The GPS Toolbox

The GPS Toolbox is dedicated to highlighting algorithms utilized by GPS engineers and scientists. If you have an interesting algorithm you would like to share with our readers or if you have a topic you would like to see covered in a future column, contact us at gps-toolbox@ngs.noaa.gov. To comment on the algorithms presented here, or to leave a request for an algorithm you may be looking for, visit our Web site (http://www.ngs.noaa.gov/gps-toolbox). ? 2000 John Wiley & Sons, Inc.     

Determination of failure probabilities of flood defence systems with improved dynamic bounds method

In this paper, we describe the computational framework of a novel method for solving the challenging problem of probabilistic finite elements. The method is called Improved Dynamic Bounds (IDB) and was developed recently to improve the efficiency of the dynamic bounds. The IDB is used in finite element numerical models to calculate time-dependent failure analyses of structures. In applications, the IDB can speed up the overall simulation process by several orders of magnitude. In applications controlled by two influential variables (e.g, two-dimensional problem), the computational efficiency is improved by a factor of 769 according to Rajabalinejad (2009). Applications of IDB indicate the method is most efficient for problems where the number of influential variables are limited. This is often the case for geotechnical and coastal flood defence systems. The IDB method is applied in this paper to the 17th Street Flood Wall, a component of the flood defence system (levee infra-structure) that failed during the Hurricane Katrina, to calculate the failure probability of an I-wall.     

Variation in Coronal Activity from Solar Cycle 24 Minimum to Maximum Using Three-Dimensional Reconstructions of the Coronal Electron Density from STEREO/COR1

Three-dimensional electron density distributions in the solar corona are reconstructed for 100 Carrington rotations (CR 2054?–?2153) during 2007/03?–?2014/08 using the spherically symmetric method from polarized white-light observations with the inner coronagraph (COR1) onboard the twin Solar Terrestrial Relations Observatory (STEREO). These three-dimensional electron density distributions are validated by comparison with similar density models derived using other methods such as tomography and a magnetohydrodynamics (MHD) model as well as using data from the Solar and Heliospheric Observatory (SOHO)/Large Angle and Spectrometric Coronagraph (LASCO)-C2. Uncertainties in the estimated total mass of the global corona are analyzed based on differences between the density distributions for COR1-A and -B. Long-term variations of coronal activity in terms of the global and hemispheric average electron densities (equivalent to the total coronal mass) reveal a hemispheric asymmetry during the rising phase of Solar Cycle 24, with the northern hemisphere leading the southern hemisphere by a phase shift of 7?–?9 months. Using 14 CR (\(\approx13\)-month) running averages, the amplitudes of the variation in average electron density between Cycle 24 maximum and Cycle 23/24 minimum (called the modulation factors) are found to be in the range of 1.6?–?4.3. These modulation factors are latitudinally dependent, being largest in polar regions and smallest in the equatorial region. These modulation factors also show a hemispheric asymmetry: they are somewhat larger in the southern hemisphere. The wavelet analysis shows that the short-term quasi-periodic oscillations during the rising and maximum phases of Cycle 24 have a dominant period of 7?–?8 months. In addition, it is found that the radial distribution of the mean electron density for streamers at Cycle 24 maximum is only slightly larger (by \(\approx30\%\)) than at cycle minimum.     

Evidence for the occurrence of hydrogeochemical processes in the groundwater of Khoy plain,northwestern Iran,using ionic ratios and geochemical modeling

Rapid population growth, industrialization, and agricultural expansion in the Khoy area (northwestern Iran) have led to its dependence on groundwater and degradation of groundwater quality. This study attempts to decipher the major processes and factors that degrade the groundwater quality of the Khoy plain. For this purpose, 54 groundwater samples from unconfined and confined aquifers of the plain were collected in July 2017 and analyzed for major cations and anions (Na, K, Ca, Mg, HCO₃, SO_4, and Cl), minor ions (NO₃ and F), and Al. Magnesium and bicarbonate were identified as the dominant cation and anion, respectively. Several ionic ratios and geochemical modeling using PHREEQC indicated that the most important hydrogeochemical processes to affect groundwater quality in the plain were weathering and dissolution of evaporitic and silicate minerals, mixing, and ion exchange. There were smaller effects from evaporation and anthropogenic factors (e.g., industries). Results showed that the high salinity of the groundwater in the northeast area of the plain was due to the high solubility of the evaporitic minerals, e.g., halite and gypsum. Reverse ion exchange and the contribution of mineral dissolution were more significant than ion exchange in the northeastern part of the plain. Elevated salinity of the groundwater in the southeast was attributed mostly to reverse ion exchange and somewhat to evaporation.     

U–Pb geochronology of gneisses and granitoids from the Danish island of Bornholm: new evidence for 1.47–1.45 Ga magmatism at the southwestern margin of the East European Craton

The Danish island of Bornholm is located at the southwestern margin of the Fennoscandian Shield, and features exposed Precambrian basement in its northern and central parts. In this paper, we present new U–Pb zircon and titanite ages for granites and orthogneisses from 13 different localities on Bornholm. The crystallization ages of the protolith rocks all fall within the range 1,475–1,445 Ma (weighted average ²⁰⁷Pb/²⁰⁶Pb ages of zircon). Minor age differences, however, may imply a multi-phase emplacement history of the granitoid complex. The presence of occasional inherited zircons (with ages of 1,700–1,800 Ma) indicates that the Bornholm granitoids were influenced by older crustal material. The east–west fabric observed in most of the studied granites and gneisses, presumably originated by deformation in close connection with the magmatism at 1,470–1,450 Ma. Most titanite U–Pb ages fall between 1,450 and 1,430 Ma, reflecting post-magmatic or post-metamorphic cooling. Granitoid magmatism at ca. 1.45 Ga along the southwestern margin of the East European Craton has previously been reported from southern Sweden and Lithuania. The ages obtained in this study indicate that the Bornholm magmatism also was part of this Mesoproterozoic event.     

Micrometeorological Observations of a Microburst in Southern Finland

On the afternoon of 3 July 2004 in Hyytiälä (Juupajoki, Finland), convective cells produced a strong downburst causing forest damage. The SMEAR II field station, situated near the damage site, enabled a unique micrometeorological analysis of a microburst with differences above and inside the canopy. At the time of the event, a squall line associated with a cold front was crossing Hyytiälä with a reflectivity maximum in the middle of the squall line. A bow echo, rear-inflow notch, and probable mesovortex were observed in radar data. The bow echo moved west-north-west, and its apex travelled just north of Hyytiälä. The turbulence data were analysed at two locations above the forest canopy and at one location at sub-canopy. At 1412 EET (Eastern European Time, UTC+2), the horizontal and vertical wind speed increased and the wind veered, reflecting the arrival of a gust front. At the same time, the carbon dioxide concentration increased due to turbulent mixing, the temperature decreased due to cold air flow from aloft and aerosol particle concentration decreased due to rain scavenging. An increase in the number concentration of ultra-fine particles (< 10 nm) was detected, supporting the new particle formation either from cloud outflow or due to rain. Five minutes after the gust front (1417 EET), strong horizontal and downward vertical wind speed gusts occurred with maxima of 22 and 15 m s^?1, respectively, reflecting the microburst. The turbulence spectra before, during and after the event were consistent with traditional turbulence spectral theory.     

Distribution of Indus born mica along Gulf of Kachchh coast: Implications in understanding current dynamics

Indus is one of the major sources of sediments to the Gulf of Kachchh. Yet only its <63 micron fraction is studied in detail with regards to the offshore current dynamics. Hence here we present our study on characteristic signature of the Indus sediment load (i.e. mica minerals) in >63 micron size fraction along the coast of Gulf of Kachchh. The spatial distribution of mica minerals along the Gulf of Kachchh coast was studied which showed in general decreasing trend as we move along the northern and southern coast of the Gulf of Kachchh but, an increase in amount near the southern mouth at Okha. The study shows that the earlier proposed tidal barrier is ineffective in restricting movement of mica across the mouth of the gulf due to its characteristic transport mechanism. Also the presence of mudflats along the gulf of Kachchh coast plays a vital role as sediment receptors in the active sediment transport processes and mica minerals prove to be a promising simple tracer in studying the Indus born sediments in the region.     

Central Europe: Active or residual tectonic stresses

The regional stress field in the Western Alps and their northern foreland has been investigated byin situ stress determinations. More than 600 strain relief measurements were made with resistance strain gages in boreholes carried out in mines, tunnels and quarries. The stresses calculated and data obtained from other papers were used to get a detailed idea of the stress conditions in Central Europe.The measurements confirm a continuous flux of compressive stress from the Alps to the northern foreland east of the Rhinegraben. The largest stresses are observed in the Central Alps, the lowest in the Rhinegraben rift system. The horizontal stresses exceed at nearly all places the vertical ones. Evidently the excess of horizontal stress is generated by active plate tectonics in the Alps. A tectonic model to explain the observed stress pattern is presented.     

Experimental Investigation of Turbulent Momentum Transfer in a Neutral Boundary Layer over a Rough Surface

The turbulent characteristics of the neutral boundary layer developing over rough surfaces are not well predicted with operational weather-forecasting models. The problem is attributed to inadequate mixing-length models, to the anisotropy of the flow and to a lack of controlled experimental data against which to validate numerical studies. Therefore, in order to address directly the modelling difficulties for the development of a neutral boundary layer over rough surfaces, and to investigate the turbulent momentum transfer of such a layer, a set of hydraulic flume experiments were carried out. In the experiments, the mean and turbulent quantities were measured by a particle image velocimetry (PIV) technique. The measured velocity variances and fluxes \({(\overline{{u_{i}^{\prime}}{u_{j}^{\prime}}})}\) in longitudinal vertical planes allowed the vertical and longitudinal gradients (?/?z and ?/?x) of the mean and turbulent quantities (fluxes, variances and third-order moments) to be evaluated and the terms of the evolution equations for ?e/?t, \({\partial \overline{u^{\prime 2}}/\partial t}\), \({\partial \overline{w^{\prime 2}}/\partial t}\) and \({\partial \overline{{u^{\prime}}{w^{\prime}}}/\partial t}\) to be quantified, where e is the turbulent kinetic energy. The results show that the pressure-correlation terms allow the turbulent energy to be transferred equitably from \({\overline{{u^{\prime}}^{2}}}\) to \({\overline{{w^{\prime}}^{2}}}\). It appears that the repartition between the constitutive terms of the budget of e, \({\overline{{u^{\prime}}^{2}}}\), \({\overline{{w^{\prime}}^{2}}}\) and \({\overline{{u^{\prime}}{w^{\prime}}}}\) is not significantly affected by the development of the rough neutral boundary layer. For the whole evolution, the transfers of energy are governed by the same terms that are also very similar to the smooth-wall case. The PIV measurements also allowed the spatial integral scales to be computed directly and to be compared with the dissipative and mixing length scales, which were also computed from the data.     

Computing the Sensitivity Kernels for 2.5-D Seismic Waveform Inversion in Heterogeneous,Anisotropic Media

2.5-D modeling and inversion techniques are much closer to reality than the simple and traditional 2-D seismic wave modeling and inversion. The sensitivity kernels required in full waveform seismic tomographic inversion are the Fréchet derivatives of the displacement vector with respect to the independent anisotropic model parameters of the subsurface. They give the sensitivity of the seismograms to changes in the model parameters. This paper applies two methods, called ‘the perturbation method’ and ‘the matrix method’, to derive the sensitivity kernels for 2.5-D seismic waveform inversion. We show that the two methods yield the same explicit expressions for the Fréchet derivatives using a constant-block model parameterization, and are available for both the line-source (2-D) and the point-source (2.5-D) cases. The method involves two Green’s function vectors and their gradients, as well as the derivatives of the elastic modulus tensor with respect to the independent model parameters. The two Green’s function vectors are the responses of the displacement vector to the two directed unit vectors located at the source and geophone positions, respectively; they can be generally obtained by numerical methods. The gradients of the Green’s function vectors may be approximated in the same manner as the differential computations in the forward modeling. The derivatives of the elastic modulus tensor with respect to the independent model parameters can be obtained analytically, dependent on the class of medium anisotropy. Explicit expressions are given for two special cases—isotropic and tilted transversely isotropic (TTI) media. Numerical examples are given for the latter case, which involves five independent elastic moduli (or Thomsen parameters) plus one angle defining the symmetry axis.