Convex optimization under inequality constraints in rank-deficient systems

Many geodetic applications require the minimization of a convex objective function subject to some linear equality and/or inequality constraints. If a system is singular (e.g., a geodetic network without a defined datum) this results in a manifold of solutions. Most state-of-the-art algorithms for inequality constrained optimization (e.g., the Active-Set-Method or primal-dual Interior-Point-Methods) are either not able to deal with a rank-deficient objective function or yield only one of an infinite number of particular solutions. In this contribution, we develop a framework for the rigorous computation of a general solution of a rank-deficient problem with inequality constraints. We aim for the computation of a unique particular solution which fulfills predefined optimality criteria as well as for an adequate representation of the homogeneous solution including the constraints. Our theoretical findings are applied in a case study to determine optimal repetition numbers for a geodetic network to demonstrate the potential of the proposed framework.     

Multivariate Prediction of Total Water Storage Changes Over West Africa from Multi-Satellite Data

West African countries have been exposed to changes in rainfall patterns over the last decades, including a significant negative trend. This causes adverse effects on water resources of the region, for instance, reduced freshwater availability. Assessing and predicting large-scale total water storage (TWS) variations are necessary for West Africa, due to its environmental, social, and economical impacts. Hydrological models, however, may perform poorly over West Africa due to data scarcity. This study describes a new statistical, data-driven approach for predicting West African TWS changes from (past) gravity data obtained from the gravity recovery and climate experiment (GRACE), and (concurrent) rainfall data from the tropical rainfall measuring mission (TRMM) and sea surface temperature (SST) data over the Atlantic, Pacific, and Indian Oceans. The proposed method, therefore, capitalizes on the availability of remotely sensed observations for predicting monthly TWS, a quantity which is hard to observe in the field but important for measuring regional energy balance, as well as for agricultural, and water resource management. Major teleconnections within these data sets were identified using independent component analysis and linked via low-degree autoregressive models to build a predictive framework. After a learning phase of 72 months, our approach predicted TWS from rainfall and SST data alone that fitted to the observed GRACE-TWS better than that from a global hydrological model. Our results indicated a fit of 79 % and 67 % for the first-year prediction of the two dominant annual and inter-annual modes of TWS variations. This fit reduces to 62 % and 57 % for the second year of projection. The proposed approach, therefore, represents strong potential to predict the TWS over West Africa up to 2 years. It also has the potential to bridge the present GRACE data gaps of 1 month about each 162 days as well as a—hopefully—limited gap between GRACE and the GRACE follow-on mission over West Africa. The method presented could also be used to generate a near-real-time GRACE forecast over the regions that exhibit strong teleconnections.     

Crustal structure along a traverse across the Middle and High Atlas mountains derived from seismic refraction studies

In 1986 explosion seismic investigations have been carried out along a traverse of about 350 km length running from NNW to SSE crossing the High and Middle Atlas. Two further profiles run E/W through the Middle and NW-SE through the High Atlas.Neither the High nor the Middle Atlas mountains have a significant root. The maximum thickness of crust with 38–39 km is found under the northern border of the High Atlas. South and north of the High Atlas the crustal thickness amounts to 35 km. The upper as well as the lower crust are stongly structurized in the vertical direction. Significant for the profiles observed is the change between high and low velocities in the whole crust. The velocities at the uppermost mantle are relatively low with values of 7.7–7.9 km/s. The average velocities for the whole crust are 6.1–6.2 km/s.

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Zusammenfassung Sprengseismische Untersuchungen wurden 1986 längs einer etwa 350 km langen Traverse durchgeführt, die NNW-SSO verläuft und den Mittleren und den Hohen Atlas quert. Zwei weitere Profile verlaufen E-W durch den Mittleren und NW-SO durch den Hohen Atlas.Der Hohe und der Mittlere Atlas besitzen keine ausgeprägte Gebirgswurzel. Die maximale Krustendicke wird mit 38–39 km unter dem nördlichen Rand des Hohen Atlas angetroffen. Südlich und nördlich vom Hohen Atlas beträgt die Krustenmächtigkeit 35 km. Sowohl die obere als auch die untere Kruste sind in vertikaler Richtung stark strukturiert. Auf allen beobachteten Profilen ist ein Wechsel zwischen hohen und niedrigen Geschwindigkeiten für die gesamte Kruste charakteristisch. Die Geschwindigkeiten im obersten Mantel sind mit Werten von 7.7–7.9 km/s relativ niedrig. Die Durchschnittsgeschwindigkeiten für die gesamte Kruste liegen bei 6.1–6.2 km/s.

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Résumé Durant l'année 1986 des sondages sismiques de tirs ont été entrepris le long d'une traverse d'environ 350 km de longueur se dirigeant du nord nordouest au sud sud-est et parcourant le Haut et Moyen Atlas. Deux autres profils parcourent le Moyen Atlas de l'est à l'ouest et le Haut Atlas du nord-ouest au sud-est.Le Haut et Moyen Atlas ne possèdent pas de racine montagneuse. L'épaisseur maximale de 38–39 km de la croûte a été localisée sous le bord nord du Haut Atlas. Au sud et au nord du Haut Atlas l'épaisseur de la croûte s'élève à 35 km. La croûte supérieure ainsi que la croûte inférieure est très structurée en direction verticale.Ce qui est caractéristique pour tous les profils observés c'est un changement rapide entre de grandes et petites vitesses et concernant le total de la croûte. Les vitesses dans le manteau le plus élevé haut sont relativement basses avec 7,7–7,9 km/s. Les vitesses moyennes pour la croûte entière sont de 6,1 à 6,2 km/s.

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1986 350 , NNW SSE . : E-W , NWSE . . , 38/39 , 35 . , . . : 7,7–7,9 /. 6,1–6,2 /.

     

First GOCE gravity field models derived by three different approaches

Three gravity field models, parameterized in terms of spherical harmonic coefficients, have been computed from 71 days of GOCE (Gravity field and steady-state Ocean Circulation Explorer) orbit and gradiometer data by applying independent gravity field processing methods. These gravity models are one major output of the European Space Agency (ESA) project GOCE High-level Processing Facility (HPF). The processing philosophies and architectures of these three complementary methods are presented and discussed, emphasizing the specific features of the three approaches. The resulting GOCE gravity field models, representing the first models containing the novel measurement type of gravity gradiometry ever computed, are analysed and assessed in detail. Together with the coefficient estimates, full variance-covariance matrices provide error information about the coefficient solutions. A comparison with state-of-the-art GRACE and combined gravity field models reveals the additional contribution of GOCE based on only 71 days of data. Compared with combined gravity field models, large deviations appear in regions where the terrestrial gravity data are known to be of low accuracy. The GOCE performance, assessed against the GRACE-only model ITG-Grace2010s, becomes superior at degree 150, and beyond. GOCE provides significant additional information of the global Earth gravity field, with an accuracy of the 2-month GOCE gravity field models of 10?cm in terms of geoid heights, and 3?mGal in terms of gravity anomalies, globally at a resolution of 100?km (degree/order 200).     

Seismic, gravity and petrological evidence for partial melt beneath the thickened Central Andean crust (21–23°S)

On the basis of seismic refraction investigations and gravimetric data we have modelled the crustal structure of the southern Central Andes (21–23°S). A pronounced variation in crustal parameters is seen in N-S- and W-E-crossing seismic profiles over the entire Andean orogene, characterized by a crustal thickness of up to 70 km under the magmatic arc and backarc, strongly reduced seismic velocities and a Bouguer minimum of −450 mGal. Anomalously low velocities of 5.9–6.0 km/s in the deeper crust of the Western Cordillera and Altiplano regions lead to an over-compensation of the Bouguer minima resulting in values of crustal densities higher than estimates based purely on seismic velocity measurements. In an attempt to reconcile these differences, the behavior of crystalline rocks based on published laboratory data was studied under varying pressure and temperature conditions up to the range of partial melting. If the temperature is increased above the melting point, a rapid decrease in seismic velocity is accompanied by a slow decrease in density. For the Central Andes, a good fit of the observed and calculated Bouguer anomalies is obtained if the densities of the rocks from the low-velocity zone (LVZ) beneath the Western Cordillera and the Altiplano are varied. Model calculations lead to a velocity-density relation for partial molten rocks that allows the melt proportions of rocks to be estimated. Model calculations indicate that 15–20 vol.% of basaltic to andesitic melt at depth is necessary to explain the LVZ and Bouguer anomaly beneath the arc and parts of the backarc. High heat flow values (100 mW/m²) support the idea that large areas of the deeper Andean crust are strongly weakened by the presence of partially molten rocks, resulting in reduced seismic velocities, with the Western Cordillera, the active volcanic arc of the Andean mountain range, acting as a ductile buffer between the two more rigid crustal blocks of the forearc and backarc regions.     

Winds of change and the threat of desertification: Case study from the Tunesian Sahara

The former agro-pastoral system which prevailed in the Tunisian Sahara had developed a pastoral resource management which preserved the rather fragile ecological equilibrium. Nowadays in the Tunisian Sahara nearly all former nomads are settled around the villages and irrigated areas which were created with the official aim of utilizing the Sahara. The sedentarization of the nomads has provoked two phenomena: a concentration and an increased density of population around the newly founded oases on the one hand and a depopulation of the more remote steppes on the other hand. Actually this demographic disequilibrium threatens the ecological equilibrium which existed in the past. It has caused a multiplication of the stocking of the actual pastures and increases the risk of degradation of the peripheral and easily accessible grazing-grounds. It exercises an excessive pressure on the environment through the intensive human activities and the satisfaction of new needs. This pressure manifests itself in the enlargement of the circles of degradation around the oases and the intensification of aeolian dynamics. A regeneration of the former organically developed eco-system is impossible without conscious reference into the local traditions and without consideration of the complexity of interelationship of the various factors.     

Volume-preserving interpolation of a smooth surface from polygon-related data

 The interpolation of continuous surfaces from discrete points is supported by most GIS software packages. Some packages provide additional options for the interpolation from 3D line objects, for example surface-specific lines, or contour lines digitized from topographic maps. Demographic, social and economic data can also be used to construct and display smooth surfaces. The variables are usually published as sums for polygonal units, such as the number of inhabitants in communities or counties. In the case of point and line objects the geometric properties have to be maintained in the interpolated surface. For polygon-based data the geometric properties of the polygon boundary and the volume should be preserved, avoiding redistribution of parts of the volume to neighboring units during interpolation. The pycnophylactic interpolation method computes a continuous surface from polygon-based data and simultaneously enforces volume preservation in the polygons. The original procedure using a regular grid is extended to surface representations based on an irregular triangular network (TIN). Received: 5 October 2000 / Accepted: 20 December 2000