are compared. The comparison is made for partial ice water columns above 12 km, following the SMR retrieval product. None of the instruments shows significant false cloud detections and a consistent view of the geographical distribution of cloud ice is obtained, but differences on the absolute levels exist. CloudSat gives the lowest values, with an overall mean of 2.12 g/m2. A comparable mean for MLS is 4.30 g/m2. This relatively high mean can be an indication of overestimation of the vertical altitude of cloud ice by the MLS retrievals. The vertical response of SMR has also some uncertainty, but this does not affect the comparison between MLS and CloudSat. SMR observations are sensitive to cloud inhomogeneities inside the footprint and some compensation is required. Results in good agreement with CloudSat, both in regard of the mean and probability density functions, are obtained for a weak compensation, while a simple characterisation of the effect indicates the need for stronger compensation. The SMR mean was found to be 1.89/2.62/4.10 g/m2 for no/selected/strongest compensation, respectively. Assumptions about the particle size distribution are a consideration for all three instruments, and constitute the dominating retrieval uncertainty for CloudSat. The comparison indicates a retrieval accuracy of about 40% (3.1±1.2 g/m2). This number is already very small compared to uncertainties of cloud ice parametrisation in atmospheric models, but can be decreased further through a better understanding of main retrieval error sources