Closed form solution to the twin P4P or the combined three dimensional resection-intersection problem in terms of Möbius barycentric coordinates

The twin perspective 4 point (twin P4P) problem – also called the combined three dimensional resection-intersection problem – is the problem of finding the position of a scene object from 4 correspondence points and a scene stereopair. While the perspective centers of the left and right scene image are positioned by means of a double three dimensional resection, the position of the scene object imaged on the left and right photograph is determined by a three dimensional intersection based upon given resected perspective centers. Here we present a new algorithm solving the twin P4P problem by means of M?bius barycentric coordinates. In the first algorithmic step we determine the distances between the perspective centers and the unknown intersected point by solving a linear system of equations. Typically, area elements of the left and right image build up the linear equation system. The second algorithmic step allows for the computation of the M?bius barycentric coordinates of the unknown intersected point which are thirdly converted into three dimensional object space coordinates {X,Y,Z} of the intersected point. Typically, this three-step algorithm based upon M?bius barycentric coordinates takes advantage of the primary double resection problem from which only distances from four correspondence points to the left and right perspective centre are needed. No orientation parameters and no coordinates of the left and right perspective center have to be made available. Received 1 May 1996; Accepted 13 September 1996     

A canonical theory for short GPS baselines. Part II: the ambiguity precision and correlation

The present contribution is the second of four parts. It considers the precision and correlation of the least-squares estimators of the carrier phase ambiguities. It is shown how the precision and correlation of the double-differenced ambiguities as well as of the widelane ambiguities are effected by the observation weights, by the number of satellites tracked, by the number of observation epochs used, and by the change over time of the relative receiver-satellite geometry. Also the ability of the widelane transformation to decorrelate and to improve the precision is investigated. Received: 16 July 1996 / Accepted: 14 November 1996     

A canonical theory for short GPS baselines. Part IV: precision versus reliability

This contribution is the last of four parts and deals with the link between baseline precision and ambiguity reliability. It is shown analytically how and to what extent the baseline-ambiguity correlation is related to the gain in baseline precision, to the volume of the ambiguity search space, and to the impact of potential integer ambiguity biases. Also, an ambiguity DOP measure is introduced together with its closed-form formulae for the three different single-baseline models. Received: 16 July 1996 / Accepted: 14 November 1996     

The spacetime gravitational field of a deformable body

The high-resolution analysis of orbit perturbations of terrestrial artificial satellites has documented that the eigengravitation of a massive body like the Earth changes in time, namely with periodic and aperiodic constituents. For the space-time variation of the gravitational field the action of internal and external volume as well as surface forces on a deformable massive body are responsible. Free of any assumption on the symmetry of the constitution of the deformable body we review the incremental spatial (“Eulerian”) and material (“Lagrangean”) gravitational field equations, in particular the source terms (two constituents: the divergence of the displacement field as well as the projection of the displacement field onto the gradient of the reference mass density function) and the `jump conditions' at the boundary surface of the body as well as at internal interfaces both in linear approximation. A spherical harmonic expansion in terms of multipoles of the incremental Eulerian gravitational potential is presented. Three types of spherical multipoles are identified, namely the dilatation multipoles, the transport displacement multipoles and those multipoles which are generated by mass condensation onto the boundary reference surface or internal interfaces. The degree-one term has been identified as non-zero, thus as a “dipole moment” being responsible for the varying position of the deformable body's mass centre. Finally, for those deformable bodies which enjoy a spherically symmetric constitution, emphasis is on the functional relation between Green functions, namely between Fourier-/ Laplace-transformed volume versus surface Love-Shida functions (h(r),l(r) versus h ^′(r),l ^′(r)) and Love functions k(r) versus k ^′(r). The functional relation is numerically tested for an active tidal force/potential and an active loading force/potential, proving an excellent agreement with experimental results. Received: December 1995 / Accepted: 1 February 1997     

Non-linear climate feedback analysis in an atmospheric general circulation model

 A method is described for evaluating the ‘partial derivatives’ of globally averaged top-of-atmosphere (TOA) radiation changes with respect to basic climate model physical parameters. This method is used to analyse feedbacks in the Australian Bureau of Meteorology Research Centre general circulation model. The parameters considered are surface temperature, water vapour, lapse rate and cloud cover. The climate forcing which produces the changes is a globally uniform sea surface temperature (SST) perturbation. The first and second order differentials of model parameters with respect to the forcing (i.e. SST changes) are estimated from quadratic least square fitting. Except for total cloud cover, variables are found to be strong functions of global SST. Strongly non-linear variations of lapse rate and high cloud amount and height appear to relate to the non-linear response in penetrative convection. Globally averaged TOA radiation differentials with respect to model parameters are also evaluated. With the exception of total cloud contributions, a high correlation is generally found to exist, on the global mean level, between TOA radiation and the respective parameter perturbations. The largest non-linear terms contributing to radiative changes are those due to lapse rate and high cloud. The contributions of linear and non-linear terms to the overall radiative response from a 4 K SST perturbation are assessed. Significant non-linear responses are found to be associated with lapse rate, water vapour and cloud changes. Although the exact magnitude of these responses is likely to be a function of the particular model as well as the imposed SST perturbation pattern, the present experiments flag these as processes which cannot properly be understood from linear theory in the evaluation of climate change sensitivity. Received: 16 January 1997/Accepted: 9 May 1997     

Modelling the short-term response of the Greenland ice-sheet to global warming

 A two-dimensional vertically integrated ice flow model has been developed to test the importance of various processes and concepts used for the prediction of the contribution of the Greenland ice-sheet to sea-level rise over the next 350 y (short-term response). The mass balance is modelled by the degree-day method and the energy-balance method. The lithosphere is considered to respond isostatically to a point load and the time evolution of the bedrock follows from a viscous asthenosphere. According to the IPCC-IS92a scenario (with constant aerosols after 1990) the Greenland ice-sheet is likely to cause a global sea level rise of 10.4 cm by 2100 AD. It is shown, however, that the result is sensitive to precise model formulations and that simplifications as used in the sea-level projection in the IPCC-96 report yield less accurate results. Our model results indicate that, on a time scale of a hundred years, including the dynamic response of the ice-sheet yields more mass loss than the fixed response in which changes in geometry are not incorporated. It appears to be important to consider sliding, as well as the fact that climate sensitivity increases for larger perturbations. Variations in predicted sea-level change on a time scale of hundred years depend mostly on the initial state of the ice-sheet. On a time scale of a few hundred years, however, the variability in the predicted melt is dominated by the variability in the climate scenarios. Received: 21 August 1996/Accepted: 12 May 1997     

A predictability study of simulated North Atlantic multidecadal variability

 The North Atlantic is one of the few places on the globe where the atmosphere is linked to the deep ocean through air–sea interaction. While the internal variability of the atmosphere by itself is only predictable over a period of one to two weeks, climate variations are potentially predictable for much longer periods of months or even years because of coupling with the ocean. This work presents details from the first study to quantify the predictability for simulated multidecadal climate variability over the North Atlantic. The model used for this purpose is the GFDL coupled ocean-atmosphere climate model used extensively for studies of global warming and natural climate variability. This model contains fluctuations of the North Atlantic and high-latitude oceanic circulation with variability concentrated in the 40–60 year range. Oceanic predictability is quantified through analysis of the time-dependent behavior of large-scale empirical orthogonal function (EOF) patterns for the meridional stream function, dynamic topography, 170 m temperature, surface temperature and surface salinity. The results indicate that predictability in the North Atlantic depends on three main physical mechanisms. The first involves the oceanic deep convection in the subpolar region which acts to integrate atmospheric fluctuations, thus providing for a red noise oceanic response as elaborated by Hasselmann. The second involves the large-scale dynamics of the thermohaline circulation, which can cause the oceanic variations to have an oscillatory character on the multidecadal time scale. The third involves nonlocal effects on the North Atlantic arising from periodic anomalous fresh water transport advecting southward from the polar regions in the East Greenland Current. When the multidecadal oscillatory variations of the thermohaline circulation are active, the first and second EOF patterns for the North Atlantic dynamic topography have predictability time scales on the order of 10–20 y, whereas EOF-1 of SST has predictability time scales of 5–7 y. When the thermohaline variability has weak multidecadal power, the Hasselmann mechanism is dominant and the predictability is reduced by at least a factor of two. When the third mechanism is in an extreme phase, the North Atlantic dynamic topography patterns realize a 10–20 year predictability time scale. Additional analysis of SST in the Greenland Sea, in a region associated with the southward propagating fresh water anomalies, indicates the potential for decadal scale predictability for this high latitude region as well. The model calculations also allow insight into regional variations of predictability, which might be useful information for the design of a monitoring system for the North Atlantic. Predictability appears to break down most rapidly in regions of active convection in the high-latitude regions of the North Atlantic. Received: 28 October 1996 / Accepted: 21 March 1997     

Reactive Nitrogen Compounds at Spitsbergen in the Norwegian Arctic

Simultaneousindependent measurements of NO_y and NO_x(NO_x= NO + NO₂) by high-sensitivitychemiluminescence systems and of PAN (peroxyacetylnitrate) and PPN (peroxypropionyl nitrate) by GC-ECDwere made at Spitsbergen in the Norwegian Arcticduring the first half year of 1994. The average mixingratio of the sum of PAN and PPN (denoted PANs)increased from around 150 pptv in early winter to amaximum of around 500 pptv in late March, whereasepisodic peak values reached 800 pptv. This occurredsimultaneously with a maximum in ozone which increasedto 45–50 ppbv in March–April. The average NO_xmixing ratio was 27 pptv and did not show any cyclethrough the period. The NO_y mixing ratio showeda maximum in late March, while the difference betweenNO_y and PAN decreased during spring. This is anindication of the dominance of PAN in the NO_ybudget in the Arctic, but possible changes in theefficiency of the NO_y converter could alsocontribute to this. Although most PAN in theArctic is believed to be due to long range transport,the observations indicate local loss and formationrates of up to 1–2 pptv h^-1 in April–May.Measurements of carbonyl compounds suggest thatacetaldehyde was the dominant, local precursor ofPAN.Now at 1.     

BLENDING HEIGHTS FOR WINDS OBLIQUE TO CHECKERBOARDS

Analysis of the two-dimensional checkerboardproblem of many alternating surfaces with different properties andvarious scales and aspect ratios is extended to the general case ofwinds oblique to the surface pattern. Relevant periodic solutions ofa three-dimensional atmospheric convection-diffusion equation arefound, and used to estimate blending heights for both concentrationand flux.Illustrative results show negligible directional effect on blendingheight for X1* (the smaller pattern wavelength) of 10 m, but a ratio of maximum/minimum heights of about 3 to 5 for X1* = 100 m. For X1* = 1000 mthe directional variation is strongly peaked with a maximum/minimum ratio of 8 to 14. At fixed X1* the maximum increases with check aspect ratio. It is associated withthe case where the wind blows along one check diagonal. In thatsingular situation, damping of the dominant harmonic varying normalto the diagonal depends purely on transverse and vertical diffusion:horizontal convection and wind shear play no part.     

The phase lag of temperature behind global solar radiation over Egypt

Summary The relation between the air temperature and the global solar radiation, which can be conveniently represented by the three characteristics: mean, amplitude and phase lag of the first harmonic of global radiation and air temperatures. A good correlation between the air temperature and the solar radiation has been found when the phase lag between them is nearly of 30 days.With 6 Figures