Geoidal Geopotential and World Height System

The geoidal geopotential value of W ₀ = 62 636 856.0 ± 0.5m ² s ^–2 , determined from the 1993 –1998 TOPEX/POSEIDON altimeter data, can be used to practically define and realize the World Height System. The W ₀ -value can also uniquely define the geoidal surface and is required for a number of applications, including General Relativity in precise time keeping and time definitions. Furthermore, the W ₀ -value provides a scale parameter for the Earth that is independent of the tidal reference system. All of the above qualities make the geoidal potential W ₀ ideally suited for official adoption as one of the fundamental constants, replacing the currently adopted semi-major axis a of the mean Earth ellipsoid. Vertical shifts of the Local Vertical Datum (LVD) origins can easily be determined with respect to the World Height System (defined by W ₀ ), in using the recent EGM96 gravity model and ellipsoidal height observations (e.g. GPS) at levelling points. Using this methodology the LVD vertical displacements for the NAVD88 (North American Vertical Datum 88), NAP (Normaal Amsterdams Peil), AMD (Australian Height Datum), KHD (Kronstadt Height Datum), and N60 (Finnish Height Datum) were determined with respect to the proposed World Height System as follows: –55.1 cm, –11.0 cm, +42.4 cm, –11.1 cm and +1.8 cm, respectively.     

Determination of Geopotential Differences between Local Vertical Datums and Realization of a World Height System

The methodology developed for connecting Local Vertical Datums (LVD) was applied to the Australian Height Datum (AHD) and the North American Vertical Datum (NAVD88). The geopotential values at AHD and NAVD88 were computed and the corresponding vertical offset of 974 mm with rms 51 mm was obtained between the zero reference surfaces defined by AHD and NAVD88. The solution is based on the four primary geodetic parameters, the GPS/levelling sites and the geopotential model EGM96. The Global Height System (or the Major Vertical Datum) can be defined by a geoidal geopotential value used in the solution as the reference value, or by the geopotential value of the LVD, e.g. NAVD88.     

Gravity potential at the Major Vertical Datum as primary geodetic constant

Summary On the basis of the fundamental relations of the Molodensky's Earth's figure theory (1945), admitting the inequality of the gravity potentials at the Major Vertical Datum W₀ and on the surface of the reference level ellipsoid U₀, and taken into account that potential W₀ enters into equations directly, it is recomended, W₀ should be adopted as a primary geodetic constant. Parameters of the best fitting ellipsoid are not needed for the solution of geodetic problems and for the investigation of the Earth's gravity field. The reason for requiring the reference and actual fields be close is only that the boundary-value problem can be solved in the linear approximation. Dedicated to the Memory of M.S. Molodensky Contribution to the I.A.G. Special Commission SC3 Fundamental Constants (SCFC).     

On The Determination Of The Earth's Model – The Mean Equipotential Surface

The sea surface cannot be used as reference for Major Vertical Datum definition because its deviations from the ideal equipotential surface are very large compared to rms in the observed quantities. The quasigeoid is not quite suitable as the surface representing the most accurate Earth's model without some additional conditions, because it depends on the reference field. The normal Earth's model represented by the rotational level ellipsoid can be defined by the geocentric gravitational constant, the difference in the principal Earth's inertia moments, by the angular velocity of the Earth's rotation and by the semimajor axis or by the potential (U ₀ ) on the surface of the level ellipsoid. After determining the geopotential at the gauge stations defining Vertical Datums, gravity anomalies and heights should be transformed into the unique vertical system (Major Vertical Datum). This makes it possible to apply Brovar's (1995) idea of determining the reference ellipsoid by minimizing the integral, introduced by Riemann as the Dirichlet principle, to reach a minimum rms anomalous gravity field. Since the semimajor axis depends on tidal effects, potential U ₀ should be adopted as the fourth primary fundamental geodetic constant. The equipotential surface, the actual geopotential of which is equal to U ₀ , can be adopted as reference for realizing the Major Vertical Datum.     

Computation of Ray Amplitudes in Inhomogeneous Media with Curved

The TOPEX/POSEIDON (T/P) satellite altimeter data from January 1, 1993 to January 3, 2001 (cycles 11–305) was used for investigating the long-term variations of the geoidal geopotential W ₀ and the geopotential scale factor R ₀ = GM÷W ₀ (GM is the adopted geocentric gravitational constant). The mean values over the whole period covered are W ₀ = (62 636 856.161 ± 0.002) m²s^-2, R ₀ = (6 363 672.5448 ± 0.0002) m. The actual accuracy is limited by the altimeter calibration error (2–3 cm) and it is conservatively estimated to be about ± 0.5 m²s^-2 (± 5 cm). The differences between the yearly mean sea surface (MSS) levels came out as follows: 1993–1994: –(1.2 ± 0.7) mm, 1994–1995: (0.5 ± 0.7) mm, 1995–1996: (0.5 ± 0.7) mm, 1996–1997: (0.1 ± 0.7) mm, 1997–1998: –(0.5 ± 0.7) mm, 1998–1999: (0.0 ± 0.7) mm and 1999–2000: (0.6 ± 0.7) mm. The corresponding rate of change in the MSS level (or R ₀) during the whole period of 1993–2000 is (0.02 ± 0.07) mm÷y. The value W ₀ was found to be quite stable, it depends only on the adopted GM, and the volume enclosed by surface W = W ₀. W ₀ can also uniquely define the reference (geoidal) surface that is required for a number of applications, including World Height System and General Relativity in precise time keeping and time definitions, that is why W ₀ is considered to be suitable for adoption as a primary astrogeodetic parameter. Furthermore, W ₀ provides a scale parameter for the Earth that is independent of the tidal reference system. After adopting a value for W ₀, the semi-major axis a of the Earth's general ellipsoid can easily be derived. However, an a priori condition should be posed first. Two conditions have been examined, namely an ellipsoid with the corresponding geopotential which fits best W ₀ in the least squares sense and an ellipsoid which has the global geopotential average equal to W ₀. It is demonstrated that both a-values are practically equal to the value obtained by the Pizzetti's theory of the level ellipsoid: a = (6 378 136.7 ± 0.05) m.     

Fundamental geodetic parameters of the earth's figure and the structure of the earth's gravity field derived from satellite data

Summary Using the geocentric constant GM=398 601.3 × 10 ⁹ m ³s ^–2 , the known value of the angular velocity of the Earth's rotation , Stokes' constants J _n ^(k) and S _n ^(k) upto n=21 (zonal), n=16 (tesseral and sectorial) [2], the geocentric co-ordinates and heights above sea-level of SAO satellite stations [2], the following will be derived: the potential on the geoid W_o, the scale factor for lengths R_o=GM/W_o, the radius-vector of the surface W=W_o, the parameters of the best-fitting Earth tri-axial ellipsoid, and the components of the deflections of the vertical with respect to the geocentric rotational IAG ellipsoid (Lucerne 1967), as well as to the best-fitting geocentric tri-axial ellipsoid. Some of the differences in the structure of the gravity field over the Northern and Southern Hemispheres will be given, and the mean values of gravity over the equatorial zone, determined from the dynamics of satellite orbits, on the one hand, and from terrestrial gravity data, on the other, will be compared.Presented at the Fifteenth IUGG General Assembly, Moscow, July 30 — August 14, 1971.     

Vertical mixing in Überlingersee (Lake Constance) traced by SF6 and heat

Vertical mixing in Überlingersee is studied by releasing sulfur hexafluoride (SF₆) as a tracer at a central hypolimnic depth of 60 m and measuring its subsequent vertical dispersion over a period of three months. The experiment started with a streaky tracer injection of 1 liter gaseous SF₆ (STP) in August 1990. At that time the lake showed a typical strong summer stratification which in a weakened form lasts until November. From the SF₆ profiles of fifteen surveys at three sampling sites vertical diffusivitiesK _z are calculated compensating internal seiche displacement and horizontal tracer loss. Except of the bottom region no sampling site or time period is marked by significant differences in the hypolimnicK _z profile. So vertical mixing in the whole Überlingersee is described by mean diffusivities decreasing from 1.7 cm²/s at 120 m depth to 0.4 cm²/s in 30 m. The minimal value of 0.3 cm²/s in the thermocline region at 20 m depth is only based on observations in autumn. For a strong summer stratification it is certainly lower. The gradient-flux-method for heat was applied to compute a meanK _z (T) profile from continuously measured temperature profiles. Significant differences resulting from the two tracers showed, that theK _z (T) values are underestimated by up to a factor of 5 if cooling by lateral exchange is neglected. Particularly, internal seiche pumping of colder water from the adjacent Lake Obersee over the separating sill of Mainau into the deep Überlingersee basin is observed in 1990 from August onward, obviously controlling the heat budget below the sill level.     

Vertical distributions of chlorophyll in deep, warm monomictic lakes

The factors affecting vertical distributions of chlorophyll fluorescence were examined in four temperate, warm monomictic lakes. Each of the lakes (maximum depth >80 m) was sampled over 2 years at intervals from monthly to seasonal. Profiles were taken of chlorophyll fluorescence (as a proxy for algal biomass), temperature and irradiance, as well as integrated samples from the surface mixed layer for chlorophyll a (chl a) and nutrient concentrations in each lake. Depth profiles of chlorophyll fluorescence were also made along transects of the longest axis of each lake. Chlorophyll fluorescence maxima occurred at depths closely correlated with euphotic depth (r ² = 0.67, P < 0.01), which varied with nutrient status of the lakes. While seasonal thermal density stratification is a prerequisite for the existence of a deep chlorophyll maximum (DCM), our study provides evidence that the depth of light penetration largely dictates the DCM depth during stratification. Reduction in water clarity through eutrophication can cause a shift in phytoplankton distributions from a DCM in spring or summer to a surface chlorophyll maximum within the surface mixed layer when the depth of the euphotic zone (z _eu) is consistently shallower than the depth of the surface mixed layer (z _SML). Trophic status has a key role in determining vertical distributions of chlorophyll in the four lakes, but does not appear to disrupt the annual cycle of maximum chlorophyll in winter.     

Mean Earth'S Equipotential Surface From Topex/Poseidon Altimetry

The geopotential value of W ₀ = (62 636 855.611 ± 0.008) m ² s ^–2 which specifies the equipotential surface fitting the mean ocean surface best, was obtained from four years (1993 - 1996) of TOPEX/POSEIDON altimeter data (AVISO, 1995). The altimeter calibration error limits the actual accuracy of W ₀ to about (0.2 - 0.5) m ² s ^–2 (2 - 5) cm. The same accuracy limits also apply to the corresponding semimajor axis of the mean Earth's level ellipsoid a = 6 378 136.72 m (mean tide system), a = 6 378 136.62 m (zero tide system), a = 6 378 136.59 m (tide-free). The variations in the yearly mean values of the geopotential did not exceed ±0.025 m ² s ^–2 (±2.5 mm).     

Determination of the Geopotential Scale Factor from TOPEX/POSEIDON Satellite Altimetry

The geopotential scale factor R _o = GM/W _o (the GM geocentric gravitational constant adopted) and/or geoidal potential Wo have been determined on the basis of the first year's (Oct 92 – Dec 93) ERS-1/TOPEX/POSEIDON altimeter data and of the POCM 4B sea surface topography model: R _o °=(6 363 672.58°±0.05) m, W _o °=(62 636 855.8°±0.05)m ² s ^–2 . The 2°–°3 cm uncertainty in the altimeter calibration limits the actual accuracy of the solution. Monitoring dW _o /dt has been projected.     

Global geoid sections determined by satellite orbit dynamics

Summary This paper ties up with [5, 6] the fundamental notations of which have been adopted. The sets of Stokes' constants (harmonic coefficients) J _n ^(k) , S _n ^(k) were adopted from [4], and the scale factor for lengths, R₀=GM/W₀, from [5]. The equations for global meridional and parallel sections of the geoid surface W=W₀ are formulated. The geoid sections are represented by best fitting ellipses, as regards the meridians always for the arcs between the equator and the pole. Address: Politickych vězňů 12, Praha 1-Nové Město.     

Vertical hydraulic exchange and the contribution of hyporheic community respiration to whole ecosystem respiration in the River Lahn (Germany)

To quantify the contribution of hyporheic community respiration to whole running-water ecosystem respiration in a cultural landscape setting, we studied the vertical hydraulic exchange in riffle–pool sequences of the River Lahn (Germany). We used flow through curves from four tracer experiments to estimate flow velocities in the surface and subsurface water. Generally, vertical exchange velocities were higher in riffle sections and a high temporal variability was observed (range of values 0.11–1.08 m day⁻¹). We then used (1) the exchange velocities and (2) time series of dissolved oxygen concentration in surface and subsurface water to calculate hyporheic respiration. Hyporheic respiration was estimated in a range of 10–50 mg O₂ m⁻³ day⁻¹ for the upper sediment layer (first 20 cm). It was much lower in the deeper sediment layer (20–40 cm), ranging from 0 to 10 mg O₂ m⁻³ day⁻¹ (volumes are volumes of interstitial water; the average porosity was 20%). We determined primary production and respiration of the biofilm growing on the sediment by modelling dissolved oxygen concentration time series for a 2,450 m long stream reach (dissolved oxygen concentrations with diurnal variations from 8 to 16 mg L⁻¹). Modelled respiration rates ranged from 2 to 21 g O₂ m² day⁻¹. All information was integrated in a system analysis with numerical simulations of respiration with and without sediments. Results indicated that hyporheic respiration accounted for 6 to 14% of whole ecosystem respiration. These values are much lower than in other whole system respiration studies on more oligotrophic river systems.     

Rupture model of the 2013 M_W 6.6 Lushan (China) earthquake constrained by a new GPS data set and its effects on potential seismic hazard

Vertical records are critically important when determining the rupture model of an earthquake, especially a thrust earthquake. Due to the relatively low fitness level of near-field vertical displacements, the precision of previous rupture models is relatively low, and the seismic hazard evaluated thereafter should be further updated. In this study, we applied three-component displacement records from GPS stations in and around the source region of the 2013 MW6.6 Lushan earthquake to re-investigate the rupture model.To improve the resolution of the rupture model, records from both continuous and campaign GPS stations were gathered, and secular deformations of the GPS movements were removed from the records of the campaign stations to ensure their reliability. The rupture model was derived by the steepest descent method(SDM), which is based on a layered velocity structure. The peak slip value was about 0.75 m, with a seismic moment release of 9.89 × 10~(18) N·m, which was equivalent to an M_W6.6 event. The inferred fault geometry coincided well with the aftershock distribution of the Lushan earthquake. Unlike previous rupture models, a secondary slip asperity existed at a shallow depth and even touched the ground surface. Based on the distribution of the co-seismic ruptures of the Lushan and Wenchuan earthquakes, post-seismic relaxation of the Wenchuan earthquake, and tectonic loading process, we proposed that the seismic hazard is quite high and still needs special attention in the seismic gap between the two earthquakes.     

Ground-Motion Scaling in the Western Alps

In order to empirically obtain the scaling relationships for the high-frequency ground motion in the Western Alps (NW Italy), regressions are carried out on more than 7500 seismograms from 957 regional earthquakes. The waveforms were selected from the database of 6 three-component stations of the RSNI (Regional Seismic network of Northwestern Italy). The events, M _W ranging between 1.2 and 4.8, were recorded within a hypocentral distance of 200 km during the time period: 1996–2001. The peak ground velocities are measured in selected narrow-frequency bands, between 0.5 and 14 Hz. Results are presented in terms of a regional attenuation function for the vertical ground motion, a set of vertical excitation terms at the reference station STV2 (hard-rock), and a set of site terms (vertical and horizontal), all relative to the vertical component of station STV2.The regional propagation of the ground motion is modeled after quantifying the expected duration of the seismic motion as a function of frequency and hypocentral distance. A simple functional form is used to take into account both the geometrical and the anelastic attenuation: a multi-variable grid search yielded a quality factor Q(f) = 310f ^0.20, together with a quadri-linear geometrical spreading at low frequency. A simpler, bi-linear geometrical spreading seems to be more appropriate at higher frequencies (f > 1.0 Hz). Excitation terms are matched by using a Brune spectral model with variable, magnitude-dependent stress drop: at M _w 4.8, we used Δσ = 50 MPa. A regional distance-independent attenuation parameter is obtained (κ₀ = 0.012 s) by modelling the average spectral decay at high frequency of small earthquakes.In order to predict the absolute levels of ground shaking in the region, the excitation/attenuation model is used through the Random Vibration Theory (RVT) with a stochastic point-source model. The expected peak-ground accelerations (PGA) are compared with the ones derived by Ambraseys et al. (1996) for the Mediterranean region and by Sabetta and Pugliese (1996) for the Italian territory.     

Seismic anisotropy of shales

Shales are a major component of sedimentary basins, and they play a decisive role in fluid flow and seismic‐wave propagation because of their low permeability and anisotropic microstructure. Shale anisotropy needs to be quantified to obtain reliable information on reservoir fluid, lithology and pore pressure from seismic data, and to understand time‐to‐depth conversion errors and non‐hyperbolic moveout. A single anisotropy parameter, Thomsen's δ parameter, is sufficient to explain the difference between the small‐offset normal‐moveout velocity and vertical velocity, and to interpret the small‐offset AVO response. The sign of this parameter is poorly understood, with both positive and negative values having been reported in the literature. δ is sensitive to the compliance of the contact regions between clay particles and to the degree of disorder in the orientation of clay particles. If the ratio of the normal to shear compliance of the contact regions exceeds a critical value, the presence of these regions acts to increase δ, and a change in the sign of δ, from the negative values characteristic of clay minerals to the positive values commonly reported for shales, may occur. Misalignment of the clay particles can also lead to a positive value of δ. For transverse isotropy, the elastic anisotropy parameters can be written in terms of the coefficients W₂₀₀ and W₄₀₀ in an expansion of the clay‐particle orientation distribution function in generalized Legendre functions. For a given value of W₂₀₀, decreasing W₄₀₀ leads to an increase in δ, while for fixed W₄₀₀, δ increases with increasing W₂₀₀. Perfect alignment of clay particles with normals along the symmetry axis corresponds to the maximum values of W₂₀₀ and W₄₀₀, given by and . A comparison of the predictions of the theory with laboratory measurements shows that most shales lie in a region of the (W₂₀₀, W₄₀₀)‐plane defined by W₄₀₀/W₂₀₀≤W^max₄₀₀/W^max₂₀₀ .     

Computer-simulation models of scoria cone degradation

Long-term erosional modifications of the relatively simple morphology of scoria (‘cinder') cones are ideally suited for study by field and computer-simulation methods. A series of temporally-distinct cones in the San Francisco and Springerville volcanic fields of Arizona provides the foundation for documenting the degradational evolution of scoria cones in a semi-arid climate. Progressive changes due to erosion are illustrated by the systematic decrease with increasing age of various morphometric parameters, including scoria cone height, cone height/width ratio (H_co/W_co), crater depth/width ratio, and slope angle. For example, Holocene–latest Pleistocene cones in the San Francisco field have a mean H_co/W_co value of 0.178±0.041, a mean maximum slope angle of 29.7±4.2°, and a mean average slope angle of 26.4±7.3°, whereas the group of Pliocene cones have values of 0.077±0.024, 20.5±5.8°, and 8.7±2.7°, respectively. Comparative morphology of scoria cones is a potentially useful dating tool for mapping volcanic fields.In order to better understand the degradational modifications of these volcanic landforms, we have developed a numerical approach to simulate the surficial processes responsible for the erosion of a typical scoria cone. The simulation algorithm can apply either a linear diffusion-equation model or a model with a nonlinear transport law. Using a finite-difference formulation, the simulation operates upon a three-dimensional scoria cone input as a matrix of elevation values. Utilizing both field and model results, the correlation between changing H_co/W_co value, cone age, and computer time step was expressed graphically to derive comprehensive values of the transport or diffusion coefficient (D_f) for both volcanic fields. For the San Francisco volcanic field, D_f had a calculated value of 21.4 m²/kyr for the linear model and 5.3 m/kyr for the nonlinear model, while for the Springerville volcanic field D_f had a calculated value of 24.4 m²/kyr for the linear model and 6.3 m/kyr for the nonlinear model.     

Application of a New Vertical Profiling Tool (ESASS) for Sampling Groundwater Quality During Hollow‐Stem Auger Drilling

A new tool called ESASS (Enhanced Screen Auger Sampling System) was developed by the U.S. Geological Survey. The use of ESASS, because of its unique U.S. patent design (U.S. patent no. 7,631,705 B1), allows for the collection of representative, depth‐specific groundwater samples (vertical profiling) in a quick and efficient manner using a 0.305‐m long screen auger during hollow‐stem auger drilling. With ESASS, the water column in the flights above the screen auger is separated from the water in the screen auger by a specially designed removable plug and collar. The tool fits inside an auger of standard inner diameter (82.55 mm). The novel design of the system constituted by the plug, collar, and A‐rod allows the plug to be retrieved using conventional drilling A‐rods. After retrieval, standard‐diameter (50.8 mm) observation wells can be installed within the hollow‐stem augers. Testing of ESASS was conducted at one waste‐disposal site with tetrachloroethylene (PCE) contamination and at two reference sites with no known waste‐disposal history. All three sites have similar geology and are underlain by glacial, stratified‐drift deposits. For the applications tested, ESASS proved to be a useful tool in vertical profiling of groundwater quality. At the waste site, PCE concentrations measured with ESASS profiling at several depths were comparable (relative percent difference <25%) to PCE concentrations sampled from wells. Vertical profiling with ESASS at the reference sites illustrated the vertical resolution achievable in the profile system; shallow groundwater quality varied by a factor of five in concentration of some constituents (nitrate and nitrite) over short (0.61 m) distances.     

Thermal equation of state of magnesite to 32 GPa and 2073 K

Pressure–volume–temperature relations have been measured to 32 GPa and 2073 K for natural magnesite (Mg_0.975Fe_0.015Mn_0.006Ca_0.004CO₃) using synchrotron X-ray diffraction with a multianvil apparatus at the SPring-8 facility. A least-squares fit of the room-temperature compression data to a third-order Birch–Murnaghan equation of state (EOS) yielded K₀ = 97.1 ± 0.5 GPa and K′ = 5.44 ± 0.07, with fixed V₀ = 279.55 ± 0.02 Å³. Further analysis of the high-temperature compression data yielded the temperature derivative of the bulk modulus (∂K_T/∂T)_P = −0.013 ± 0.001 GPa/K and zero-pressure thermal expansion α = a₀ + a₁T with a₀ = 4.03 (7) × 10⁻⁵ K⁻¹ and a₁ = 0.49 (10) × 10⁻⁸ K⁻². The Anderson–Grüneisen parameter is estimated to be δ_T = 3.3. The analysis of axial compressibility and thermal expansivity indicates that the c-axis is over three times more compressible (K_Tc = 47 ± 1 GPa) than the a-axis (K_Tc = 157 ± 1 GPa), whereas the thermal expansion of the c-axis (a₀ = 6.8 (2) × 10⁻⁵ K⁻¹ and a₁ = 2.2 (4) × 10⁻⁸ K⁻²) is greater than that of the a-axis (a₀ = 2.7 (4) × 10⁻⁵ K⁻¹ and a₁ = −0.2 (2) × 10⁻⁸ K⁻²). The present thermal EOS enables us to accurately calculate the density of magnesite to the deep mantle conditions. Decarbonation of a subducting oceanic crust containing 2 wt.% magnesite would result in a 0.6% density reduction at 30 GPa and 1273 K. Using the new EOS parameters we performed thermodynamic calculations for magnesite decarbonation reactions at pressures to 20 GPa. We also estimated stability of magnesite-bearing assemblages in the lower mantle.     

Vertical mixing in Überlinger See,western part of Lake Constance

Depth variable vertical eddy diffusion coefficients for heat (K _z) were calculated from continuously measured temperature profiles in Überlinger See (western part of Lake Constance). The temperatures were averaged over vertical intervals of 10 m yielding 14 discrete values (maximum depth of Überlinger See: 147 m). A linear fit from 10 June to 29 September 1987 was used to smooth the significant temperature fluctuations caused by internal seiches of Lake Constance.Assuming horizontal homogeneity for the smoothed data the Gradient-Flux-Method was applied to compute vertical diffusion coefficientsK _z at different depths using the depth variable volumes and surfaces of the 14 layers. The resulting mean diffusion coefficients for the period from June to September are 0.04 cm²/s near the thermocline and up to 0.8 cm²/s in deeper strata (accuracy: ± 50%). It is shown that horizontal mixing between Überlinger See and Obersee (main lake) alters the computation ofK _z by less than 50%.A relationship betweenK _z and stability (Brunt-Väisälä) frequencyN is found which corresponds well to the theory of internal wave induced turbulence.Combining the diffusion coefficients with measured phosphorus profiles, a phosphorus flux from the hypolimnion to the epilimnion of (0.7 ± 0.4) mg P m^–2 d^–1 was calculated, corresponding to about 20% of the average external loading per area of Lake Constance in 1986.