Optimising 2-parameter Lambert Conformal Conic projections for ground-to-grid distortions

[EN] A Lambert Conformal Conic (LCC) projection with two true-scale parallels of latitudes phi(l)and phi(u)can be recast in a LCC projection with one standard parallel of latitude phi(0)and scalek(0), having the practical advantage that the same type of definition can be used for the two conformal projections universally used: LCC and Transverse Mercator (TM). While equations giving phi(0)andk(0)in terms phi(l)and phi(u)can be found in the literature, inverse relationships are not readily found. They are derived in the present paper. These may be necessary in views of the planned future definition of the United States State Plane Coordinate System (SPCS) 2022 for the users of particular mapping software requiring to specify the two latitude values instead of the central latitude and central scale. While map projection parameters are customary selected to minimise ellipsoid-to-grid distortions for a region, in some cases it could be more convenient to study and minimise ground-to-grid distortions. Also bearing in mind the design of SPCS 2022, we discuss the advantages and disadvantages of working with each type of distortion definition.Baselga Moreno, S. (2021). Optimising 2-parameter Lambert Conformal Conic projections for ground-to-grid distortions. Survey Review. 53(380):415-421. https://doi.org/10.1080/00396265.2020.17973394154215338

Fibonacci lattices for the evaluation and optimization of map projections

[EN] Latitude-longitude grids are frequently used in geosciences for global numerical modelling although they are remarkably inhomogeneous due to meridian convergence. In contrast, Fibonacci lattices are highly isotropic and homogeneous so that the area represented by each lattice point is virtually the same. In the present paper we show the higher performance of Fibonacci versus latitude-longitude lattices for evaluating distortion coefficients of map projections. In particular, we obtain first a typical distortion for the Lambert Conformal Conic projection with their currently defined parameters and geographic boundaries for Europe that has been adopted as standard by the INSPIRE directive. Further, we optimize the defining parameters of this projection, lower and upper standard parallel latitudes, so that the typical distortion for Europe is reduced a 10% when they are set to 36 degrees and 61.5 degrees, respectively. We also apply the optimization procedure to the determination of the best standard parallels for using this projection in Spain, whose values remained unspecified by the National decree that commanded its official adoption, and obtain optimum values of 37 degrees and 42 degrees and a resulting typical distortion of 828 ppm.Baselga Moreno, S. (2018). Fibonacci lattices for the evaluation and optimization of map projections. Computers & Geosciences. 117:1-8. https://doi.org/10.1016/j.cageo.2018.04.012S1811

TestGrids: Evaluating and Optimizing Map Projections

[EN] In the study of map projections, it is relatively simple to obtain meaningful estimators of distortion for a small area. The definition and especially the evaluation of global distortion measures (i.e., estimators representing the distortion worldwide or in a continent-like area) are undoubtedly more troublesome. Therefore, it is relatively common to find that recommendations for the parameters to use in a particular map projection, be it devised for a continent or a country, are based on simple rules (like the one-sixth rule of thumb for conic projections), with no possibility of further improvement in terms of resulting distortions and sometimes even with no knowledge at all of the sizes of these distortions. Although the choice of map defining parameters is normally made for reasons other than distortion minimization, such as ease of use (e.g., integer or half-integer numbers may be preferable), preservation of conventional or traditional definitions, and uniformity of parameters between neighboring regions, it is always worthwhile to know the optimal set of parameters in terms of minimal distortion. Then, the cartographer may mindfully deviate from this optimal set, documenting the differences in defining parameters and in the resulting distortions. The present research provides a means to do this by extending a related work presented in a previous contribution, where the evaluation and optimization of distortions were studied for a single map projection and only two areas of interest. To this end, a new tool has been developed and presented in this paper. This tool allows users to evaluate several measures of distortion for the most common conformal and equal-area projections within user-defined geographic boundaries of interest. Also embedded in the tool and transparent to users are global optimization techniques operating on Fibonacci grids, which permit the optimization of parameters for the particular map projection and area of interest under two possible criteria: minimization of typical distortion or minimization of extreme distortions. This tool and the associated techniques are applied to several official projections to analyze their original performance and to propose new parameters that significantly improve the resulting distortions while leaving room for users to easily evaluate and optimize the tool for the lowest distortions of these projections within their regions of interest.Baselga Moreno, S. (2019). TestGrids: Evaluating and Optimizing Map Projections. Journal of Surveying Engineering. 145(3):1-8. https://doi.org/10.1061/(ASCE)SU.1943-5428.0000279S18145

Two Conformal Projections for Constant-Height Surface to Plane Mapping

[EN] Regions at high elevations may require specific mapping solutions other than the conventional ellipsoid-to-grid projections, which produce large discrepancies between ground and projected distances. These particular solutions are known as low-distortion projections (LDPs). They can be realized by using an elevated ellipsoid (EE) or a constant-height surface (ChS) above the ellipsoid as the reference surface, or by means of a scaled projection. No conformal projections have been derived thus far for the ChS-to-plane transformation. This work aims to solve this problem by deriving the formulation of direct and transverse Mercator-type projections for ChS-to-plane conformal mapping.Baselga Moreno, S. (2021). Two Conformal Projections for Constant-Height Surface to Plane Mapping. Journal of Surveying Engineering. 147(2):1-7. https://doi.org/10.1061/(ASCE)SU.1943-5428.000034517147

Automated detection of discontinuities in EUREF permanent GNSS network stations due to earthquake events

[EN] The EUREF Permanent GNSS Network (EPN) provides the users with data and products such as station coordinate time series. These are subject to possible discontinuities and trend changes, being earthquake events one of the possible natural causes for these variations. We present here a fully automated tool for the analysis of the coordinate time series of EPN stations located in the desired neighbourhood of an earthquake epicentre. The tool is made freely available to the public and applied here to two significant earthquake events occurred in Europe in recent years, where several trend changes and jumps are revealed.Baselga Moreno, S.; Najder, J. (2022). Automated detection of discontinuities in EUREF permanent GNSS network stations due to earthquake events. Survey Review. 54(386):420-428. https://doi.org/10.1080/00396265.2021.19642304204285438

Intersection and point-to-line solutions for geodesics on the ellipsoid

[EN] The paper presents two algorithms for the computation of intersection of geodesics and minimum distance from a point to a geodesic on the ellipsoid, respectively. They are based on the iterative use of direct and inverse problems of geodesy by means of their implementations with machine-precision accuracy in GeographicLib. The algorithms yield the same results as those obtained by Karney¿s approach based on the use of auxiliary ellipsoidal gnomonic projections, with the advantage on our side that the algorithms are not limited to distances below 10000 km. This results in our algorithm being the only general solution for the problem of minimum distance from a point to a geodesic on the ellipsoid.Baselga Moreno, S.; Martínez Llario, JC. (2018). Intersection and point-to-line solutions for geodesics on the ellipsoid. Studia Geophysica et Geodaetica. 62(3):353-363. https://doi.org/10.1007/s11200-017-1020-zS353363623JaVaWa, 2017. JAVAWA GPS Tools. (http://www.javawa.nl/coordcalc_en.html).Karney C.F.F., 2011a. Geodesics on an Ellipsoid of Revolution. Technical Report. SRI International (http://arxiv.org/abs/1102.1215v1).Karney C.F.F., 2011b. Intersection between Two Geodesic Lines. (https://sourceforge.net/p/geographiclib/discussion/1026621/thread/21aaff9f).Karney C.F.F., 2013. Algorithms for geodesics. J. Geodesy, 87, 43–55.Karney C.F.F., 2017. GeographicLib, version 1.47. (http://geographiclib.sourceforge.net).Patrikalakis N.M., Maekawa T. and Cho W., 2009. Shape Interrogation for Computer Aided Design and Manufacturing. (http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/mathe.html).PostGIS Development Group, 2017. PostGIS 2.3.3dev Manual. (http://postgis.net/docs/index.html).Sjöberg L.E., 2002. Intersections on the sphere and ellipsoid. J. Geodesy, 76, 115–120.Sjöberg L.E., 2006. Direct and indirect geodetic problems on the ellipsoid. Z. Vermess., 131, 35–39.Sjöberg L.E., 2008. Geodetic intersection on the ellipsoid. J. Geodesy, 82, 565–567.Sjöberg L.E., 2009. New solutions to classical geodetic problems on the ellipsoid. In: Sideris M.G. (Ed.), Observing our Changing Earth. International Association of Geodesy Symposia 133, 781–784, Springer-Verlag, Berlin, Germany.Sjöberg L.E. and Shirazian M, 2012. Solving the direct and inverse geodetic problems on the ellipsoid by numerical integration. J. Surv. Eng., 138, 9–16.Todhunter I., 1886. Spherical Trigonometry, 5th Edition. MacMillan and Co., London, U.K. (http://www.gutenberg.org/ebooks/19770).Vincenty T., 1975. Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations. Surv. Rev., 23, 88–93

Optimal combination and reference functions of signal-to-noise measurements for GNSS multipath detection

[EN] Multipath is the most limiting factor in many GNSS positioning applications, where it inevitably degrades the attainable precision. Among the different proposals to identify observations affected by multipath, Strode and Groves have recently proposed a method based on the comparison of GPS signal-to-noise (SNR) actual measurements with suitable reference functions previously computed in a low-multipath environment. We have found significant issues with its application to our particular GNSS experiments, however. In particular, we discuss whether the reference functions that are needed to be computed for low-multipath environments after tedious and time consuming field campaigns can be used for a future occasion, or not, as well as the possibility of applying the method to other GNSS global constellations (Galileo and GLONASS). Additionally, we elaborate on an alternative idea consisting in the use of the best combination of SNR measurements for the different signals in the different constellations in order to obtain a multipath estimator that is unbiased, universal and performs better than the use of reference functions.Pánik, P.; García-Asenjo Villamayor, L.; Baselga Moreno, S. (2019). Optimal combination and reference functions of signal-to-noise measurements for GNSS multipath detection. Measurement Science and Technology. 30(4):1-13. https://doi.org/10.1088/1361-6501/ab05aeS11330

Accurate Algorithms for Spatial Operations on the Spheroid in a Spatial Database Management System

[EN] Some of the most powerful spatial analysis software solutions (Oracle, Google Earth Engine, PostgreSQL + PostGIS, etc.) are currently performing geometric calculations directly on the ellipsoid (a quadratic surface that models the earth shape), with a double purpose: to attain a high degree of accuracy and to allow the full management of large areas of territory (countries or even continents). It is well known that both objectives are impossible to achieve by means of the traditional approach using local mathematical projections and Cartesian coordinates. This paper demonstrates in a quantitative methodological way that most of the spatial analysis software products make important deviations in calculations regarding to geodesics, being the users unaware of the magnitude of these inaccuracies, which can easily reach meters depending on the distance. This is due to the use of ellipsoid calculations in an approximate way (e.g., using a sphere instead of an ellipsoid). This paper presents the implementation of two algorithms that solve with high accuracy (less than 100 nm) and efficiently (few iterations) two basic geometric calculations on the ellipsoid that are essential to build more complex spatial operators: the intersection of two geodesics and the minimum distance from a point to a geodesic.Martínez Llario, JC.; Baselga Moreno, S.; Coll-Aliaga, E. (2021). Accurate Algorithms for Spatial Operations on the Spheroid in a Spatial Database Management System. Applied Sciences. 11(11):1-21. https://doi.org/10.3390/app11115129121111

Assessment of Cranial Deformation Indices by Automatic Smartphone-Based Photogrammetric Modelling

[EN] This paper presents research carried out to assess the accuracy of a fully automatic smartphone-based photogrammetric solution (PhotoMeDAS) to obtain a cranial diagnostic based on the 3D head model. The rigorous propagation of the coordinate measurement uncertainty to the infant's derived cranial deformation indices is demonstrated. The cranial anthropometric parameters and cranial deformation indices that PhotoMeDAS calculates automatically were analysed based on the estimated accuracy and uncertainty. To obtain both accuracy and uncertainty, a dummy head was measured 54 times under different conditions. The same head was measured with a top-of-the-line coordinate-measuring machine (CMM), and the results were used as ground-truth data. It is demonstrated that the PhotoMeDAS 3D models are an average of 1.01 times bigger than the corresponding ground truth, and the uncertainties are around 1 mm. Even assuming uncertainties in the coordinates of up to 1.5 mm, the error in the derived deformation index uncertainties is around 1%. In conclusion, the PhotoMeDAS solution improves the uncertainty obtained in an ordinary paediatric consultation and can be recommended as a tool for doctors to establish an adequate medical diagnosis based on comprehensive cranial deformation indices, which is much more precise and complete than the information obtained by existing analogue devices (measuring tapes and callipers) and easier to use and less expensive than radiological imaging (CT and MRI).Baselga Moreno, S.; Mora Navarro, JG.; Lerma, JL. (2022). Assessment of Cranial Deformation Indices by Automatic Smartphone-Based Photogrammetric Modelling. Applied Sciences. 12(22):1-15. https://doi.org/10.3390/app122211499115122

GBDM+: an improved methodology for a GNSS-based distance meter

[EN] The determination of distances consistent with the definition of the base unit of length in the International System of Units (SI), the SI meter, with uncertainties of less than 1 ppm up to 5 km in the open air is a current challenge that is being increasingly required for different applications, including the determination of local ties, calibration baselines, and high precision geodetic metrology in singular scientific and engineering projects. The required knowledge of the index of refraction of the propagating medium at the same level of 1 ppm is a hard limit to the use of precise electronic distance meters (EDMs), which has motivated the recent development of new two-color, refractivity compensated, EDM prototypes. As an alternative, the use of global navigation satellite systems (GNSS) could benefit from their high scale stability although the lack of appropriate estimation of the uncertainties in their sources of error and their unknown propagation into the final result during the data processing has prevented a rigorous uncertainty analysis and, therefore, the use of GNSS for absolute distance determination. Stemming from our initial methodology for a GNSS-based distance meter (GBDM) that was restricted to relatively horizontal baselines and distances up to 1 km only, we have improved the method so that its application range is extended to baselines of up to 5 km with a possibly significant height difference so that it provides the final baseline distance with the corresponding uncertainty derived from the uncertainties in the different error sources rigorously propagated through the equations by which the distance is finally determined. This improved methodology, named as GBDM+, constitutes a significant step forward in the application of GNSS to open air length metrology.The work leading to this paper was performed within the 18SIB01 GeoMetre project of the European Metrology Programme for Innovation and Research (EMPIR). This project has received funding from the EMPIR programme co-financed by the Participating States and from the European Union's Horizon 2020 research and innovation programme, funder ID: 10.13039/100014132. Raquel Lujan acknowledges the funding from the Programa de Ayudas de Investigacion y Desarrollo (PAID-01-20) de la Universitat Politecnica de Valencia.Baselga Moreno, S.; García-Asenjo Villamayor, L.; Garrigues Talens, P.; Luján, R. (2022). GBDM+: an improved methodology for a GNSS-based distance meter. Measurement Science and Technology. 33(8):1-16. https://doi.org/10.1088/1361-6501/ac6f4511633