Ensemble filter methods with perturbed observations applied to nonlinear problems

The ensemble Kalman filter (EnKF) has been shown repeatedly to be an effective method for data assimilation in large-scale problems, including those in petroleum engineering. Data assimilation for multiphase flow in porous media is particularly difficult, however, because the relationships between model variables (e.g., permeability and porosity) and observations (e.g., water cut and gas–oil ratio) are highly nonlinear. Because of the linear approximation in the update step and the use of a limited number of realizations in an ensemble, the EnKF has a tendency to systematically underestimate the variance of the model variables. Various approaches have been suggested to reduce the magnitude of this problem, including the application of ensemble filter methods that do not require perturbations to the observed data. On the other hand, iterative least-squares data assimilation methods with perturbations of the observations have been shown to be fairly robust to nonlinearity in the data relationship. In this paper, we present EnKF with perturbed observations as a square root filter in an enlarged state space. By imposing second-order-exact sampling of the observation errors and independence constraints to eliminate the cross-covariance with predicted observation perturbations, we show that it is possible in linear problems to obtain results from EnKF with observation perturbations that are equivalent to ensemble square-root filter results. Results from a standard EnKF, EnKF with second-order-exact sampling of measurement errors that satisfy independence constraints (EnKF (SIC)), and an ensemble square-root filter (ETKF) are compared on various test problems with varying degrees of nonlinearity and dimensions. The first test problem is a simple one-variable quadratic model in which the nonlinearity of the observation operator is varied over a wide range by adjusting the magnitude of the coefficient of the quadratic term. The second problem has increased observation and model dimensions to test the EnKF (SIC) algorithm. The third test problem is a two-dimensional, two-phase reservoir flow problem in which permeability and porosity of every grid cell (5,000 model parameters) are unknown. The EnKF (SIC) and the mean-preserving ETKF (SRF) give similar results when applied to linear problems, and both are better than the standard EnKF. Although the ensemble methods are expected to handle the forecast step well in nonlinear problems, the estimates of the mean and the variance from the analysis step for all variants of ensemble filters are also surprisingly good, with little difference between ensemble methods when applied to nonlinear problems.     

Hopf bifurcations with fluctuating gain

A Hopf bifurcation describes a transition from a steady state to an oscillatory one. This report describes how the presence of white Gaussian noise in the term driving the instability (the gain) affects the character of such a transition. We find that for small noise intensities the effective critical value of the bifurcation parameter increases linearly with the noise intensity. This result is used to quantitatively describe the onset of the magneto‐rotational instability in a laboratory experiment using liquid metal with turbulent background flow. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Enrichment of Rare Earth Elements during magmatic and post-magmatic processes: a case study from the Loch Loyal Syenite Complex, northern Scotland

Concern about security of supply of critical elements used in new technologies, such as the Rare Earth Elements (REE), means that it is increasingly important to understand the processes by which they are enriched in crustal settings. High REE contents are found in syenite-dominated alkaline complexes intruded along the Moine Thrust Zone, a major collisional zone in north-west Scotland. The most northerly of these is the Loch Loyal Syenite Complex, which comprises three separate intrusions. One of these, the Cnoc nan Cuilean intrusion, contains two mappable zones: a Mixed Syenite Zone in which mafic melasyenite is mixed and mingled with leucosyenite and a Massive Leucosyenite Zone. Within the Mixed Syenite Zone, hydrothermal activity is evident in the form of narrow altered veins dominated by biotite and magnetite; these are poorly exposed and their lateral extent is uncertain. The REE mineral allanite is relatively abundant in the melasyenite and is extremely enriched in the biotite–magnetite veins, which have up to 2 % total rare earth oxides in bulk rock analyses. An overall model for development of this intrusion can be divided into three episodes: (1) generation of a Light Rare Earth Element (LREE)-enriched parental magma due to enrichment of the mantle source by subduction of pelagic carbonates; (2) early crystallisation of allanite in melasyenite, due to the saturation of the magma in the LREE; and (3) hydrothermal alteration, in three different episodes identified by petrography and mineral chemistry, generating the intense enrichment of REE in the biotite–magnetite veins. Dating of allanite and titanite in the biotite–magnetite veins gives ages of c. 426 Ma, overlapping with previously published crystallisation ages for zircon in the syenite.     

A new approach for detecting and mapping sewage impacts

Increased nitrogen loading has been implicated in eutrophication occurrences worldwide. Much of this loading is attributable to the growing human population along the world's coastlines. A significant component of this nitrogen input is from sewage effluent, and delineation of the distribution and biological impact of sewage-derived nitrogen is becoming increasingly important. Here, we show a technique that identifies the source, extent and fate of biologically available sewage nitrogen in coastal marine ecosystems. This method is based on the uptake of sewage nitrogen by marine plants and subsequent analysis of the sewage signature (elevated delta 15N) in plant tissues. Spatial analysis is used to create maps of delta 15N and establish coefficient of variation estimates of the mapped values. We show elevated delta 15N levels in marine plants near sewage outfalls in Moreton Bay, Australia, a semi-enclosed bay receiving multiple sewage inputs. These maps of sewage nitrogen distribution are being used to direct nutrient reduction strategies in the region and will assist in monitoring the effectiveness of environmental protection measures.     

Crustal structure in the southwestern part of the Iberian Peninsula

Seismic investigations to determine the crustal structure in the southwestern part of the Iberian Peninsula have been initiated in 1970. First experiments were carried out during July 1970, when a series of ten shots was fired off Cabo de Sines (Portugal) in shallow water and recorded up to distances of 185 km along a SE-profile towards Huelva (Spain). The profile was reversed in December 1970, when a series of twelve shots was fired off the south coast near Fuzeta (east of Faro) and recorded up to distances of about 260 km along a NW-profile towards Cabo da Roca west of Lisboa. A considerable increase in the seismic efficiency of the explosions could be achieved by generating standing waves in the water.

The structure deduced exhibits some peculiar features. Below the Palaeozoic sediments a fairly high velocity of 6.4 km/sec is found for the dome-shaped basement in that area. The lower crust, which is separated from the upper crust by a distinct velocity inversion (with a minimum velocity of about 5.3–5.6 km/sec), is characterized by a velocity of 7.1 km/sec. From the geological evidence and the sequence of seismic velocities it must be concluded that the upper crustal block in the southwestern part of the Iberian Peninsula has been uplifted by about 2–5 km since Permo-Triassic time, thus emphasizing the significance of vertical movement in tectonic activity.

The top of the upper mantle (8.15 km/sec) was detected at a depth of 30 km close to the Atlantic coast in the west, while near the Algarve coast in the south the depth to the M-discontinuity is about 34–35 km. This result in conjunction with studies of earthquake focal mechanisms confirms the suggestion that the Iberian block is being underthrust under the African plate.     

Petrographic observations suggestive of microbial mats from Rampur Shale and Bijaigarh Shale, Vindhyan basin, India

Petrographic observations of two Vindhyan black shales (Rampur Shale of the Semri Group and Bijaigarh Shale of the Kaimur Group) revealed the following features:

The electrokinetic properties and flotation behaviour of apatite and calcite in the presence of dodecylamine chloride

Electrokinetic and flotation studies on apatites and calcite show that under certain conditions these minerals are floatable with dodecylamine chloride (DDACl) and the possible mode of DDACl adsorption is due to Coulombic and Van der Waals forces. The results indicate hemi-micelle formation of dodecylamine ions and suggest involvement of neutral molecular amine in the hemi-micelle structure, the critical hemi-micelle concentration being influenced by the nature of charge at the mineral surface.     

Meridional propagation of large-scale monsoon convective zones

Summary Observational studies indicate that the convective activity of the monsoon systems undergo intraseasonal variations with multi-week time scales. The zone of maximum monsoon convection exhibits substantial transient behavior with successive propagating from the North Indian Ocean to the heated continent. Over South Asia the zone achieves its maximum intensity. These propagations may extend over 3000 km in latitude and perhaps twice the distance in longitude and remain as coherent entities for periods greater than 2–3 weeks. Attempts to explain this phenomena using simple ocean-atmosphere models of the monsoon system had concluded that the interactive ground hydrology so modifies the total heating of the atmosphere that a steady state solution is not possible, thus promoting lateral propagation. That is, the ground hydrology forces the total heating of the atmosphere and the vertical velocity to be slightly out of phase, causing a migration of the convection towards the region of maximum heating. Whereas the lateral scale of the variations produced by the Webster (1983) model were essentially correct, they occurred at twice the frequency of the observed events and were formed near the coastal margin, rather than over the ocean.Webster's (1983) model used to pose the theories was deficient in a number of aspects. Particularly, both the ground moisture content and the thermal inertia of the model were severely underestimated. At the same time, the sea surface temperatures produced by the model between the equator and the model's land-sea boundary were far too cool. Both the atmosphere and the ocean model were modified to include a better hydrological cycle and ocean structure. The convective events produced by the modified model possessed the observed frequency and were generated well south of the coastline.The improved simulation of monsoon variability allowed the hydrological cycle feedback to be generalized. It was found that monsoon variability was constrained to lie within the bounds of a positive gradient of aconvective intensity potential (I). The function depends primarily on the surface temperature, the availability of moisture and the stability of the lower atmosphere which varies very slowly on the time scale of months. The oscillations of the monsoon perturb the mean convective intensity potential causing local enhancements of the gradient. These perturbations are caused by the hydrological feedbacks, discussed above, or by the modification of the air-sea fluxes caused by variations of the low level wind during convective events. The final result is the slow northward propagation of convection within an even slower convective regime. The ECMWF analyses show very similar behavior of the convective intensity potential. Although it is considered premature to use the model to conduct simulations of the African monsoon system, the ECMWF analysis indicates similar behavior in the convective intensity potential suggesting, at least, that the same processes control the low frequency structure of the African monsoon. The implications of the hypotheses on numerical weather prediction of monsoon phenomenon are discussed.     

Growth Of The Summer Daytime Convective Boundary Layer At Anand

The heights of the daytime convective boundary layer (CBL), computed by a one-dimensional model for a bare soil surface at a semi-arid station,Anand, during the dry and hot summer month of May 1997, are presented. As input, the model requires surface heat flux, friction velocity and air temperature as functions of time. Temperature data at the one-metre level from a tower and sonic anemometer data at 9.5 m collected during the period 13–17 May 1997 in the Land Surface Processes Experiment (LASPEX-97) are used to compute hourly values of surface heat flux, friction velocity and Obukhov length following the operational method suggested by Holtslag and Van Ulden [J. Climate Appl. Meteorol. 22,517–529 (1983)]. The model has been tested with different values for the potential temperature gradient ( ) above the inversion. The model-estimated CBL heights comparefavourably with observed heights obtained from radiosonde ascents.