Quantum Space-Time and Noncommutative Gauge Field Theories

Arguments arising from quantum mechanics and gravitation theory as well as from string theory, indicate that the description of space-time as a continuous manifold is not adequate at very short distances. An important candidate for the description of space-time at such scales is provided by noncommutative space-time where the coordinates are promoted to noncommuting operators. Thus, the study of quantum field theory in noncommutative space-time provides an interesting interface where ordinary field theoretic tools can be used to study the properties of quantum spacetime. The three original publications in this thesis encompass various aspects in the still developing area of noncommutative quantum field theory, ranging from fundamental concepts to model building. One of the key features of noncommutative space-time is the apparent loss of Lorentz invariance that has been addressed in different ways in the literature. One recently developed approach is to eliminate the Lorentz violating effects by integrating over the parameter of noncommutativity. Fundamental properties of such theories are investigated in this thesis. Another issue addressed is model building, which is difficult in the noncommutative setting due to severe restrictions on the possible gauge symmetries imposed by the noncommutativity of the space-time. Possible ways to relieve these restrictions are investigated and applied and a noncommutative version of the Minimal Supersymmetric Standard Model is presented. While putting the results obtained in the three original publications into their proper context, the introductory part of this thesis aims to provide an overview of the present situation in the field.

Gauge Field Theories, Quantum Space-Time and Some Applications

In this thesis, the possibility of extending the Quantization Condition of Dirac for Magnetic Monopoles to noncommutative space-time is investigated. The three publications that this thesis is based on are all in direct link to this investigation. Noncommutative solitons have been found within certain noncommutative field theories, but it is not known whether they possesses only topological charge or also magnetic charge. This is a consequence of that the noncommutative topological charge need not coincide with the noncommutative magnetic charge, although they are equivalent in the commutative context. The aim of this work is to begin to fill this gap of knowledge. The method of investigation is perturbative and leaves open the question of whether a nonperturbative source for the magnetic monopole can be constructed, although some aspects of such a generalization are indicated. The main result is that while the noncommutative Aharonov-Bohm effect can be formulated in a gauge invariant way, the quantization condition of Dirac is not satisfied in the case of a perturbative source for the point-like magnetic monopole.

Noncommutative Quantum Field Theories and UV/IR Mixing

Our present-day understanding of fundamental constituents of matter and their interactions is based on the Standard Model of particle physics, which relies on quantum gauge field theories. On the other hand, the large scale dynamical behaviour of spacetime is understood via the general theory of relativity of Einstein. The merging of these two complementary aspects of nature, quantum and gravity, is one of the greatest goals of modern fundamental physics, the achievement of which would help us understand the short-distance structure of spacetime, thus shedding light on the events in the singular states of general relativity, such as black holes and the Big Bang, where our current models of nature break down. The formulation of quantum field theories in noncommutative spacetime is an attempt to realize the idea of nonlocality at short distances, which our present understanding of these different aspects of Nature suggests, and consequently to find testable hints of the underlying quantum behaviour of spacetime. The formulation of noncommutative theories encounters various unprecedented problems, which derive from their peculiar inherent nonlocality. Arguably the most serious of these is the so-called UV/IR mixing, which makes the derivation of observable predictions especially hard by causing new tedious divergencies, to which our previous well-developed renormalization methods for quantum field theories do not apply. In the thesis I review the basic mathematical concepts of noncommutative spacetime, different formulations of quantum field theories in the context, and the theoretical understanding of UV/IR mixing. In particular, I put forward new results to be published, which show that also the theory of quantum electrodynamics in noncommutative spacetime defined via Seiberg-Witten map suffers from UV/IR mixing. Finally, I review some of the most promising ways to overcome the problem. The final solution remains a challenge for the future.

Noncommutative Quantum Field Theory : Problem of Time and Some Applications

In this thesis the current status and some open problems of noncommutative quantum field theory are reviewed. The introduction aims to put these theories in their proper context as a part of the larger program to model the properties of quantized space-time. Throughout the thesis, special focus is put on the role of noncommutative time and how its nonlocal nature presents us with problems. Applications in scalar field theories as well as in gauge field theories are presented. The infinite nonlocality of space-time introduced by the noncommutative coordinate operators leads to interesting structure and new physics. High energy and low energy scales are mixed, causality and unitarity are threatened and in gauge theory the tools for model building are drastically reduced. As a case study in noncommutative gauge theory, the Dirac quantization condition of magnetic monopoles is examined with the conclusion that, at least in perturbation theory, it cannot be fulfilled in noncommutative space.

Sneutrino-antisneutrino oscillations at colliders

Modern elementary particle physics is based on quantum field theories. Currently, our understanding is that, on the one hand, the smallest structures of matter and, on the other hand, the composition of the universe are based on quantum field theories which present the observable phenomena by describing particles as vibrations of the fields. The Standard Model of particle physics is a quantum field theory describing the electromagnetic, weak, and strong interactions in terms of a gauge field theory. However, it is believed that the Standard Model describes physics properly only up to a certain energy scale. This scale cannot be much larger than the so-called electroweak scale, i.e., the masses of the gauge fields W^+- and Z^0. Beyond this scale, the Standard Model has to be modified. In this dissertation, supersymmetric theories are used to tackle the problems of the Standard Model. For example, the quadratic divergences, which plague the Higgs boson mass in the Standard model, cancel in supersymmetric theories. Experimental facts concerning the neutrino sector indicate that the lepton number is violated in Nature. On the other hand, the lepton number violating Majorana neutrino masses can induce sneutrino-antisneutrino oscillations in any supersymmetric model. In this dissertation, I present some viable signals for detecting the sneutrino-antisneutrino oscillation at colliders. At the e-gamma collider (at the International Linear Collider), the numbers of the electron-sneutrino-antisneutrino oscillation signal events are quite high, and the backgrounds are quite small. A similar study for the LHC shows that, even though there are several backrounds, the sneutrino-antisneutrino oscillations can be detected. A useful asymmetry observable is introduced and studied. Usually, the oscillation probability formula where the sneutrinos are produced at rest is used. However, here, we study a general oscillation probability. The Lorentz factor and the distance at which the measurement is made inside the detector can have effects, especially when the sneutrino decay width is very small. These effects are demonstrated for a certain scenario at the LHC.

Noncommutative Gravitation as a Gauge Theory of Twisted Poincaré Symmetry

Gravitaation kvanttiteorian muotoilu on ollut teoreettisten fyysikkojen tavoitteena kvanttimekaniikan synnystä lähtien. Kvanttimekaniikan soveltaminen korkean energian ilmiöihin yleisen suhteellisuusteorian viitekehyksessä johtaa aika-avaruuden koordinaattien operatiiviseen ei-kommutoivuuteen. Ei-kommutoivia aika-avaruuden geometrioita tavataan myös avointen säikeiden säieteorioiden tietyillä matalan energian rajoilla. Ei-kommutoivan aika-avaruuden gravitaatioteoria voisi olla yhteensopiva kvanttimekaniikan kanssa ja se voisi mahdollistaa erittäin lyhyiden etäisyyksien ja korkeiden energioiden prosessien ei-lokaaliksi uskotun fysiikan kuvauksen, sekä tuottaa yleisen suhteellisuusteorian kanssa yhtenevän teorian pitkillä etäisyyksillä. Tässä työssä tarkastelen gravitaatiota Poincarén symmetrian mittakenttäteoriana ja pyrin yleistämään tämän näkemyksen ei-kommutoiviin aika-avaruuksiin. Ensin esittelen Poincarén symmetrian keskeisen roolin relativistisessa fysiikassa ja sen kuinka klassinen gravitaatioteoria johdetaan Poincarén symmetrian mittakenttäteoriana kommutoivassa aika-avaruudessa. Jatkan esittelemällä ei-kommutoivan aika-avaruuden ja kvanttikenttäteorian muotoilun ei-kommutoivassa aika-avaruudessa. Mittasymmetrioiden lokaalin luonteen vuoksi tarkastelen huolellisesti mittakenttäteorioiden muotoilua ei-kommutoivassa aika-avaruudessa. Erityistä huomiota kiinnitetään näiden teorioiden vääristyneeseen Poincarén symmetriaan, joka on ei-kommutoivan aika-avaruuden omaama uudentyyppinen kvanttisymmetria. Seuraavaksi tarkastelen ei-kommutoivan gravitaatioteorian muotoilun ongelmia ja niihin kirjallisuudessa esitettyjä ratkaisuehdotuksia. Selitän kuinka kaikissa tähänastisissa lähestymistavoissa epäonnistutaan muotoilla kovarianssi yleisten koordinaattimunnosten suhteen, joka on yleisen suhteellisuusteorian kulmakivi. Lopuksi tutkin mahdollisuutta yleistää vääristynyt Poincarén symmetria lokaaliksi mittasymmetriaksi --- gravitaation ei-kommutoivan mittakenttäteorian saavuttamisen toivossa. Osoitan, että tällaista yleistystä ei voida saavuttaa vääristämällä Poincarén symmetriaa kovariantilla twist-elementillä. Näin ollen ei-kommutoivan gravitaation ja vääristyneen Poincarén symmetrian tutkimuksessa tulee jatkossa keskittyä muihin lähestymistapoihin.

On the Geometry of Infinite-Dimensional Grassmannian Manifolds and Gauge Theory

This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.

Applications of Gauge/Gravity Duality to QCD and Heavy Ion Collisions

The description of quarks and gluons, using the theory of quantum chromodynamics (QCD), has been known for a long time. Nevertheless, many fundamental questions in QCD remain unanswered. This is mainly due to problems in solving the theory at low energies, where the theory is strongly interacting. AdS/CFT is a duality between a specific string theory and a conformal field theory. Duality provides new tools to solve the conformal field theory in the strong coupling regime. There is also some evidence that using the duality, one can get at least qualitative understanding of how QCD behaves at strong coupling. In this thesis, we try to address some issues related to QCD and heavy ion collisions, applying the duality in various ways.

The ozone transfer between atmosphere and vegetation. A study on Scots pine in the field

Ozone (O3) is a reactive gas present in the troposphere in the range of parts per billion (ppb), i.e. molecules of O3 in 109 molecules of air. Its strong oxidative capacity makes it a key element in tropospheric chemistry and a threat to the integrity of materials, including living organisms. Knowledge and control of O3 levels are an issue in relation to indoor air quality, building material endurance, respiratory human disorders, and plant performance. Ozone is also a greenhouse gas and its abundance is relevant to global warming. The interaction of the lower troposphere with vegetated landscapes results in O3 being removed from the atmosphere by reactions that lead to the oxidation of plant-related components. Details on the rate and pattern of removal on different landscapes as well as the ultimate mechanisms by which this occurs are not fully resolved. This thesis analysed the controlling processes of the transfer of ozone at the air-plant interface. Improvement in the knowledge of these processes benefits the prediction of both atmospheric removal of O3 and its impact on vegetation. This study was based on the measurement and analysis of multi-year field measurements of O3 flux to Scots pine (Pinus sylvestris L.) foliage with a shoot-scale gas-exchange enclosure system. In addition, the analyses made use of simultaneous CO2 and H2O exchange, canopy-scale O3, CO2 and H2O exchange, foliage surface wetness, and environmental variables. All data was gathered at the SMEAR measuring station (southern Finland). Enclosure gas-exchange techniques such as those commonly used for the measure of CO2 and water vapour can be applied to the measure of ozone gas-exchange in the field. Through analysis of the system dynamics the occurring disturbances and noise can be identified. In the system used in this study, the possible artefacts arising from the ozone reactivity towards the system materials in combination with low background concentrations need to be taken into account. The main artefact was the loss of ozone towards the chamber walls, which was found to be very variable. The level of wall-loss was obtained from simultaneous and continuous measurements, and was included in the formulation of the mass balance of O3 concentration inside the chamber. The analysis of the field measurements in this study show that the flux of ozone to the Scots pine foliage is generated in about equal proportions by stomatal and non-stomatal controlled processes. Deposition towards foliage and forest is sustained also during night and winter when stomatal gas-exchange is low or absent. The non-stomatal portion of the flux was analysed further. The pattern of flux in time was found to be an overlap of the patterns of biological activity and presence of wetness in the environment. This was seen to occur both at the shoot and canopy scale. The presence of wetness enhanced the flux not only in the presence of liquid droplets but also during existence of a moisture film on the plant surfaces. The existence of these films and their relation to the ozone sinks was determined by simultaneous measurements of leaf surface wetness and ozone flux. The results seem to suggest ozone would be reacting at the foliage surface and the reaction rate would be mediated by the presence of surface wetness. Alternative mechanisms were discussed, including nocturnal stomatal aperture and emission of reactive volatile compounds. The prediction of the total flux could thus be based on a combination of a model of stomatal behaviour and a model of water absorption on the foliage surfaces. The concepts behind the division of stomatal and non-stomatal sinks were reconsidered. This study showed that it is theoretically possible that a sink located before or near the stomatal aperture prevents or diminishes the diffusion of ozone towards the intercellular air space of the mesophyll. This obstacle to stomatal diffusion happens only under certain conditions, which include a very low presence of reaction sites in the mesophyll, an extremely strong sink located on the outer surfaces or stomatal pore. The relevance, or existence, of this process in natural conditions would need to be assessed further. Potentially strong reactions were considered, including dissolved sulphate, volatile organic compounds, and apoplastic ascorbic acid. Information on the location and the relative abundance of these compounds would be valuable. The highest total flux towards the foliage and forest happens when both the plant activity and ambient moisture are high. The highest uptake into the interior of the foliage happens at large stomatal apertures, provided that scavenging reactions located near the stomatal pore are weak or non-existent. The discussion covers the methodological developments of this study, the relevance of the different controlling factors of ozone flux, the partition amongst its component, and the possible mechanisms of non-stomatal uptake.

Effects of seed origin latitude on the timing of height growth cessation and field performance of silver birch

The aim of this thesis was to increase our knowledge about the effects of seed origin on the timing of height growth cessation and field performance of silver birch from different latitudes, with special attention paid to the browsing damage by moose in young birch plantations. The effect of seed origin latitude and sowing time on timing of height growth cessation of first-year seedlings was studied in a greenhouse experiment with seven seed origins (lat. 58º - 67ºN). Variation in critical night length (CNL) for 50 % bud set within two latitudinally distant stands (60º and 67ºN) was studied in three phytotron experiments. Browsing by moose on 5-11 -year-old silver birch saplings from latitudinally different seed origins (53º - 67ºN) was studied in a field experiment in southern Finland. Yield and stem quality of 22-year-old silver birch trees of Baltic, Finnish and Russian origin (54º - 63ºN) and the effect of latitudinal seed transfers were studied in two provenance trials at Tuusula, southern and Viitasaari, central Finland. The timing of height growth cessation depended systematically on latitude of seed origin and sowing date. The more northern the seed origin, the earlier the growth cessation and the shorter the growth period. Later sowing dates delayed growth cessation but also shortened the growth period. The mean CNL of the southern ecotype was longer, 6.3 ± 0.2 h (95 % confidence interval), than that of the northern ecotype, 3.1 ± 0.3 h. Within-ecotype variance of the CNL was higher in the northern ecotype (0.484 h2) than in the southern ecotype (0.150 h2). Browsing by moose decreased with increasing latitude of seed origin and sapling height. Origins transferred from more southern latitudes were more heavily browsed than the more northern native ones. Southern Finnish seed origins produced the highest volume per unit area in central Finland (lat. 63º11'N). Estonian and north Latvian stand seed origins, and the southern Finnish plus tree origins, were the most productive ones in southern Finland (lat. 60º21'N). Latitudinal seed transfer distance had a significant effect on survival, stem volume/ha and proportion of trees with a stem defect. The relationship of both survival and stem volume/ha to the latitudinal seed transfer distance was curvilinear. Volume was increased by transferring seed from ca. 2 degrees of latitude from the south. A longer transfer from the south, and transfer from the north, decreased the yield. The proportion of trees with a stem defect increased linearly in relation to the latitudinal seed transfer distance from the south.

Nitrogen dynamics and nitrogen use efficiency of spring cereals under Finnish growing conditions

Nitrogen (N) is one of the main inputs in cereal cultivation and as more than half of the arable land in Finland is used for cereal production, N has contributed substantially to agricultural pollution through fertilizer leaching and runoff. Based on this global phenomenon, the European Community has launched several directives to reduce agricultural emissions to the environment. Trough such measures, and by using economic incentives, it is expected that northern European agricultural practices will, in the future, include reduced N fertilizer application rates. Reduced use of N fertilizer is likely to decrease both production costs and pollution, but could also result in reduced yields and quality if crops experience temporary N deficiency. Therefore, more efficient N use in cereal production, to minimize pollution risks and maximize farmer income, represents a current challenge for agronomic research in the northern growing areas. The main objective of this study was to determine the differences in nitrogen use efficiency (NUE) among spring cereals grown in Finland. Additional aims were to characterize the multiple roles of NUE by analysing the extent of variation in NUE and its component traits among different cultivars, and to understand how other physiological traits, especially radiation use efficiency (RUE) and light interception, affect and interact with the main components of NUE and contribute to differences among cultivars. This study included cultivars of barley (Hordeum vulgare L.), oat (Avena sativa L.) and wheat (Triticum aestivum L.). Field experiments were conducted between 2001 and 2004 at Jokioinen, in Finland. To determine differences in NUE among cultivars and gauge the achievements of plant breeding in NUE, 17-18 cultivars of each of the three cereal species released between 1909 and 2002 were studied. Responses to nitrogen of landraces, old cultivars and modern cultivars of each cereal species were evaluated under two N regimes (0 and 90 kg N ha-1). Results of the study revealed that modern wheat, oat and barley cultivars had similar NUE values under Finnish growing conditions and only results from a wider range of cultivars indicated that wheat cultivars could have lower NUE than the other species. There was a clear relationship between nitrogen uptake efficiency (UPE) and NUE in all species whereas nitrogen utilization efficiency (UTE) had a strong positive relationship with NUE only for oat. UTE was clearly lower in wheat than in other species. Other traits related to N translocation indicated that wheat also had a lower harvest index, nitrogen harvest index and nitrogen remobilisation efficiency and therefore its N translocation efficiency was confirmed to be very low. On the basis of these results there appears to be potential and also a need for improvement in NUE. These results may help understand the underlying physiological differences in NUE and could help to identify alternative production options, such as the different roles that species can play in crop rotations designed to meet the demands of modern agricultural practices.

Ammonia emissions from pig and cattle slurry in the field and utilization of slurry nitrogen in crop production

Volatilization of ammonia (NH3) from animal manure is a major pathway for nitrogen (N) losses that cause eutrophication, acidification, and other environmental hazards. In this study, the effect of alternative techniques of manure treatment (aeration, separation, addition of peat) and application (broadcast spreading, band spreading, injection, incorporation by harrowing) on ammonia emissions in the field and on nitrogen uptake by ley or cereals was studied. The effect of a mixture of slurry and peat on soil properties was also investigated. The aim of this study was to find ways to improve the utilization of manure nitrogen and reduce its release to the environment. Injection into the soil or incorporation by harrowing clearly reduced ammonia volatilization from slurry more than did the surface application onto a smaller area by band spreading or reduction of the dry matter of slurry by aeration or separation. Surface application showed low ammonia volatilization, when pig slurry was applied to tilled bare clay soil or to spring wheat stands in early growth stages. Apparently, the properties of both slurry and soil enabled the rapid infiltration and absorption of slurry and its ammoniacal nitrogen by the soil. On ley, however, surface-applied cattle slurry lost about half of its ammoniacal nitrogen. The volatilization of ammonia from surface-applied peat manure was slow, but proceeded over a long period of time. After rain or irrigation, the peat manure layer on the soil surface retarded evaporation. Incorporation was less important for the fertilizer effect of peat manure than for pig slurry, but both manures were more effective when incorporated. Peat manure applications increase soil organic matter content and aggregate stability. Stubble mulch tillage hastens the effect in surface soil compared with ploughing. The apparent recovery of ammoniacal manure nitrogen in crop yield was higher with injection and incorporation than with surface applications. This was the case for leys as well as for spring cereals, even though ammonia losses from manures applied to cereals were relatively low with surface applications as well. The ammoniacal nitrogen of surface-applied slurry was obviously adsorbed by the very surface soil and remained mostly unavailable to plant roots in the dry soil. Supplementing manures with inorganic fertilizer nitrogen, which adds plant-available nitrogen to the soil at the start of growth, increased the overall recovery of applied nitrogen in crop yields.

Wind drift of snowfall between the radar volume and ground

This study evaluates how the advection of precipitation, or wind drift, between the radar volume and ground affects radar measurements of precipitation. Normally precipitation is assumed to fall vertically to the ground from the contributing volume, and thus the radar measurement represents the geographical location immediately below. In this study radar measurements are corrected using hydrometeor trajectories calculated from measured and forecasted winds, and the effect of trajectory-correction on the radar measurements is evaluated. Wind drift statistics for Finland are compiled using sounding data from two weather stations spanning two years. For each sounding, the hydrometeor phase at ground level is estimated and drift distance calculated using different originating level heights. This way the drift statistics are constructed as a function of range from radar and elevation angle. On average, wind drift of 1 km was exceeded at approximately 60 km distance, while drift of 10 km was exceeded at 100 km distance. Trajectories were calculated using model winds in order to produce a trajectory-corrected ground field from radar PPI images. It was found that at the upwind side from the radar the effective measuring area was reduced as some trajectories exited the radar volume scan. In the downwind side areas near the edge of the radar measuring area experience improved precipitation detection. The effect of trajectory-correction is most prominent in instant measurements and diminishes when accumulating over longer time periods. Furthermore, measurements of intensive and small scale precipitation patterns benefit most from wind drift correction. The contribution of wind drift on the uncertainty of estimated Ze (S) - relationship was studied by simulating the effect of different error sources to the uncertainty in the relationship coefficients a and b. The overall uncertainty was assumed to consist of systematic errors of both the radar and the gauge, as well as errors by turbulence at the gauge orifice and by wind drift of precipitation. The focus of the analysis is error associated with wind drift, which was determined by describing the spatial structure of the reflectivity field using spatial autocovariance (or variogram). This spatial structure was then used with calculated drift distances to estimate the variance in radar measurement produced by precipitation drift, relative to the other error sources. It was found that error by wind drift was of similar magnitude with error by turbulence at gauge orifice at all ranges from radar, with systematic errors of the instruments being a minor issue. The correction method presented in the study could be used in radar nowcasting products to improve the estimation of visibility and local precipitation intensities. The method however only considers pure snow, and for operational purposes some improvements are desirable, such as melting layer detection, VPR correction and taking solid state hydrometeor type into account, which would improve the estimation of vertical velocities of the hydrometeors.