Accuracy evaluation of pendulum gravity measurements of Robert Daublebsky von Sterneck Alena Pešková, Jan Holešovský Department of Geomatics, Faculty of Civil Engineering Czech Technical University in Prague Thákurova 7, 166 29 Prague 6, Czech Republic alena.peskova@fsv.cvut.cz, jan.holesovsky@fsv.cvut.cz Abstract The accuracy of first pendulum gravity measurements in the Czech territory was determined using both original surveying notebooks of Robert Daublebsky von Ster- neck and modern technologies. Since more accurate methods are used for gravity measurements nowadays, the work [3] is mostly important from the historical point of view. In previous works [5], the accuracy of Sterneck’s gravity measurements was determined using only a small dataset. Here we process all Sterneck’s mea- surements from the Czech territory (a dataset ten times larger than in the previous works [5]), and we complexly assess the accuracy of these measurements. Locations of the measurements were found with the help of original notebooks. Gravity in the site was interpolated using gravity model EGM08, resultant gravity is in actual system S–Gr10. Finally, the accuracy of Sterneck’s measurements was evaluated on the base of the differences between the measured and interpolated gravity. Keywords: Robert Daublebsky von Sterneck, relative pendulum measurements, gravity. 1. Introduction Robert Daublebsky von Sterneck (* 7.2.1839, † 2.11.1910) was born in Prague, he acted as geodesist, astronomer and geophysicist. He was Head of Astronomical Observatory Institute in Vienna in 1880-1884 and also he was the first to make gravimetrical measurements in the Austria-Hungary. Although he worked in army all his life, he also did various surveying and astronomical measurements. His work was recognized and his name is famous also nowadays. The pendulum instrument built by Sterneck himself was used for gravity measurements, and its improved version was also used in other countries in Europe. Daublebsky used a relative method to measure gravity. Only the time of swing of the pendulum was measured with four implemented corrections. The initial gravity point was located in the cellar of Military Geographical Institute in Vienna, with value g = 980 876 mGal [2]. We divided Sterneck’s measurements to two datasets. The first rule for division was different localities of the measurements (measurements on hilltops near trigonometrical points, and measurements in buildings in towns). The second rule was the time of measurements (there is a 3 year gap between the two datasets). Geoinformatics FCE CTU 14(1), 2015, doi:10.14311/gi.14.1.3 39 http://orcid.org/0000-0002-3338-3779 http://orcid.org/0000-0003-0503-7569 http://dx.doi.org/10.14311/gi.14.1.3 http://creativecommons.org/licenses/by/4.0/ A. Pešková and J. Holešovský: Accuracy of Pendulum gravity measurements 2. Localization of Sterneck’s gravity measurements The original Daublebsky’s surveying notebooks [6] and a summary of results in technical report [4] were used for gravity measurement localization. The technical report contains approximate astronomical coordinates of the measurements, whereas detailed information about the measurement process and locations is given in the notebooks. From the technical report, were used these informations: year of measurement, number and title of point (Czech and Germany), latitude and longitude, elevation and the measured value of gravity. Only the details about the locations were used from the notebooks. These details were not registered for all measured points, - 15 points measured in towns haven’t had any information about their location (these points were locallized only by approximate coordinates and heights). The measurements were divided into two groups: both by measurement location and by the time measurement. In 1889 – 1895, - 106 points were determined in the Czech territory, as is shown in Figure 1. The first group of points is located on hilltops close to know trigonometric points – hilltop dataset (blue circles in Figure 1). In 1889 – 1891 were determined 35 points in close trigonometric points and 6 points with differently locations in the Czech territory. In 1894 – 1895 (after a 3 year gap), the second group of 65 points was measured in buildings inside towns in the Moravian territory – building dataset (green squares in Figure 1). Figure 1: Locations of Sterneck’s gravity measurements. 3. Determination of the gravity differences We used the ArcMap program to determine the coordinates of the measurements with joint WMS provided by The Czech Office for Surveying, Mapping and Cadastre (ČÚZK). Coordi- nates of the locations with error estimates and corrections for heights (e.g. measurement in a building or on top of a lookout tower) was provided by The Department of Gravimetry, Land Survey Office (ZÚ). They intepolated the complete Bouguer anomaly using the methods of ordinary kriging. The results of interpolation are the most probable values of gravity for the Geoinformatics FCE CTU 14(1), 2015 40 A. Pešková and J. Holešovský: Accuracy of Pendulum gravity measurements referenced locations, given with their upper and lower estimate limits. The gravity value is found in this interval with 95% probability. The limits are affected by the uncertainty in elevation and position. The estimated interval isn’t symmetrical and it is different for each of the measured points. Throughout this work, only the most probable gravity values were used. The gravity differences are calculated as the difference between Sterneck’s measured gravity and the interpolated gravity. These differences were used to evaluate the accuracy of Daublebsky’s pendulum gravity measurements. 4. Data analysis The differences between the measured and interpolated gravity values are distinctly different for the hilltop and the building dataset. The differences gravity in the building dataset show a systematical offset +21.7 mGal, shown in Figure 2. This displacement represents a 72 meters error in elevation. The cause of this displacement isn’t known, therefore both datasets were processed separately. A surprising fact about building dataset is that the gravity differences for points without precise location information (only approximate coordinates and heights) and points with these information weren’t significantly different. This is illustrated in Figure 3. -50 -30 -10 10 30 50 70 hilltop dataset building dataset hilltop dataset, building dataset m e a su re d g ra vi ty - in te rp o la te d g ra vi ty [ m G a l] Figure 2: Differences between measured and interpolated gravity for both datasets. The datasets were tested for data quality. Dependencies between various quantities were tested for this purpose using hypothesis verification. The computed correlation coefficient was compared with its critical value. The tested hypotheses are: gravity falls with growing elevation – (H1), gravity grows with growing latitude – (H2), and gravity and longitude are independent – (H3). All three hypotheses were verified for the hilltop dataset. In the building dataset, H2 and H3 were also verified, but H1 not. Because all of the tested quantities in the building dataset are all right, we think that the elevation values are also affected by an error different from Gaussian noise. Still, the building dataset was used in other processing. The accuracy of Sterneck’s measurements was evaluated by several methods. First, we deter- Geoinformatics FCE CTU 14(1), 2015 41 A. Pešková and J. Holešovský: Accuracy of Pendulum gravity measurements 370 375 380 385 390 395 400 405 410 415 420 425 430 435 440 -20 -10 0 10 20 30 40 50 60 70 pointsvwithvlocationvinformation pointsvwithoutvlocationvinformation pointvnumberv m e a su re d vg ra vi ty v- vin te rp o la te d vg ra vi ty v[ m G a l] Figure 3: Differences between measured and interpolated gravity for the building dataset. mined the value mean gravity difference. This value shows the magnitude of the difference between Sterneck’s measured gravity and the interpolated gravity. The hilltop dataset has mean gravity difference +11.6 mGal, and +33.3 mGal is for the building dataset. These means are apparently affected by an unknown displacement of the used gravity systems. However, the computed differences are only valid with the assumption of null displacement between the gravity systems. If we want to compare the accuracy of the past and recent measurements, we must calculate the mean difference from absolute value of gravity difference. The datasets are characterized by the mean absolute value of gravity difference of 12.9 mGal for the hill- top dataset and 33.3 mGal for the building dataset. The second method is to evaluate the precision of the measurements using standard deviation of the mean gravity difference. This value shows precision of the measurement method and removes the systematic displacement between the two datasets. Both datasets have identical value of standard deviation equal to 10.3 mGal. The conclusion is that both datasets have identical measurement accuracy, although they were determined with different conditions and in a different environment. 5. Discussion and conclusion Sterneck’s measurements were divided into two dataset differing by both the type of the mea- surement locations and the time of their acquisition. The statistical processing and evaluation was done separately because of these differences. The building dataset is displaced system- atically by about +21.7 mGal from the hilltop dataset (mean gravity difference 11.6 mGal for the hilltop dataset and 33.3 mGal for the building dataset). The cause of this systematic displacement is unknown. The building dataset was determined after 3 year gap. During this time some parameters of the pendulum instrument or some changes in way of calculating corrections could be changed. These changes probably can cause the systematic displace- ment between both of datasets. Therefore the accuracy of Sterneck’s measurements is better assessed by standard deviation of the mean difference. That is 10.3 mGal and is identi- cal for both datasets. This value can be compared with Sterneck’s precision estimate of Geoinformatics FCE CTU 14(1), 2015 42 A. Pešková and J. Holešovský: Accuracy of Pendulum gravity measurements 10 mGal [4]. The mean of gravity difference 11.6 mGal for the hilltop dataset and 33.3 mGal for the building dataset can be compared to measurements in Hungary where the errors of Sterneck’s measurements are up to ±20 mGal [5], (but the difference for some points is up to 25 mGal [1]). Acknowledgement The authors thank the employees of The Department of Gravimetry, Land Survey Office (ZÚ) in Prague; Martin Lederer, who borrowed the original surveying notebooks of Robert Daublebsky von Sterneck and Otakar Nesvadba, who interpolated the gravity values. References [1] Alexandr Drbal and Milan Kocáb. “Významný rakouský generálmajor Dr.h.c. Robert Daublebsky von Sterneck”. In: Geodetický a kartografický obzor 56(98).2 (2010), pp. 40– 46. url: http://archivnimapy.cuzk.cz/zemvest/cisla/Rok201002.pdf. [2] Martin Lederer. “Historie kyvadlových měření na území České republiky”. In: Geodetický a kartografický obzor 58(100).6 (2012), pp. 129–133. url: http://archivnimapy.cuzk. cz/zemvest/cisla/Rok201206.pdf. [3] Alena Pešková. “Hodnocení přesnosti kyvadlových tíhových měření R. Sternecka”. Mas- ter thesis. Czech Technival University in Prague, 2015. url: http://geo.fsv.cvut. cz/proj/dp/2015/alena-peskova-dp-2015.pdf. [4] Zdeněk Šimon. Kyvadlová měření v letech 1956 - 1962. Tech. rep. Geodetický a to- pografický ústav v Praze, 1962. [5] V. B. Staněk and J. Potoček. “Vývoj a způsob měření intensity tíže v Čechách a na Moravě”. In: Zeměměřičský obzor 1(28).6 (1940), pp. 81–87. [6] Robert Sterneck. “Měřické sešity 1889 - 1895”. Vojenský zeměpisný ústav ve Vídni. Unpublished. Geoinformatics FCE CTU 14(1), 2015 43 http://archivnimapy.cuzk.cz/zemvest/cisla/Rok201002.pdf http://archivnimapy.cuzk.cz/zemvest/cisla/Rok201206.pdf http://archivnimapy.cuzk.cz/zemvest/cisla/Rok201206.pdf http://geo.fsv.cvut.cz/proj/dp/2015/alena-peskova-dp-2015.pdf http://geo.fsv.cvut.cz/proj/dp/2015/alena-peskova-dp-2015.pdf A. Pešková and J. Holešovský: Accuracy of Pendulum gravity measurements Table 1: Input – Part 1 Year of Number of Latitude Longitude Altitude Measured gravity measurement point [° ’] from Ferro [° ’] [m] [mGal] 1889 49 49 24 32 38 738 980 856 50 49 36 32 20 712 980 887 51 49 55 32 27 545 980 938 52 50 44 33 24 1602 980 762 53 50 08 32 08 356 981 016 54 50 33 31 36 835 980 924 55 50 08 32 39 213 981 070 56 49 57 32 51 470 980 952 57 50 22 31 57 205 981 076 58 50 23 31 57 459 981 019 59 50 25 31 40 202 981 060 60 50 26 31 41 417 980 998 61 50 25 31 41 250 981 055 1890 62 49 14 31 58 624 980 846 63 49 22 31 29 585 980 851 64 49 39 31 31 842 980 855 65 49 48 31 45 659 980 911 66 49 49 31 20 716 980 893 67 50 01 30 40 822 980 922 68 50 12 31 25 534 980 983 69 50 34 31 08 921 980 920 70 50 48 31 47 748 980 963 71 50 44 32 39 1010 980 915 72 50 25 32 59 430 981 016 73 50 32 32 23 565 980 989 74 49 58 30 10 939 980 862 75 49 40 30 39 537 980 937 76 49 26 30 52 724 980 877 78 49 00 31 29 1362 980 663 79 48 52 31 57 1084 980 716 80 48 46 32 15 869 980 760 1890 81 49 39 32 59 709 980 849 82 49 47 33 24 662 980 895 1891 85 49 30 33 30 693 980 881 86 49 19 33 11 732 980 873 87 49 05 32 51 731 980 819 88 49 10 33 22 710 980 861 89 49 22 33 45 639 980 841 90 49 11 33 56 513 980 846 91 49 05 34 16 201 981 004 92 48 52.0 34 19.0 550 980 853 Geoinformatics FCE CTU 14(1), 2015 44 A. Pešková and J. Holešovský: Accuracy of Pendulum gravity measurements Table 2: Input – Part 2 Year of Number of Latitude Longitude Altitude Measured gravity measurement point [° ’] from Ferro [° ’] [m] [mGal] 1894 371 48 51.3 34 47.7 160 980 943 372 49 00.6 34 47.8 193 980 917 373 48 59.7 34 31.5 226 980 943 374 48 58.9 34 11.3 181 980 957 375 49 03.0 33 58.8 246 980 961 376 48 59.1 33 44.5 355 980 937 377 49 03.3 33 28.5 465 980 925 1895 378 50 26.3 33 01.3 273 981 057 379 50 14.5 33 09.5 228 981 068 380 50 02.3 33 26.8 214 981 076 381 49 54.6 33 03.5 263 981 054 382 49 36.5 33 14.7 428 980 946 383 49 45.7 33 34.3 569 980 935 378 50 26.3 33 01.3 273 981 057 379 50 14.5 33 09.5 228 981 068 380 50 02.3 33 26.8 214 981 076 381 49 54.6 33 03.5 263 981 054 382 49 36.5 33 14.7 428 980 946 383 49 45.7 33 34.3 569 980 935 384 49 42.9 33 55.9 555 980 955 385 49 57.3 33 49.7 287 981 030 386 49 11.7 34 16.5 235 980 962 387 49 02.3 34 17.1 191 980 979 388 48 59.9 33 01.0 506 980 911 1895 389 49 23.7 33 15.5 514 980 940 390 49 21.3 33 40.7 425 980 955 391 49 33.7 33 36.6 574 980 922 392 49 31.4 33 55.5 554 980 942 393 49 21.0 34 05.3 270 980 999 394 49 29.3 34 19.7 396 980 969 395 49 35.4 34 33.3 410 980 953 396 49 35.4 34 55.3 225 981 026 397 49 16.7 34 40.0 254 981 001 398 49 21.5 35 02.3 200 980 983 399 49 06.3 35 03.7 209 980 958 400 49 01.4 35 18.8 248 980 932 401 49 08.4 35 40.7 390 980 892 402 49 13.7 35 20.2 231 980 959 403 49 24.0 35 20.5 316 980 972 404 49 32.9 35 24.2 256 981 010 405 49 20.4 35 39.7 340 980 954 Geoinformatics FCE CTU 14(1), 2015 45 A. Pešková and J. Holešovský: Accuracy of Pendulum gravity measurements Table 3: Input – Part 3 Year of Number of Latitude Longitude Altitude Measured gravity measurement point [° ’] from Ferro [° ’] [m] [mGal] 1895 406 49 21.9 35 58.5 510 980 906 407 50 33.8 33 34.9 415 981 052 408 50 39.8 33 29.1 610 981 045 409 50 36.7 33 10.4 462 981 052 410 50 24.3 33 21.0 335 981 039 411 50 30.8 33 41.0 359 981 097 412 50 35.2 33 59.9 405 981 085 413 50 25.1 33 49.8 337 981 069 414 50 09.9 33 56.6 321 981 014 415 50 02.2 34 10.0 368 981 007 416 49 54.8 34 16.8 387 981 002 417 50 05.1 34 25.6 567 980 972 418 50 09.8 34 36.8 536 980 969 419 49 53.0 34 32.3 301 981 000 420 49 45.5 34 19.9 350 981 002 421 49 46.3 34 47.3 235 981 025 422 50 04.2 34 45.6 489 981 005 423 50 13.9 34 52.5 441 981 023 424 50 23.5 34 40.4 339 981 043 425 50 16.5 35 22.9 238 981 081 426 50 07.4 35 03.1 519 981 003 427 49 47.7 35 06.6 550 980 944 428 49 58.0 35 16.2 550 980 999 429 50 05.4 35 22.7 313 981 041 433 49 32.9 35 52.9 406 980 973 435 49 45.1 36 18.3 308 980 972 436 49 34.7 36 26.0 386 980 973 Geoinformatics FCE CTU 14(1), 2015 46