Geophys. J. Int. (2010) 181, 154–172
doi: 10.1111/j.1365-246X.2010.04521.x
GJI Geodynamics and tectonics
Combining geophysical data sets to study the dynamics of shallow evaporites in urban environments: application to Hamburg, Germany Torsten Dahm,1 Daniela K¨uhn,2 Matthias Ohrnberger,3 Jens Kr¨oger,4 Helga Wiederhold,5 Claus-Dieter Reuther,6 Ali Dehghani7 and Frank Scherbaum3 1 Institut
f¨ur Geophysik, University of Hamburg, Hamburg, Germany. E-mail:
[email protected] PO Box 53 2027 Kjeller, Norway 3 Institut f¨ ur Erd- und Umweltwissenschaften, Universit¨at Potsdam, Potsdam, Germany 4 Geologisches Landesamt Hamburg, Hamburg, Germany 5 Leibniz-Institut f¨ ur Angewandte Geophysik, Hannover, Germany 6 Inst. f¨ ur Geol. & Pal¨aont., Universit¨at Hamburg, Hamburg, Germany 7 Institut f¨ ur Geophysik, Universit¨at Hamburg, Hamburg, Germany 2 NORSAR
Accepted 2010 January 13. Received 2010 January 13; in original form 2008 November 28
SUMMARY Shallowly situated evaporites in built-up areas are of relevance for urban and cultural development and hydrological regulation. The hazard of sinkholes, subrosion depressions and gypsum karst is often difficult to evaluate and may quickly change with anthropogenic influence. The geophysical exploration of evaporites in metropolitan areas is often not feasible with active industrial techniques. We collect and combine different passive geophysical data as microgravity, ambient vibrations, deformation and hydrological information to study the roof morphology of shallow evaporites beneath Hamburg, Northern Germany. The application of a novel gravity inversion technique leads to a 3-D depth model of the salt diapir under study. We compare the gravity-based depth model to pseudo-depths from H/V measurements and depth estimates from small-scale seismological array data. While the general range and trend of the diapir roof is consistent, a few anomalous regions are identified where H/V pseudo-depths indicate shallower structures not observed in gravity or array data. These are interpreted by shallow residual caprock floaters and zones of increased porosity. The shallow salt structure clearly correlates with a relative subsidence in the order of 2 mm yr −1 . The combined interpretation of roof morphology, yearly subsidence rates, chemical analyses of groundwater and of hydraulic head in aquifers indicates that the salt diapir beneath Hamburg is subject to significant ongoing dissolution that may possibly affect subrosion depressions, sinkhole distribution and land usage. The combined analysis of passive geophysical data may be exemplary for the study of shallow evaporites beneath other urban areas. Key words: Gravity anomalies and Earth structure; Hydrogeophysics; Surface waves and free oscillations; Sedimentary basin processes; Diapir and diapirism.
1 I N T RO D U C T I O N Accumulated sediments in the North European Basin are up to 10–12 km thick and are embedding the sequences of Zechstein evaporites, a so-called ‘salt giant’ of several hundred metres thickness (see Jaritz 1987; Warren 2006; Scheck-Wenderoth et al. 2008, for a description of Zechstein cycles and salt stratigraphy in NGermany). The Zechstein formation in North Germany is situated at a depth of about 4.5 km. From Triassic times on vertical flow of salt (halokinese) was triggered during periods of changing tectonic stress and led to the formation of several salt walls and diapirs, often along the strike direction of deeper faults (Geluk et al. 2007;
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Scheck-Wenderoth et al. 2008). Today, a considerable portion of the North European Basin and several built-up urban regions are influenced or affected by shallow evaporites. The larger area of Hamburg, N-Germany, is underlayn by about nine salt diapirs of different shape and depth (Fig. 1), which belong to the northern salt domain characterized by parallel elongated salt structures trending in NNE direction (Jaritz 1987; Warren 2006). The shallowest one, the Othmarschen Langelfelde diapir (OLD), has a length of about 20 km and caprocks a few metres beneath the surface. The OLD is used here for an exemplary exploration of shallowly situated evaporites under urban conditions. Conducting C 2010 The Authors C 2010 RAS Journal compilation
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Figure 1. Overview of Hamburg area and the existing salt diapirs and pillows (see Reinhold et al. 2008). The dashed brown line bounding the Othmarschen-Langenfeld diapir (OLD) gives a gravity-based outline of the diapir from Grube (1973). Small red-filled polygons on the OLD show known sinkhole structures. Dashed grey lines declare Tertiary faults at about 10 km depth. The solid grey contour lines denote the 10 m elevation indicating the Alster and Pinnau depression. The black star shows the basis reference point at the Institute of Geophysics to which all gravity data have been related.
active, large-scale geophysical experiments is often difficult if not impossible in heavily build-up areas. We investigate the feasibility of different passive geophysical methods to map the roof morphology and the dynamics of shallow evaporites. This is a first step to better quantify salt-related hazard. The study comprises own data as densely distributed microgravity, shallow reflection seismic and ambient vibration measurements. Additional geophysical and geological data such as borehole stratigraphies, surface deformation, salt concentration, temperature and hydraulic head in aquifers have been compiled and compared to our own data. Based on gravity and structural data a 3-D gravity model was inverted applying a novel inversion code developed in-house. This 3-D model is discussed to understand the structure and dynamics of the roof morphology of the OLD. We integrated previous knowledge and studies on the OLD. Whether and to which extend the OLD salt wall is controlled by local tectonic, deeply rooted faults is controversy and poorly studied (see also Ehlers 1995). One possible lineament at the southernmost border of the OLD is the Elbe line, which belongs to a deeply rooted system and has been interpreted as the termination point of the thinning Baltic crust (East European craton) against the North German Basin (e.g. Bayer et al. 2002). Basement faults beneath the OLD striking N–NE are indicated from the geometry of the salt wall but have not yet been verified by independent methods like reflection seismics. Two flank depressions east (Pinnau Niederung) and west of the OLD (Außenalster) have developed and indicate the far reaching influence of the OLD on surface features (Fig. 1). Only few previous studies dealed with the roof morphology of the OLD. Gripp (1920) compiled information on caprocks and residual gypsum at the OLD and other diapirs in North Germany and discussed rising or deflating salt levels, defined as the depth where solution brine and fresh water are in equilibrium. He gives indications that the caprock head of the OLD may currently deflate due to subrosion and salt dissolution. C
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Niedermayer (1962) discussed the thickness of caprock layers by means of borehole stratigraphies in the southern part of the OLD (see also Koch 1955). He estimated the post Upper Miocene uplift in the range of 100–120 m, but is at odd with the estimate of 250 m by Gripp (1920) including the effect of simultaneous subrosion subsidence. Niedermayer (1962) and Grube (1973) compiled and discussed the knowledge on depth and thickness of residual gypsum layers and positions of subsidence depressions, sinkholes or deep void space found in borehole profiles. Plaumann (1979) was the first to derive a gravity based geometrical model of the diapir based on own measurements and the integration of older data. Major features of the gravity anomaly (gravity disturbance) were described for the first time and led to the recognition that the salt diapir is NS oriented and not EW as previously thought. He identified two parallel NS striking gravity highs at the border regions of the OLD and excluded, mainly based on borehole information, that these are solely caused by uplifted Buntsandstein slivers. Instead they indicate a bowl-shaped roof structure of the diapir. Prexl (1997) made an effort to use high resolution seismic data to study the morphology and structure of the roof region of the OLD. The bowl-shaped structure was confirmed. Sedimentation and layer thickness above the salt roof indicates that the diapir had a convex shape during the time of formation of the Hamburger Ton (lithographic clay layer about 20 Mio years old), but that a subrosion-related depression in the central part formed later-on (Prexl 1997). Increased subrosion rates in the central part of the OLD were indicated as well by comparison with other diapirs in North Germany (e.g. Gorleben). These are explained by the availability of highly soluble components in the central part of the diapir, while less soluble components are present at the rims such that rim highs develop together with thick residual caprock layers. Prexl (1997) also recognized the difference in the geometrical shape of the eastern and western border and documented that the OLD dips southward beneath the Elbe river. Seismic sections further indicate a smooth flexure and deformation of sediments at the western border and faulting, possibly subrosion-induced and partly with offsets of 100 m, at the eastern border of the OLD. Although difficult to resolve, Prexl (1997) suggested ongoing formation of a central graben in the eastern part of the OLD. One question of controversy is whether the deeper body of the OLD is still rising. An early work on stratigraphy, sedimentation and hydrology of the OLD suggests that diapir uplift was continuous from Miocene to present (Gripp 1920), however, this view is not established or commonly accepted until today. The OLD is not only of interest from a tectonic, structural or hydrological point of view, but also because of its possible hazard for the metropolitan area of Hamburg. Subrosion of the diapir roof leads to formation of thick residual caprock, a mixture of less soluble components like gypsum, anhydrite, clay and other residual sediments. Shallow blocks or layers of gypsum possibly experience karst formation that poses a hazard for subsidence, sinkhole formation and small collapse earthquakes (see Warren 2006, for a general review). Specific regions above the OLD have repeatedly experienced sinkhole events or collapse earthquakes in the past, as will be discussed below (see also Dahm et al. 2010). Damage of houses related to subsidence and sinkholes has been documented (e.g. Buurman 2010), but may most often remain unknown or undocumented since insurances for private owners are usually not existing. The distribution and extend of gypsum karst and its related hazard is not yet understood, and it is controversial whether the karst system has formed during palaeoperiods (e.g. during Upper Miocene
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Figure 2. Left-hand panel: locations of gravity (circles) and elevation measurements (crosses). Red circles declare gravity points measured by the Geophysical Institute of Hamburg University, and yellow circles represent older measurements by the GGA-Insitute Hannover. The black line indicates the outline of the OLD diapir. Right-hand panel: locations of ambient vibration seismic measurements by the Geophysical Institutes of Hamburg and Potsdam Universities (red circles) and 2D seismic reflection lines across the OLD (grey lines, GGA-Institute Hannover). Borehole data have been provided by the Geologisches Landesamt Hamburg and are indicated by yellow triangles.
as postulated by Gripp 1920) or is an ongoing process. Cases of man-induced sinkholes or collapse earthquakes are not given for Hamburg to our knowledge, but it is well known from other urban areas above gypsum karst that human-induced or human-controlled sinkhole events may pose a significant additional hazard (e.g. Benito et al. 1995; Soriano & Simon 1995, 2002; Waltham et al. 2005; Johnson 2008). 2 D ATA B A S E A N D P R O C E S S I N G Within the course of this project a large amount of new data has been collected and compiled, consisting of microgravity, ambient vibration single station and array measurements, and trends in surface subsidence from repeated levelling (Fig. 2). Existing seismic reflection lines, older gravity data, borehole stratigraphies and hydrological borehole data have been integrated in the study, as well as ground based (levelling) and novel satellite-based data on subsidence rates in Hamburg [processed and kindly provided by Sch¨affer, Bundesanstalt f¨ur Geowissenschaften und Rohstoffe (BGR), Hannover, 2007]. The 3-D salt model resulting from our study is mainly based on the inversion of gravity data. 2.1 Elevation data Fig. 2 gives an overview over measured gravity points. In most cases the elevation data and coordinates have been provided by ‘Landesbetrieb Geoinformation und Vermessung in Hamburg’. Usually, two or three measuring times (epochs) have been performed between 1974 and 1994 at the individual levelling point and analysed to calculate uplift trends. These levelling-based uplift rates have been used to verify and calibrate satellite-derived data (PS-InSAR) provided by BGR (see e.g. Sch¨affer 2009; K¨uhn et al. 2009a; K¨uhn et al. 2009b). The measuring period is from 1993 April 25 to 2005 May 11 and comprises 46 ERS-1/2 and 6 ENVISAT-ASAR satellite data sets. 52 sequences have been processed (software PSI-GENESIS, PSIC4)
and 57 282 persistent scatterers (PS) have been identified with a PS density of 53 PS km−2 . The standard deviation of the annual motion rate is 0.9 mm yr −1 , while 80 per cent of the scatterers showed a variability within the interval ±1.5 mm yr −1 . Only scatterer points with a coherence larger than 0.8 and a standard deviation below 1 mm yr −1 have been considered for our gridding and comparison. When necessary, additional differential GPS height measurements (LEICA GPS 1200 system of the University Potsdam, infield SAPOS/ASCOS correction services by mobile phone) were performed to process gravity points in between existing levelling points or outside the area of Hamburg. WGS84 ellipsoid heights were transformed to heights on the Bessel ellipsoid. The accuracy of the GPS-based height measurements was checked at reference points and was sufficient to integrate these gravity data in the data set (±5 cm and thus useful to resolve Bouguer anomalies larger than about ±0.01 mGal, which is about 1 per cent of the observed anomaly).
2.2 Reflection seismic lines Fig. 2 shows in addition the positions of seismic reflection lines that were acquired in 1991 and 1992 by the Nieders¨achsisches Landesamt f¨ur Bodenforschung (NLfB, Hannover). With a P-wave Vibroseis source, sweep frequencies of 24–96 Hz and 14 s length as well as a seismograph (DFSV) with 48 channels and 10 m geophonegroup spacing, a penetration depth of up to 1000 m was reached. The common-midpoint traces have a coverage of at least 12-fold and the trace spacing is 5 m. The data processing includes trace editing, refraction statics, filtering and deconvolution and velocity analysis. After performing a common-midpoint stacking, a time migration was applied. The nine seismic sections of about 30-km-length reveal reflections from the top of the salt diapir as well as from Tertiary and Quaternary cover (Fig. 3). The bulge of the salt at the eastern rim is proven (Fig. 3 top panel) as well as the different geometrical shape of the western rim (Fig. 3 bottom panel). The central line on top of the OLD shows well the descent of top salt to the North (Fig. 3 C
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Figure 3. Seismic sections of three profiles (see Fig. 2 right-hand panel; time migrated and depth converted; datum level is mean sea level; intersection with other profiles is marked). Top panel: northernmost W–E profile (No. P6) showing the eastern shoulder of the OLD. Middle panel: N–S profile (No. P4) on top of the OLD. Bottom panel: southernmost W–E profile (No. P5) showing the western shoulder of the OLD.
middle panel). This profile will be compared below to gravity and H/V data.
2.3 Ambient seismic vibration measurements: H/V single station method Between 2006 January and 2007 August, almost 1000 locations across the OLD have been used for ambient seismic vibration H/V measurements, thereof approximately 650 for array and 350 for single station measurements. As instrument, Lennartz 5 s threecomponent sensors have been used. Single station measurement sites are visible in Fig. 2. A grid with 1 km spacing underlies all positions. At points of interest, mainly in the South where the salt dome is situated near to the surface, the grid has been refined to distances of only a few tens of metres. Further, the spacing is C
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minimized at array measurement sites since individual array stations can be treated as single stations. The single station measurements were analysed by the H/V method. This method, also known as Nakamura’s technique, was first introduced by Nogoshi & Igarashi (1971) and developed further by Nakamura (1989). The horizontal-to-vertical (H/V) spectral ratio of microtremors, observed at a site using a three-component sensor, may yield information on impedance contrasts in the subsurface. The time traces having a typical length of 30–45 min are subdivided in time windows of 60 s. In the next step, power spectra are calculated for the components and ratios between east and vertical component as well as north and vertical component are computed. Finally, both ratios are averaged to the so-called H/V spectrum. The processing (see K¨uhn et al. 2010, for more details) has been performed using the GEOPSY software (see Wathelet 2005, and http://www.geopsy.org/).
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2.5 Gravity data and gravity disturbance
Figure 4. H/V spectra for two ambient vibration measurements; top panel: outside salt diapir, bottom panel: above salt diapir in the southern region. The thin grey lines correspond to spectra gained from single time-windows, while the thick black lines (straight/dashed) give mean and standard deviations. The grey vertical bars indicate automatically estimated standard deviations as defined from the peak frequencies of each time-window.
If a strong impedance contrast exists in the subsurface, a spectral peak is found in the H/V spectrum (e.g. Faeh et al. 2001, for theoretical models). The peak frequency depends on the depth of the impedance contrast. A peak at low frequencies is caused by a structure situated at greater depths than a peak at higher frequencies. In Fig. 4, the H/V spectra from two measurement sites are shown, one is located outside the OLD and the other on top of the southern part of the OLD. Peak frequencies estimated with this approach have been identified only above the OLD and are limited between 0.14 and 2.7 Hz. The minimum of 0.14 Hz is found in the north. High peak frequencies have been observed in the southern part of the OLD and correspond to salt dome elevations. Peak frequencies may be transformed to the depth of a strong impedance contrast by means of empirical relations. Below, we derive an empirical relation for the OLD region and discuss the results in terms of caprock thickness in the southern part of the OLD, since the impedance contrast there is most shallow and resolution was highest. A more detailed discussion of the H/V spectra and their interpretation, along with a comparison to small-scale array measurements and the inversion of phase velocity curves, is given in K¨uhn et al. (2010).
2.4 Borehole-derived stratigraphic and hydrological data Fig. 2 shows the positions of boreholes which data could be used for our study. Borehole measurements in Hamburg are collected and archived since more than 100 yr at the Geologisches Landesamt Hamburg. Most borehole data have been retrieved during exploitation of fresh water or the analysis and monitoring of water quality. These types of borehole data comprise water chemistry and temperature data. Additionally, borehole stratigraphic data are typically collected as obligatory presite survey if larger buildings or structures are planned. Some other boreholes have been drilled to investigate the sinkhole potential, especially in the South region of the OLD and in relation to the construction of the Elbe highway
Altogether, about 670 data points distributed over the area have been collected with a LaCoste Romberg gravimeter (Type G, No 260). A standard Bouguer reduction was applied with a reference base point outside the anomaly1 (star in Fig. 2). Topographic correction is estimated in the range of only 0.001 mGal and therefore neglected. The influence of high buildings or building structures in different areas of the city was not measurable or of minor influence (e.g. Milson 1996) and was therefore not corrected. In addition, older gravity data along four profiles (see Plaumann 1979) have been provided by the GGA-Insitute Hannover and were incorporated in our data set. An important step in the data processing was the removal of a regional bi-linear trend from the corrected values. The regional gravity decreases in SW direction and is explained by a deeply rooted, different basin filling south of Hamburg (Hamburger Loch), possibly related to the crustal thinning of the East European craton towards the North German Basin (e.g. Plaumann 1979; Hoffmann et al. 1996). This long wavelength trend overprints the anomaly from the shallow portion of the OLD and was removed to find the final Bouguer anomaly gB . Fig. 5 shows the detrended Bouguer anomaly, which is interpreted to be purely caused by the OLD structure. This interpretation is justified by the fact that the anomaly is geometrically constrained to the previously known region of the OLD. The anomaly extends in EW direction about 6 km and in NS direction about 14–16 km. The strike of the axisymmetric structure is about 10◦ . The anomaly covers a range of 4–5 mGal and decreases in general northward, where the roof of the OLD is diving to deeper levels. The anomaly has a short wavelength high above the salt structure with its maximum close to the northward border of the Elbe river (Fig. 5). This high is riding on a longer wavelength low (depression) caused by the deeper structure of the diapir. The long wavelength low is better identified in regional gravity data (e.g. Plaumann 1979). Both anomalies are approximately axisymmetric to the NS striking trend of the OLD. The general pattern can be explained by a mass excess at shallow levels located on top of a deeply rooted mass deficit. At depths larger than about 100 m, salt (halite) has lost its porosity and has a density of 2200 kg m −3 , which remains nearly constant down to a depth of about 6–8 km (Warren 2006). In contrary, the non-evaporitic sideburden experiences compression at depth and thus an increasing density. The density-depth function of unconsolidated sediments depends strongly on porosity and age and is often approximated by a linear, exponential (e.g. Granser 1987) or polynomial laws (e.g. Hermes 1986; Rao et al. 1993; Garcia-Abdeslem 2005; Chakravarthi & Sundararajan 2006). At shallow levels the sedimental overburden has a density of about 1900 kg m−3 . The crossover density (level of neutral buoyancy, LNB) is typically between 1.2 and 1.5 km (e.g. Warren 2006, and own 1 The term Bouguer anomaly is used here to describe the gravity disturbance, in a relative sense, which is based on ellipsoidal heights (see e.g. Fairhead & Green 2003). C
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Figure 5. Detrended final gravity disturbance (termed Bouguer anomaly in the following; gridded with GMT tool near-neighbour N4). Regions with sparse data coverage have been cut. The anomalies along three given profiles have been projected from an orthogonal distance to the profiles of ±2 km and low pass filtered (median filter with a length of 0.7 km and not considering gaps larger than 0.5 km). They represent average values in the southern, middle and northern part of the salt structure. The dashed line indicates a NS profile along which density model and H/V data have been calibrated.
estimates). Thus, the negative density difference at larger depths generates a negative, long-wavelength anomaly, while the positive density difference of the shallow salt structure generates the positive, central short-wavelength anomaly (see also Lerche & O’Brian 1987). Our interpretation and inversion focuses on the shallow roof region and morphology of the OLD. A characteristic feature of the positive anomaly is the nearly continuous bulge of high values along the expected rim of the OLD. This feature was already identified by early gravity studies (Plaumann 1979) and can be explained by a bowl-like morphology of the diapir roof. 3 C A L I B R AT I O N T E S T S Two independent data sets are used to estimate the morphology of the OLD: (1) gravity anomaly and (2) H/V peak frequencies. Both ‘inversions’ need to be calibrated. 3.1 Gravity method and density model calibration We used a 3-D gravity inversion code developed in-house for interpreting the Bouguer anomaly. The differential density ρ(z) is parametrized in rectangular prisms with a fixed bottom level at depth z = z 2 = const and variable upper level z 1 . The differential density within each prism follows a surface-based, depth-dependent cubic law, ρ(z) = c1 + c2 z + c3 z 2 + c4 z 3 , where ck , k = 1, 4, are four coefficients. A cubic polynomial form ρ(z) is flexible and well suited to fit a non-linear density-depth function. The differential density defines the difference between the C
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density of the non-evaporitic overburden, ρ s (z), and the salt density ρ h . Outside the diapir, ρ(z) ≈ 0, and within the diapir ρ(z) = ρ h − ρ s (z) = 0. This differential formulation leads to a finite model space so that only few measurements outside the diapir are sufficient to study the shallow roof morphology. The inversion may be applied in two modes: (1) by keeping the geometry fixed and solving for the coefficients ck , or (2) by keeping ck constant and solving in a least squares sense for the upper level z 1 of the prisms. The density function problem (1) is linear and solved in a least squares sense, with the option to impose additional inequality constraints (e.g. d ρ/d z ≤ 0, or d 2 ρ/ d z 2 ≥ 0). The 3-D morphology problem (2) is non-linear and solved with a linearized, iterative gradient method (Gauß method). More details and basic equations of method and procedure are described in Appendix A. A novel aspect of the method is that the gravimetric singleinterface problem, that is usually taken to study regional basin morphology, is adopted here for differential density models and the study of the morphology of the shallow salt structure. The singleinterface problem is unique if an appropriate, depth-dependent density model can be assumed for the non-evaporitic overburden, and the accuracy will depend on the fidelity of this density model. To estimate a density function and to demonstrate the method we chose a NS profile along which the interface depth was constrained by borehole and seismic data between approximately 70 and 400 m (Figs 6 and 2). Since the regional trend removed during preprocessing likewise had a NS direction, our derived density model may compensate a possible bias from the preprocessing. We projected Bouguer anomaly measurements, salt depths derived from migrated seismic lines and borehole lithological constraints from ±500 m to the profile. The numerous Bouguer values
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Figure 6. Density and H/V pseudo-depth calibration along NS profile 4. The left-hand figure shows the fit to Bouguer anomaly data (top left-hand panel) and the associated density model ρ(z) (bottom left-hand panel, coloured). Black filled circles indicate depth constraints for the salt top from seismic sections. Grey squares indicate pseudo-depths and their errors from H/V peak frequencies when projected to the profile (not used for density-inversion). The crossover density (dashed) is at 1.5 km depth. The bottom right-hand figure shows the the constant salt density of ρ h = 2200 kg m−3 and the estimated sediment density ρ s (z). The solid line represents the cubic density model, and the dashed line the squared density model. Note that the inverted density is ρ(z) = ρ h − ρ s (z). The figure in the top right-hand panel shows the borehole lithology data used for H/V pseudo-depth calibration, that is, H/V peaks at positions of boreholes with indications for salt/gyspum depth; only peak frequencies f 0 of highest (filled circles) and good (open circles) quality have been used.
were further smoothed to obtain an averaged, single-valued 2-D anomaly along this profile. The cells in this 2-D application were defined by prism of 100 m length along x and within −6 km ≤ y ≤ + 6 km and z 1 ≤ z ≤ 4 km. The density inversion (mode 1) is performed under the constraint that the non-evaporitic density increases with depth (ρ(z) decreases with z, see Lawson & Hanson 1974, for implementation). The increasing density assumption is useful because our data set does not impose sufficient information at larger depth below 1.5 km, and polynomial functions may easily produce unphysical oscillating solutions. On the other hand, within the first kilometre the polynomial law is more flexible than a quadratic, linear or exponential law, and thus potentially leads to a better approximation of the density column. The 2-D inversion had 158 degrees of freedom and five model parameters (a static shift of Bouguer values is al−13 m2 s−4 lowed). The average squared residual was χ (5) ν = 5.9 × 10 (Fig. 6). Using three (linear) or four parameters (squared law), the −13 m2 s−4 . average squared residuals increased to χ (4) ν = 15.4 × 10 An F-test for one additional model parameter (e.g. Bevington 1969) indicates that the cubic law is significant and can be resolved. Applying a quadratic model without inequality constraints lead to a similar fit but decreasing densities below 2 km depth. The final density model is
at the surface (e.g. Warren 2006). ρ s (z) continuously increases to a value of about 2300 kg m−3 at 4 km depth. We find a density crossover depth (level of neutral buoyancy) between sediments and salt at about 1.5 km, which is only slightly deeper than assumed as typical by Warren (2006). Although densities below 1.5 km should be taken with care, since no control points were available at this depth, it is notable that the derived density column is very similar to other findings in North Germany (e.g. Hermes 1986) or to other sedimentary basins (e.g. Garcia-Abdeslem 2005). If we apply the density inversion to the complete 3-D data set, we derive very similar coefficients as
z z2 ρ(z) = 294 − 0.290 + 0.755 × 10−4 2 −3 (kg m ) (m) (m ) 3 z − 6.924 × 10−9 3 , 0 ≤ z ≤ 4 km. (m )
H/V peaks are directly transformed to pseudo-depth by means of an empirical relation, which was estimated by linear regression at points where borehole stratigraphies or shear wave velocity profiles from dispersion curve inversion (K¨uhn et al. 2010) were available. Most of the borehole data stem from the southern part of the OLD, where the interface depth between tertiary sediments and compact gypsum could be extracted. The regression of frequency-depth data
ρ(z) and ρ s (z) are plotted in Fig. 6. At the surface, ρ s = 1906 kg m−3 , a value typically observed for unconsolidated sediments
2 z ρ(z) −4 z = 296 − 0.290 + 0.754 × 10 (kg m3 ) (m) (m2 ) 3 z − 6.908 × 10−9 3 , 0 ≤ z ≤ 4 km (m )
with the same crossover depth. We therefore conclude that the density inversion with cubic law is very stable if inequality constraints as dρ s /dz > 0 or d2 ρ s /dz 2 < 0 are imposed.
3.2 H/V pseudo-depth calibration
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Figure 7. Grid of the initial (left-hand panel) and final depth model (right-hand panel). Each rectangular prism is represented by a filled circle where colours indicate the depth of the prism surface. Depths beyond the given scale have been clipped for plotting purposes (affects only deep levels). Two types of depth constraints have been considered: (1) high weight constraints are indicated by small black-filled circles and are implemented to avoid oscillations of the unconstrained deep morphology outside the anomaly. Triangles mark positions where either seismic profiles or boreholes data indicate the depth of the salt body.
was performed in logarithmic space and led to an empirical relation to interpolate pseudo-depths z H/V from normalized peak frequencies f 0 /(1 Hz) as f0 b with a = 125.29 ± 13.91 m z H/V = a 1 Hz and b = −1.07 ± 0.15. This relation has been derived for the sedimentary overburden of the OLD in Hamburg and peak frequencies between 0.15 and 2.72 Hz and covers the depth range between 50 and 580 m (K¨uhn et al. 2010, and Fig. 6). Fig. 6 compares the estimated pseudo-depths from H/V peaks to depth constraints of seismic lines and the gravity 2-D calibration.
4 3 - D S A LT M O D E L F R O M M O D E 2 INVERSION The 3-D inversion of the whole Bouguer anomaly (mode 2) is performed with fixed coefficients ck . Only the top surface of the salt, z 1 (x, y), is inverted for. A fixed density model is justified since the geological situation does not vary substantially over the region under study. The deeper volume of the diapir cannot be resolved so that only z 1 controls the anomaly. If the four parameters ck number of unknowns of the 3-D inversion problem is equal to the number of prisms. We parametrize the model space by quadratic prisms of equal size. The side length is 500 m. 17 and 25 grid cells are chosen along x and y direction rotated to a line of symmetry of the anomaly such that altogether 425 unknown prism depths are present. A smoothness constraint as well as ridge regression has been applied in order to stabilize the damped generalized inversion. The generalized matrix has full rank and the problem is formally overdetermined (973 independent equations and 426 unknowns), although some individ C
2010 The Authors, GJI, 181, 154–172 C 2010 RAS Journal compilation
ual prisms may by physically underdetermined. Damping constants have been estimated by trial and error so that both a sufficient smoothness and a convergence has been observed. Fig. 7 shows the starting model (initial model) and the final minimum misfit salt depth model. The model is ‘non-gridded’ to show the parametrization and the geometrical resolution obtained. We start with an inclined box-like salt model and iterate with larger damping in the beginning, until a more realistic starting model has been retrieved. The starting depth of weakly resolved prisms was slightly modified by trial and error. Then, damping and smoothness constraints were slightly relaxed and the model was ‘fine-tuned’ until a minimum misfit model was retrieved and step-lengths of model parameter have been small or zero. The root mean square residuals are 2.1 × 10−14 m2 s−4 . The mean error of predicted to measured Bouguer anomaly is 0.015 mGal. The convergence close to the minimum was relatively slow, and we could observe a set of end-member models with nearly similar misfit. This is interpreted in terms of a non-uniqueness of the gravity problem. Fig. 8 shows the residual (observed minus theoretical) Bouguer anomaly for the best 3-D model. The residuals in the central and southern part of the OLD are mostly
