Short guide to direct gravitational field modelling with Hotine’s equations

This paper presents a unified approach to the least squares spherical harmonic analysis of the acceleration vector and Eötvös tensor (gravitational gradients) in an arbitrary orientation. The Jacobian matrices are based on Hotine’s equations that hold in the Earth-fixed Cartesian frame and do not need any derivatives of the associated Legendre functions. The implementation was confirmed through closed-loop tests in which the simulated input is inverted in the least square sense using the rotated Hotine’s equations. The precision achieved is at the level of rounding error with RMS about $10^{-12}{-}10^{-14}$  m in terms of the height anomaly. The second validation of the linear model is done with help from the standard ellipsoidal correction for the gravity disturbance that can be computed with an analytic expression as well as with the rotated equations. Although the analytic expression for this correction is only of a limited accuracy at the submillimeter level, it was used for an independent validation. Finally, the equivalent of the ellipsoidal correction, called the effect of the normal, has been numerically obtained also for other gravitational functionals and some of their combinations. Most of the numerical investigations are provided up to spherical harmonic degree 70, with degree 80 for the computation time comparison using real GRACE data. The relevant Matlab source codes for the design matrices are provided.     

Estimation of mass change trends in the Earth’s system on the basis of GRACE satellite data, with application to Greenland

The Gravity Recovery and Climate Experiment (GRACE) satellite mission measures the Earth’s gravity field since March 2002. We propose a new filtering procedure for post-processing GRACE-based monthly gravity field solutions provided in the form of spherical harmonic coefficients. The procedure is tuned for the optimal estimation of linear trends and other signal components that show a systematic behavior over long time intervals. The key element of the developed methodology is the statistically optimal Wiener-type filter which makes use of the full covariance matrices of noise and signal. The developed methodology is applied to determine the mass balance of the Greenland ice sheet, both per drainage system and integrated, as well as the mass balance of the ice caps on the islands surrounding Greenland. The estimations are performed for three 2-year time intervals (2003–2004, 2005–2006, and 2007–2008), as well as for the 6-year time interval (2003–2008). The study confirms a significant difference in the behavior of the drainage systems over time. The average 6-year rate of mass loss in Greenland is estimated as 165 ± 15 Gt/year. The rate of mass loss of the ice caps on Ellesmere Island (together with Devon Island), Baffin Island, Iceland, and Svalbard is found to be 22 ± 4, 21 ± 6, 17 ± 9, and 6 ± 2 Gt/year, respectively. All these estimates are corrected for the effect of glacial isostatic adjustment.     

Spatio-temporal divergence in the responses of Finland’s boreal forests to climate variables

Spring greening in boreal forest ecosystems has been widely linked to increasing temperature, but few studies have attempted to unravel the relative effects of climate variables such as maximum temperature (TMX), minimum temperature (TMN), mean temperature (TMP), precipitation (PRE) and radiation (RAD) on vegetation growth at different stages of growing season. However, clarifying these effects is fundamental to better understand the relationship between vegetation and climate change. This study investigated spatio-temporal divergence in the responses of Finland’s boreal forests to climate variables using the plant phenology index (PPI) calculated based on the latest Collection V006 MODIS BRDF-corrected surface reflectance products (MCD43C4) from 2002 to 2018, and identified the dominant climate variables controlling vegetation change during the growing season (May–September) on a monthly basis. Partial least squares (PLS) regression was used to quantify the response of PPI to climate variables and distinguish the separate impacts of different variables. The study results show the dominant effects of temperature on the PPI in May and June, with TMX, TMN and TMP being the most important explanatory variables for the variation of PPI depending on the location, respectively. Meanwhile, drought had an unexpectedly positive impact on vegetation in few areas. More than 50 % of the variation of PPI could be explained by climate variables for 68.5 % of the entire forest area in May and 87.7 % in June, respectively. During July to September, the PPI variance explained by climate and corresponding spatial extent rapidly decreased. Nevertheless, the RAD was found be the most important explanatory variable to July PPI in some areas. In contrast, the PPI in August and September was insensitive to climate in almost all of the regions studied. Our study gives useful insights on quantifying and identifying the relative importance of climate variables to boreal forest, which can be used to predict the possible response of forest under future warming.     

Exploring the spatiotemporal behavior of Helsinki’s housing prices with fractal geometry and co-integration

Fractal geometry and co-integration are combined for exploring spatial morphological aspects of quarterly dwelling prices in Helsinki’s region from 1977 to 2011. Curves of fractal scaling behavior are first employed to measure the fractal dimensions of high- and low-price/m² spatial clusters at multiple scales. Subsequently, the fractal dimensions at indicative neighborhood and citywide scales are modeled with vector error correction specifications. The results identify long-run joint equilibria between the fractal geometries of high- and low-price/m² clusters at both spatial scales. High-price/m² clusters exhibit consistently higher fractal dimensions than their low-value counterparts at the neighborhood scale, while this long-run relation is reversed at the citywide scale. Short-run disequilibria and subsequent adjustments are also scale sensitive. The fractal geometry of high-price/m² clusters leads the dynamics at the neighborhood scale, while low-price/m² clusters lead at the citywide scale. The system’s responses to exogenous shocks take longer time to stabilize at the neighborhood scale compared to the citywide scale, but in both scales the non-stationary nature of fractal behavior is evident. These elements indicate that a closer look on spatial economic behavior at more than one spatial and temporal scale at a time can reveal non-trivial information in the context of urban research and policy analysis.     

An experimental performance analysis of real-time kinematic positioning with NASA’s Internet-Based Global Differential GPS

The Global Differential GPS (GDGPS) system developed by JPL aims at seamless global real-time positioning at the dm accuracy level for dual-frequency receivers either fixed or mobile, anywhere and at any time. The GDGPS system relies on GPS data transmitted in real-time to a central processing center at JPL from a global network of permanently operating GPS dual-frequency receivers. At the processing center, the Internet-based Global Differential GPS (IGDG) system, the heart of JPLs GDGPS, generates and disseminates over the open Internet special 1-s global differential corrections (IGDG corrections) to the GPS broadcast ephemerides. The IGDG corrections enhance the accuracy of GPS broadcast orbits and clocks down to the dm level and serve as the key-factor for high-precise real-time positioning of a stand-alone receiver. An experimental verification of the dm positional accuracy of the IGDG system was carried out in the Netherlands, by means of both a static and a kinematic test. During the static test GPS data were collected for 5 consecutive days using a fixed immobile receiver and processed as if in real-time. Within the framework of the kinematic test, an experiment was carried out using a kinematic platform. Our tests confirmed the dm accuracy of stand-alone receiver positioning with IGDG. The standard deviation for positioning both in static and kinematic mode appears to be 10 cm in each horizontal component and 20 cm in the vertical component. More than 99% of the IGDG corrections were received with the expected 1-s interval in the field via mobile communication, the latency of the corrections was generally from 7 to 8 s.     

Bertin’s matrix concepts reconsidered: transformations of semantics and semiotics to support geovisualization use

A system for managing and visualizing archaeological finds and their spatio-temporal properties is presented. The system is an attempt to combine archaeologists' interest to keep track of time with interests related to the representation and visualization of the spatio-temporal component of archaeological excavations, with the help of cartographic and GIS methods and techniques. Several parameters of a prehistoric excavation such as finds, their thematic characteristics, and their spatio-temporal distributions are examined. A three-dimensional GIS/cartographic environment and its user interface to the excavation database have been developed as a tool for enhancing archaeological interpretation in an exploratory environment. The development of the system is an ongoing task; the article describes its current use as a prototype.     

Cross-Checking: A Method to Test the Comprehensiveness of Pigot’s Nineteenth-Century Plans of Manchester & Salford

The long sequence of Pigot’s plans of Manchester and Salford is used to test the concept that the dates of churches and chapels can be used as a valuable indicator of the completeness of the coverage of large-scale nineteenth-century town plans. The approach appears to hold some promise and suggests that Pigot’s plans were surprisingly comprehensive. This may reflect not merely his drawing on existing surveys but, more interestingly, may be the incidental product of collecting data for his town directories. The methodology could usefully be extended to explore the value of directory plans of other towns.     

Responses of landscape pattern of China’s two largest freshwater lakes to early dry season after the impoundment of Three-Gorges Dam

The effects of hydrologic cycle change (caused by human activity and global climate change) on ecosystems attract the increasing attention around the world. As a result of impounding of the Three Gorges Dam (TGD), climate change and sand mining, the dry season of Poyang Lake and Dongting Lake (China’s two largest freshwater lakes) came early after the TGD impoundment. It was the primary cause of the increasing need for sluice/dam construction to store water in the Lakes and attracted increasing attention. In this paper, we compared the landscape pattern between three hydrologic years with early dry season (EY) and three normal hydrologic years (NY) of each lake by remote sensing technology, to reveal the effect of early dry season on landscape pattern. The results showed that early dry season caused expanding of Phalaris to mudflat zone in Poyang Lake, while caused expanding of Carex to Phalaris zone and expanding of Phalaris to mudflat zone in Dongting Lake. In landscape level, there was no significant difference in landscape grain size, landscape grain shape, habitat connectivity and landscape diversity between EY and NY in the two lakes. While in habitat class level, there were significant changes in area of mudflat and Phalaris and grain size of mudflat in Poyang Lake, and in area of Carex, grain size of Phalaris and grain shape of Carex and Phalaris in Dongting Lake. These changes will impact migrating birds of East Asian and migratory fishes of Yangtze River.     

Transformation of amplitudes and frequencies of precession and nutation of the earth’s rotation vector to amplitudes and frequencies of diurnal polar motion

The temporal change of the rotation vector of a rotating body is, in the first order, identical in a space-fixed system and in a body-fixed system. Therefore, if the motion of the rotation axis of the earth relative to a space-fixed system is given as a function of time, it should be possible to compute its motion relative to an earth-fixed system, and vice versa. This paper presents such a transformation. Two models of motion of the rotation axis in the space-fixed system are considered: one consisting only of a regular (i.e., strictly conical) precession and one extended by circular nutation components, which are superimposed upon the regular precession. The Euler angles describing the orientation of the earth-fixed system with respect to the space-fixed system are derived by an analytical solution of the kinematical Eulerian differential equations. In the first case (precession only), this is directly possible, and in the second case (precession and nutation), a solution is achieved by a perturbation approach, where the result of the first case serves as an approximation and nutation is regarded as a small perturbation, which is treated in a linearized form. The transformation by means of these Euler angles shows that the rotation axis performs in the earth-fixed system retrograde conical revolutions with small amplitudes, namely one revolution with a period of one sidereal day corresponding to precession and one revolution with a period which is slightly smaller or larger than one sidereal day corresponding to each (prograde or retrograde) circular nutation component. The peculiar feature of the derivation presented here is the analytical solution of the Eulerian differential equations.     

‘I’ve got my Sat Nav,it’s alright’: Users’ Attitudes towards,and Engagements with,Technologies of Navigation

The use of qualitative research techniques in a largely quantitative cartographic domain is opening up myriad ways to explore users’ engagements technologies of navigation. This study draws on young UK-based students’ real words and life experiences as they engage with Satellite Navigation and other wayfinding technologies during first-time visits to new places to reflect on the nature of the changing relationships between self, navigational object, space and place.     

On the equivalence of spherical splines with least-squares collocation and Stokes’s formula for regional geoid computation

This work is an investigation of three methods for regional geoid computation: Stokes’s formula, least-squares collocation (LSC), and spherical radial base functions (RBFs) using the spline kernel (SK). It is a first attempt to compare the three methods theoretically and numerically in a unified framework. While Stokes integration and LSC may be regarded as classic methods for regional geoid computation, RBFs may still be regarded as a modern approach. All methods are theoretically equal when applied globally, and we therefore expect them to give comparable results in regional applications. However, it has been shown by de Min (Bull Géod 69:223–232, 1995. doi: 10.1007/BF00806734) that the equivalence of Stokes’s formula and LSC does not hold in regional applications without modifying the cross-covariance function. In order to make all methods comparable in regional applications, the corresponding modification has been introduced also in the SK. Ultimately, we present numerical examples comparing Stokes’s formula, LSC, and SKs in a closed-loop environment using synthetic noise-free data, to verify their equivalence. All agree on the millimeter level.     

Spatiotemporal filtering of regional GNSS network’s position time series with missing data using principle component analysis

The existing spatiotemporal analysis methods suppose that the involved time series are complete and have the same data interval. However missing data inevitably occur in the position time series of Global Navigation Satellite Systems networks for many reasons. In this paper, we develop a modified principal component analysis to extract the Common Mode Error (CME) from the incomplete position time series. The principle of the proposed method is that a time series can be reproduced from its principle components. The method is equivalent to the method of Dong et al. (J Geophys Res 111:3405–3421, 2006) in case of no missing data in the time series and to the extended ‘stacking’ approach under the assumption of a uniformly spatial response. The new method is first applied to extract the CME from the position time series of the Crustal Movement Observation Network of China (CMONOC) over the period of 1999–2009 where the missing data occur in all stations with the different gaps. The results show that the CMEs are significant in CMONOC. The size of the first principle components for the North, East and Up coordinates are as large as 40, 41 and 37 % of total principle components and their spatial responses are not uniform. The minimum amplitudes of the first eigenvectors are only 41, 15 and 29 % for the North, East and Up coordinate components, respectively. The extracted CMEs of our method are close to the data filling method, and the Root Mean Squared error (RMS) values computed from the differences of maximum CMEs between two methods are only 0.31, 0.52 and 1.55 mm for North, East and Up coordinates, respectively. The RMS of the position time series is greatly reduced after filtering out the CMEs. The accuracies of the reconstructed missing data using the two methods are also comparable. To further comprehensively test the efficiency of our method, the repeated experiments are then carried out by randomly deleting different percentages of data at some stations. The results show that the CMEs can be extracted with high accuracy at the non missing-data epochs. And at the missing-data epochs, the accuracy of extracted CMEs has a strong dependence on the number of stations with missing data.     

Reflection on the development of the tool kits of Bertin’s methods

ABSTRACT

Bertin’s first book, Semiology of Graphics, was published in 1967. His second book, Graphics and Graphic Information Processing, was subsequently published in 1977. The word “processing” in the title of the second book is interesting because in those days there were no personal computers with an interactive display system. But in Bertin’s laboratory there were many kinds of tool kits – basically manually developed thematic maps and data analysis. Bertin’s methods were concerned with making a thematic map and data visualization. Maps, and more generally graphics, were represented by sets of cartographic symbols. Thus, they are abstractions that demand both theoretical and technical literacy to represent and understand them. If the representation is systematic, a sort of tool kit might be necessary, because the representation demands consistency based on the theory. Otherwise a cartographer faces the risk of an unstable and unintelligible representation. In this paper, we discuss the discrimination between tool kits intended either for an automated system or a process assisting system. The latter process might be useful and necessary to develop a graphic way of thinking. This investigation refers to Bertin’s books, materials conserved at the National Archives in Paris, and other related software developed later.

Abbreviation: EHESS: Ecole des Hautes Etudes en Sciences Sociales inherited Ecole Pratique des Hautes Etudes since 1975     

Conversion of time-varying Stokes coefficients into mass anomalies at the Earth’s surface considering the Earth’s oblateness

Time-varying Stokes coefficients estimated from GRACE satellite data are routinely converted into mass anomalies at the Earth’s surface with the expression proposed for that purpose by Wahr et al. (J Geophys Res 103(B12):30,205–30,229, 1998). However, the results obtained with it represent mass transport at the spherical surface of 6378 km radius. We show that the accuracy of such conversion may be insufficient, especially if the target area is located in a polar region and the signal-to-noise ratio is high. For instance, the peak values of mean linear trends in 2003–2015 estimated over Greenland and Amundsen Sea embayment of West Antarctica may be underestimated in this way by about 15%. As a solution, we propose an updated expression for the conversion of Stokes coefficients into mass anomalies. This expression is based on the assumptions that: (i) mass transport takes place at the reference ellipsoid and (ii) at each point of interest, the ellipsoidal surface is approximated by the sphere with a radius equal to the current radial distance from the Earth’s center (“locally spherical approximation”). The updated expression is nearly as simple as the traditionally used one but reduces the inaccuracies of the conversion procedure by an order of magnitude. In addition, we remind the reader that the conversion expressions are defined in spherical (geocentric) coordinates. We demonstrate that the difference between mass anomalies computed in spherical and ellipsoidal (geodetic) coordinates may not be negligible, so that a conversion of geodetic colatitudes into geocentric ones should not be omitted.     

The Problem of Double Longitudes on Glavač’s Map

Abstract

Exploring large volumes of geospatial data is difficult. This paper presents an approach that combines visual and computational analysis to make this process easier. This approach is based on the effective application of computational algorithms, such as the Self-Organizing Map (SOM). These are used to uncover the structure, patterns, relationships and trends in the data, and for the creation of abstractions where conventional methods may be limited. In addition, graphical representations are applied to portray extracted patterns in a visual form that allows for better understanding of the derived structures and possible geographical processes, and should facilitate knowledge construction.     

Accuracy assessment of the National Geodetic Survey’s OPUS-RS utility

OPUS-RS is a rapid static form of the National Geodetic Survey’s On-line Positioning User Service (OPUS). Like OPUS, OPUS-RS accepts a user’s GPS tracking data and uses corresponding data from the U.S. Continuously Operating Reference Station (CORS) network to compute the 3-D positional coordinates of the user’s data-collection point called the rover. OPUS-RS uses a new processing engine, called RSGPS, which can generate coordinates with an accuracy of a few centimeters for data sets spanning as little as 15 min of time. OPUS-RS achieves such results by interpolating (or extrapolating) the atmospheric delays, measured at several CORS located within 250 km of the rover, to predict the atmospheric delays experienced at the rover. Consequently, standard errors of computed coordinates depend highly on the local geometry of the CORS network and on the distances between the rover and the local CORS. We introduce a unitless parameter called the interpolative dilution of precision (IDOP) to quantify the local geometry of the CORS network relative to the rover, and we quantify the standard errors of the coordinates, obtained via OPUS-RS, by using functions of the form