Technical Note: Trend estimation from irregularly sampled, correlated data

Estimation of a trend of an atmospheric state variable is often performed by fitting a linear regression line to a set of data of this variable sampled at different times. Often these data are irregularly sampled in space and time and clustered in a sense that 5 error correlations among data points cause a similar error of data points sampled at similar times. Since this can affect the estimated trend, we suggest to take the full error covariance matrix of the data into account. Superimposed periodic variations can be jointly fitted in a straight forward manner, even if the shape of the periodic function is not known. Global data sets, particularly satellite data, can form the basis to estimate 10 the error correlations

Technical Note: Trend estimation from irregularly sampled, correlated data

Estimation of a trend of an atmospheric state variable is usually performed by fitting a linear regression line to a set of data of this variable sampled at different times. Often these data are irregularly sampled in space and time and clustered in a sense that error correlations among data points cause a similar error of data points sampled at similar times. Since this can affect the estimated trend, we suggest to take the full error covariance matrix of the data into account. Superimposed periodic variations can be jointly fitted in a straightforward manner, even if the shape of the periodic function is not known. Global data sets, particularly satellite data, can form the basis to estimate the error correlations. State-dependent amplitudes of superimposed periodic corrections result in a non-linear optimization problem which is solved iteratively

Direct inversion of circulation and mixing from tracer measurements - Part 1: Method

From a series of zonal mean global stratospheric tracer measurements sampled in altitude vs. latitude, circulation and mixing patterns are inferred by the inverse solution of the continuity equation. As a first step, the continuity equation is written as a tendency equation, which is numerically integrated over time to predict a later atmospheric state, i.e., mixing ratio and air density. The integration is formally performed by the multiplication of the initially measured atmospheric state vector by a linear prediction operator. Further, the derivative of the predicted atmospheric state with respect to the wind vector components and mixing coefficients is used to find the most likely wind vector components and mixing coefficients which minimize the residual between the predicted atmospheric state and the later measurement of the atmospheric state. Unless multiple tracers are used, this inversion problem is under-determined, and dispersive behavior of the prediction further destabilizes the inversion. Both these problems are addressed by regularization. For this purpose, a first-order smoothness constraint has been chosen. The usefulness of this method is demonstrated by application to various tracer measurements recorded with the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS). This method aims at a diagnosis of the Brewer–Dobson circulation without involving the concept of the mean age of stratospheric air, and related problems like the stratospheric tape recorder, or intrusions of mesospheric air into the stratosphere

Influences of the pulsed power supply on corona streamer appearance

Pulsed positive corona streamers in air are studied by images obtained with an intensified CCD camera. Using a switched capacitor power supply, thin streamers are observed that branch. A power supply consisting of a 4-stage transmission line transformer gives pulses of much higher current to the same gap. In this case, the number of streamers is less, they are wider and they hardly branch. The current density is roughly 1 A/mm2 in both type of streamer

A Tale of Two Set Theories

We describe the relationship between two versions of Tarski-Grothendieck set theory: the first-order set theory of Mizar and the higher-order set theory of Egal. We show how certain higher-order terms and propositions in Egal have equivalent first-order presentations. We then prove Tarski's Axiom A (an axiom in Mizar) in Egal and construct a Grothendieck Universe operator (a primitive with axioms in Egal) in Mizar

Carbon monoxide distributions from the upper troposphere to the mesosphere inferred from 4.7 μm non-local thermal equilibrium emissions measured by MIPAS on Envisat

We present global distributions of carbon monoxide (CO) from the upper troposphere to the mesosphere observed by the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS) on Envisat. Vertically resolved volume mixing ratio profiles have been retrieved from 4.7 μm limb emission spectra under consideration of non-local thermodynamic equilibrium. The precision of individual CO profiles is typically 5–30 ppbv (15–40% for altitudes greater than 40 km and lower than 15 km and 30–90% within 15–40 km). Estimated systematic errors are in the order of 8–15%. Below 60 km, the vertical resolution is 4–7 km. The data set which covers 54 days from September 2003 to March 2004 has been derived with an improved retrieval version including (i) the retrieval of log(vmr), (ii) the consideration of illumination-dependent vibrational population gradients along the instrument's line of sight, and (iii) joint-fitted vmr horizontal gradients in latitudinal and longitudinal directions. A detailed analysis of spatially resolved CO distributions during the 2003/2004 Northern Hemisphere major warming event demonstrate the potential of MIPAS CO observations to obtain new information on transport processes during dynamical active episodes, particularly on those acting in the vertical. From the temporal evolution of zonally averaged CO abundances, we derived extraordinary polar winter descent velocities of 1200 m per day inside the recovered polar vortex in January 2004. Middle stratospheric CO abundances show a well established correlation with the chemical source CH<sub>4</sub>, particularly in the tropics. In the upper troposphere, a moderate CO decrease from September 2003 to March 2004 was observed. Upper tropospheric CO observations provide a detailed picture of long-range transport of polluted air masses and uplift events. MIPAS observations taken on 9–11 September 2003 confirm the trapping of convective outflow of polluted CO-rich air from Southeast Asia into the Asian monsoon anticyclone, which has been described in previous studies. Upper tropospheric CO plumes, observed by MIPAS on this day, were predominantly located in the Northern Hemisphere. Most of these plumes could be related to Southeast Asian pollution by means of backward trajectory calculations. During 20–22 October, southern hemispheric biomass burning was the most likely source of the major CO plumes observed over the Southern Atlantic and Indian Ocean

Weak-coupling expansions for the attractive Holstein and Hubbard models

Weak-coupling expansions (conserving approximations) are carried out for the attractive Holstein and Hubbard models (on an infinite-dimensional hypercubic lattice) that include all bandstructure and vertex correction effects. Quantum fluctuations are found to renormalize transition temperatures by factors of order unity, but may be incorporated into the superconducting channel of Migdal-Eliashberg theory by renormalizing the phonon frequency and the interaction strength.Comment: 10 pages, (five figures available from the author by request) typeset with ReVTeX, preprint NSF-ITP-93-10

Observed temporal evolution of global mean age of stratospheric air for the 2002 to 2010 period

An extensive observational data set, consisting of more than 10<sup>6</sup> SF<sub>6</sub> vertical profiles from MIPAS measurements distributed over the whole globe has been condensed into monthly zonal means of mean age of air for the period September 2002 to January 2010, binned at 10° latitude and 1–2 km altitude. The data were analysed with respect to their temporal variation by fitting a regression model consisting of a constant and a linear increase term, 2 proxies for the QBO variation, sinusoidal terms for the seasonal and semi-annual variation and overtones for the correction of the shapes to the observed data set. The impact of subsidence of mesospheric SF<sub>6</sub>-depleted air and in-mixing into non-polar latitudes on mid-latitudinal absolute age of air and its linear increase was assessed and found to be small. <br><br> The linear increase of mean age of stratospheric air was found to be positive and partly larger than the trend derived by Engel et al. (2009) for most of the Northern mid-latitudes, the middle stratosphere in the tropics, and parts of the Southern mid-latitudes, as well as for the Southern polar upper stratosphere. Multi-year decrease of age of air was found for the lowermost and the upper stratospheric tropics, for parts of Southern mid-latitudes, and for the Northern polar regions. Analysis of the amplitudes and phases of the seasonal variation shed light on the coupling of stratospheric regions to each other. In particular, the Northern mid-latitude stratosphere is well coupled to the tropics, while the Northern lowermost mid-latitudinal stratosphere is decoupled, confirming the separation of the shallow branch of the Brewer-Dobson circulation from the deep branch. We suggest an overall increased tropical upwelling, together with weakening of mixing barriers, especially in the Northern Hemisphere, as a hypothetical model to explain the observed pattern of linear multi-year increase/decrease, and amplitudes and phase shifts of the seasonal variation