Assimilating spatially dense data for subsurface applications—balancing information and degrees of freedom

The degrees of freedom (DOF) in standard ensemble-based data assimilation is limited by the ensemble size. Successful assimilation of a data set with large information content (IC) therefore requires that the DOF is sufficiently large. A too small number of DOF with respect to the IC may result in ensemble collapse, or at least in unwarranted uncertainty reduction in the estimation results. In this situation, one has two options to restore a proper balance between the DOF and the IC: to increase the DOF or to decrease the IC. Spatially dense data sets typically have a large IC. Within subsurface applications, inverted time-lapse seismic data used for reservoir history matching is an example of a spatially dense data set. Such data are considered to have great potential due to their large IC, but they also contain errors that are challenging to characterize properly. The computational cost of running the forward simulations for reservoir history matching with any kind of data is large for field cases, such that a moderately large ensemble size is standard. Realization of the potential in seismic data for ensemble-based reservoir history matching is therefore not straightforward, not only because of the unknown character of the associated data errors, but also due to the imbalance between a large IC and a too small number of DOF. Distance-based localization is often applied to increase the DOF but is example specific and involves cumbersome implementation work. We consider methods to obtain a proper balance between the IC and the DOF when assimilating inverted seismic data for reservoir history matching. To decrease the IC, we consider three ways to reduce the influence of the data space; subspace pseudo inversion, data coarsening, and a novel way of performing front extraction. To increase the DOF, we consider coarse-scale simulation, which allows for an increase in the DOF by increasing the ensemble size without increasing the total computational cost. We also consider a combination of decreasing the IC and increasing the DOF by proposing a novel method consisting of a combination of data coarsening and coarse-scale simulation. The methods were compared on one small and one moderately large example with seismic bulk-velocity fields at four assimilation times as data. The size of the examples allows for calculation of a reference solution obtained with standard ensemble-based data assimilation methodology and an unrealistically large ensemble size. With the reference solution as the yardstick with which the quality of other methods are measured, we find that the novel method combining data coarsening and coarse-scale simulations gave the best results. With very restricted computational resources available, this was the only method that gave satisfactory results.     

Geochemical and H-O-Sr-Nd isotope evidence for magmatic processes and meteoric-water interactions in the basal complex of La Gomera, Canary Islands

The plutonic rocks of the Basal Complex of La Gomera, Canary Islands, Spain, were studied by means of major and trace element contents and by H-O-Sr-Nd isotope compositions in order to distinguish primary magmatic characteristics and late-stage alteration products. Deciphering the effects of alteration allowed us to determine primary, plume-related compositions that indicated D- and ¹⁸O-depletion relative to normal upper mantle, supporting the conclusions of earlier studies on the plutonic rocks of Fuerteventura and La Palma. Late-stage alteration took place during the formation of the intrusive series induced by interaction with meteoric water. Inferred isotopic compositions of the meteoric water indicate that the water infiltrated into the rock edifice at a height of about 1500 m above sea level, suggesting the existence of a subaerial volcano which was active during the intrusive activity and that it has been either distroyed or remain buried by later volcanic and landslide events.     

Geochemical characteristics of the high- and low-Ti basaltic rocks from the uplifted shoulder of the Ohře (Eger) Rift,Western Bohemia

High-Ti melanephelinite (3.8–5.9 wt% TiO₂), medium-Ti (phono)tephrite (2.7–3.1 wt% TiO₂), and low-Ti olivine melanephelinite/basanite (1.9–2.3 wt.% TiO₂) are subordinate rock types in the central European Cenozoic Volcanic Province. A contrasting melanephelinite to (phono)tephrite series occurs in the Lou?ná–Oberwiesenthal Volcanic Centre (37–28 Ma) and also as satellite volcanic bodies (26–12 Ma) together with olivine melanephelinite/basanite (17–20 Ma) on the southwestern periphery of the Kru?né hory mountains (Erzgebirge). The volcanic rocks intrude the Variscan basement of the uplifted shoulder of the Oh?e/Eger Rift in the Kru?né hory mountains of the Bohemian Massif. Low Mg# (44–59) and Cr, Ni contents and enrichment of LILE, Zr, Hf, Nb, Ta, U, Th and LREE in the high-Ti melanephelinites contrast with the composition of primitive low-Ti olivine melanephelinites/basanites displaying high Mg# (63–74) and high contents of compatible elements. The high-Ti melanephelinites reveal a wide range in initial ⁸⁷Sr/⁸⁶Sr of ca. 0.7034–0.7038 and ε_Nd of 2.4–4.9. The low-Ti melanephelinites show an overlapping range of initial ⁸⁷Sr/⁸⁶Sr of ca. 0.7035–0.7036 and ε_Nd of 4.3–5.5. The large variation in initial ⁸⁷Sr/⁸⁶Sr ratios at similar ε_Nd values in those rock types is interpreted as evidence for melting of metasomatized lithospheric mantle sources comprising K-bearing phases with radiogenic Sr. Modification of the olivine-free alkali basaltic magmas by differentiation or crustal contamination could give rise to the medium-Ti (phono) tephrites. The initial isotope ratios of all samples are consistent with HIMU-mantle sources and contributions from lithospheric mantle. The olivine-free melanephelinitic rocks often contain alkali pyroxenite–ijolite xenoliths with initial ⁸⁷Sr/⁸⁶Sr ratios of ca. 0.7036 and ε_Nd of 3.0. We interpret these xenoliths as samples of an intra-crustal alkali complex derived from similar mantle sources as those for the basaltic volcanic rocks.     

Relationship between deformation, equilibration temperatures, REE and radiogenic isotopes in mantle xenoliths (Ray Pic, Massif Central, France): an example of plume-lithosphere interaction?

Anhydrous spinel peridotite xenoliths from the Ray Pic Quaternary alkali basalt volcano (French Massif Central) show a wide range of mineralogical and geochemical compositions, reflecting significant heterogeneities in the shallow sub-continental lithospheric mantle. Variations in modal mineralogy, mineral chem istry, REE patterns and radiogenic isotope data suggest that depletion by partial melting and enrichment by cryptic metasomatism were the major mantle processes which caused the heterogeneity. The lithospheric mantle beneath Ray Pic contains two contrasting types of peridotite: (i) lherzolites with LREE-depleted compositions, high ¹⁴³Nd/¹⁴⁴Nd, low ⁸⁷Sr/⁸⁶Sr and unradiogenic Pb isotope ratios; (ii) lherzolites, harzburgites and a wehrlite with LREE-enriched patterns, low ¹⁴³Nd/¹⁴⁴Nd, high ⁸⁷Sr/⁸⁶Sr and radiogenic Pb isotope ratios. The former closely resemble depleted MORB-source mantle. The latter are related to enrichment by recent infiltration of small degree partial melts or fluids from the asthenospheric mantle, possibly related to the “low velocity component” observed by Hoernle et al. (1995) in European Neogene alkaline magmas. Thus, the Ray Pic peridotite xenoliths represent interaction between asthenospheric mantle-derived melts/fluids and depleted lithospheric mantle. This is probably linked to the upwelling mantle plume imaged beneath the Massif Central (Granet et al. 1995). A relationship between textural deformation, equilibration temperature and geochemistry of the xenoliths suggests that the hotter (> 900 °C) undeformed regions are LREE-enriched and tend to have more enriched isotope ratios, whereas the cooler (< 900 °C) regions have undergone more deformation and are more depleted both in LREE and in isotope compositions. Received: 27 July 1996 / Accepted: 25 November 1996     

Entropy analysis of in situ particle size spectra

Entropy analysis has been used to classify in situ particle (floc) size spectra of suspended particles into groups based on similar distribution characteristics. Results revealed that the in situ spectra sorted into groups that reflected different forcing conditions (e.g. variations in turbulence). The different forcing conditions were not necessarily reflected in other commonly used distribution measures such as median floc diameter. This suggests that entropy analysis may be an effective approach for investigating the effect of changes in forcing conditions on floc size. It is hypothesized that it may be possible to derive the average shape of floc size spectra from measurement of the forcing conditions alone and subsequently derive parameters such as floc fraction, floc density, floc settling velocity and the optical properties of the water column from the average spectra.     

Latest precambrian (Cadomian) zircon ages,Nd isotopic systematics and P-T evolution of granitoid orthogneisses of the Erzgebirge,Saxony and Czech Republic

Single zircons from two orthogneiss complexes, the Grey Gneiss and Red Gneiss, the lowermost tectonic units in the Erzgebirge, were dated. The grey Freiberg Gneiss is of igneous origin and has a ²⁰⁷Pb/²⁰⁶Pb emplacement age of 550±7 Ma. A quartz monzonite from Lauenstein contains idiomorphic zircons with a mean ²⁰⁷Pb/²⁰⁶Pb age of 555±7 Ma as well as xenocrysts ranging in age between 850 and 1910 Ma. Red gneisses from the central Erzgebirge contain complex zircon populations, including numerous xenocrysts up to 2464 Ma in age. The youngest, idiomorphic, zircons in all samples yielded uniform ²⁰⁷Pb/²⁰⁶Pb ages between 550±9 and 554±10 Ma. Nd isotopic data support the interpretation of crustal anatexis for the origin of both units. _Nd(t) values for the grey gneisses are –7.5 and –6.0 respectively, (mean crustal residence ages of 1.7–1.8 Ga). The red gneisses have a wider range in _Nd(t) values from –7.7 to –2.8 (T _DM ages of 1.4–1.8 Ga). The zircon ages document a distinct late Proterozoic phase of granitoid magmatism, similar in age to granitoids in the Lusatian block farther north-east. However, Palaeozoic deformation as well as medium pressure metamorphism ( 8 kbar/600–650° C) are identical in both gneiss units and distinguish these rocks from the Lusatian granitoids. The grey and red gneisses were overthrust by units with abundant high-pressure relicts and a contrasting P-T evolution. Zircon xenocryst and Nd model ages in the range 1000–1700 Ma are similar to those in granitoid rocks of Lusatia and the West-Sudetes, and document a pre-Cadomian basement in parts of east-central Europe that, chronologically, has similarities with the Sveconorwegian domain in the Baltic Shield.     

Post-collisional Variscan lamprophyres (Black Forest, Germany): Ar/Ar phlogopite dating, Nd, Pb, Sr isotope, and trace element characteristics

Subsequent to high-pressure and temperature metamorphism of the axial zone of the Variscan foldbelt in central Europe at ca. 340 to 330 Ma, formation of lamprophyre dikes during transtensional tectonics may be viewed as the beginning of the post-collisional stage of the orogeny. We report the results of ⁴⁰Ar/³⁹Ar mica dating, major and trace element data, and isotope compositions for lamprophyre and rhyodacite samples from the southern Black Forest. The chemical compositions of these rocks shed light on the upper mantle and crust at the end of the Variscan orogeny. ⁴⁰Ar/³⁹Ar plateau ages for four phlogopite–biotite separates from lamprophyres indicate emplacement at 332 to 314 Ma. This event coincides with melting of the crust as indicated by ⁴⁰Ar/³⁹Ar biotite plateau ages of ca. 332 Ma for rhyodacite dikes which are probably related to coeval undeformed granites. Incompatible trace element patterns of the lamprophyre samples reveal the characteristics of evolved continental crust and are interpreted as evidence for melting of sediment in a subduction-modified mantle. Nd, Sr, and Pb isotope compositions indicate an enriched mantle source with _Nd-values of −1.5 to −6.8 which is similar to Variscan crust. Significant contamination of the lamprophyric melts by Variscan crust can be ruled out as mantle-derived phlogopite phenocrysts have similar _Nd-values as in the whole-rock samples. We propose that the isotope compositions and incompatible trace element characteristics of the lamprophyres were predominantly inherited from melted sediment. The isotope compositions of Variscan lamprophyres from western Europe suggest that enriched upper mantle was only partly delaminated when ascending hot mantle triggered melting of the lower crust, as has been invoked for the origin of post-collisional granites. The isotope compositions of Tertiary basalts and mantle xenoliths indicate a depleted upper mantle under western Europe, implying that the enriched Variscan material was efficiently removed and mixed into the convecting mantle.     

Evolution of the Pan-African Wadi Haimur metamorphic sole, Eastern Desert, Egypt

By comparison with the general features of metamorphic soles (e.g. vertical and lateral extension, metamorphic grade and diagnostic mineral parageneses, deformation and dominant rock types), it is inferred that the amphibolites, metagabbros and hornblendites of the Wadi Um Ghalaga–Wadi Haimur area in the southern part of the Eastern Desert of Egypt represent the metamorphic sole of the Wadi Haimur ophiolite belt. The overlying ultramafic rocks represent overthrusted mantle peridotite. Mineral compositions and thermobarometric studies indicate that the rocks of the metamorphic sole record metamorphic conditions typical of such an environment. The highest P – T conditions ( c . 700 °C and 6.5–8.5 kbar) are preserved in clinopyroxene amphibolites and garnet amphibolites from the top of the metamorphic sole, which is exposed in the southern part of the study area. The massive amphibolites and metagabbros further north (Wadi Haimur) represent the basal parts of the sole and show the lowest P – T  conditions (450–620 °C and 4.7–7.8 kbar). The sole is the product of dynamothermal metamorphism associated with the tectonic displacement of ultramafic rocks. Heat was derived mainly from the hot overlying mantle peridotites, and an inverted P – T  gradient was caused by dynamic shearing during ophiolite emplacement. Sm/Nd dating of whole-rock–metamorphic mineral pairs yields similar ages of c . 630 Ma for clinopyroxene and hornblende, which is interpreted as a lower age limit for ophiolite formation and an upper age limit for metamorphism. A younger Sm/Nd age for a garnet-bearing rock ( c . 590 Ma) is interpreted as reflecting a meaningful cooling age close to the metamorphic peak. Hornblende K/Ar ages in the range 570–550 Ma may reflect thermal events during late orogenic granite magmatism.     

Zircon ages,geochemistry, and Nd isotopic systematics of pre-Variscan orthogneisses from the Erzgebirge,Saxony (Germany), and geodynamic interpretation

High grade granitoid orthogneisses occur in several metamorphic units of the Erzgebirge in the Saxothuringian Zone of the Variscan Belt. The determination of protolith ages and the geochemical characterization of these rocks permit a reconstruction of the Neoproterozoic to early Palaeozoic magmatic and geodynamic history of the Erzgebirge. Single zircon Pb-Pb evaporation and SHRIMP ages combined with major and trace element data and Sm-Nd isotope systematics indicate at least two discrete magmatic events concealed in the so-called red gneisses, one at ~550 Ma in rocks of the medium pressure—medium temperature (MP-MT) unit and the other at ~500–480 Ma in rocks of the high pressure units. The transition zones comprise both Neoproterozoic granitoids and early Palaeozoic metarhyolites. The granitoid gneisses represent Neoproterozoic calc-alkaline granitoids with REE patterns similar to those produced in Andean-type continental margins. The early Palaeozoic muscovite gneisses are geochemically distinct from the older granitoids and may be derived from melts generated in a back-arc setting. Initial _Nd values in all samples overlap and range from –4.1 to –9.2, corresponding to crustal sources with average residence times of 1.5 to 1.9 Ga. Zircon xenocryst ages as old as 2992 Ma provide evidence for Grenvillian, Svecofennian-Birimian-Aazonian and older age components and suggest an association of the Erzgebirge with Avalonia.B. Mingram and A. Kröner have shared senior authorship