GEOID COMPUTATIONS BASED ON
advertisement
GEOID COMPUTATIONS BASED ON TORSION BALANCE MEASUREMENTS Lajos VÖLGYESI Technical University of Budapest Department of Geodesy Abstract A very large part of the area of Hungary - first of all flatland and hilly areas of moderate height - was covered with a network of torsion balance stations. As far as torsion balance measurements are considered Hungary is the most well measured country in the world. This gives us a good possibility to interpolate a very dense net of deflections of the vertical from gravity gradients and applying astronomical levelling to compute geoid heights. Results of the computations in a Hungarian test area show mean square errors below a decimetre of geoid undulations computed in this way. Introduction Nowadays based on up-to-date geodetic measurement results, contour line maps of the main geoid forms for the whole Earth are available. These global geoid forms - having a few metres accuracy - however, do not contain the "fine structure" of the geoid, while we need at least centimetres accuracy for the practical geodetic purposes (e.g. determination of height above see level from GPS measurement). A dense net of deflections of the vertical interpolated from gravity gradients measured by torsion balance gives us a good possibility to determine a detailed local geoid map. As known, there exists a definite mathematical relationship between the geoid undulations and the deflections of the vertical (BIRÓ, 1982). Between any points Pi and Pk the geoid height change is k N ik cos sin ds (1) i where is the azimuth of the line of length s connecting the two points. For using this line integral we would have to know the function of deflections of the vertical between the points. In practice the integral (1) is to be evaluated by a numerical integration: k k N ik i cos ik i sin ik sik . 2 2 (2) Generally we want to compute the difference N not only between two points but we need a detailed geoid map for a larger territory. If we want to apply astronomical levelling for a larger territory, the deflection components must be given at enough stations such that the interpolation between these stations can be done reliably. In practice we have a sparser net of astronomical stations, and this astrogeodetic net is interpolated by different methods. There is a very good possibility to interpolate deflections of the vertical from gravity gradients where torsion balance measurements are available. A very simple relationship based on potential theory can be written for the changes of ik and ik between arbitrary points i and k of the deflection of the vertical components and as well as for gravity gradients W and Wxy measured by torsion balance: ik sin ki ik cos ki nik W U i W U k sin 2 ki Wxy U xy i Wxy U xy k 2 cos 2 ki 4g (3) where nik is the distance between points i and k , g is the average value of gravity between them, U and U xy are gravity gradients in the normal gravity field, whereas ki is the direction azimuth between the two points (VÖLGYESI, 1993, 1995). The computation being fundamentally an integration, practically possible only by approximation, in deriving (3) it had to be supposed that the change of gravity gradients between points i and k , measurable by torsion balance, was linear thus the equality sign in (3) is valid only for this case (VÖLGYESI, 1993). If we have a denser net of deflections of the vertical, and we want to apply astronomical levelling for a larger territory, we should first interpolate deflection of the vertical components and from an arbitrary shaped network points to grid points of a square-shaped network. Then applying equation (2) we can compute differences of geoid heights, along sets of points of the square-shaped network in both east-west and northsouth profiles. Going around each square of the square-shaped network and summarising N differences we can get misclosures. These misclosures can be adjusted as if they were the misclosures of the well known geometrical levelling. Results of this computation will be the adjusted N differences between the adjacent square-shaped grid points. If an 2 initial geoid height N 0 is given at an arbitrary point of the grid we can get the final geoid heights N i at point i , summarising the suitable N differences. A computer program package developed by us is able to determine deflections of the vertical based on torsion balance measurements either along triangulation chains or in networks covering arbitrary areas using any of the interpolation methods fully described in (VÖLGYESI, 1993). It can plot the interpolation network and vector diagram of interpolated deflections of the vertical, calculate geoid heights by astronomic levelling and also plot either a perspective or an isoline map of the geoid for the area. Test computations A characteristic Hungarian area measured by torsion balance extended over some 1200 2 km was chosen for the purpose of our test computation. The torsion balance measurement points' location in our test area is displayed in Fig. 1. Stations were not located with the same density because the observations were carried out with a greater density of points in "disturbed" areas of rugged topography. In Figs. 2 and 3 gravity gradients W and Wxy measured by torsion balance were plotted on isoline maps. 30000 10 10 872 876 438 440 424420 423 427 428432 422316 415419421 426 318 314 288 417 430434320 411413 728 727 726 725 724 723 710 707 706 698 694 735 729 731 732 722 721 709 27 705 699 695 636 692 612 553 737 736 730 774 773 713 712 711 704 700 696 637 632 633 625 611 852 5957 624 610 613 641 608 686 776 775 796 772 714 760 703 701 697 715 638 631 777 607 853 807 797 639 640 630 609 614 575 574 765 763 761 750 719 717 716 20 5956 5958 577 576 518 615 806 798 804 764 762 749 720 745 718 666 771 579 578 5955 595 623 622 619 616 570 5951 779 805 803 758 759 748 747 746 753 770 596 5950 594 621 620 617 814 757 751 754 752 767 769 598 597 5952 5954 811 802 5930 5949 13 756 755 786 766 676 812 810 813 20000 5931 10000 5934 5878 5935 5876 30 14 5933 5936 5938 5945 5937 5939 5947 5953 820 815 817 808 788 787 785 5946 821 819 818 816 789 855 5944 5940 0 0 10000 20000 30000 40000 50000 Fig. 1. Torsion balance stations Six points can be found in the test area where x , h deflections of the vertical are known. Each of these points is such that gravimetric deflections of the vertical are available 3 based on gravity data; four of them (points labelled 10, 20, 30, and 27) are astrogeodetic points. Points 10, 20 and 30 are fixed points of interpolation, points 13, 14 and 27 served the purpose of checking interpolated values. The accuracy of relative deflections of the vertical at the astrogeodetic stations 10, 20, 30 and 27 can be described by the standard error of astronomic position determinations, which is o o 0.2 . The interpolation network in Fig. 1 has 206 points in all and 203 of these are points with unknown deflections. Since there are two unknown components of deflection of the vertical at each point there are 406 unknowns for which 558 equations can be written. In Figs. 4 and 5 x and h components of deflections of the vertical are visualized in isoline maps that resulted from the computation. Besides this it is given in Table 1 how large deviations arose at checkpoints between computed and known x , h values. Standard m 0.60 m 0.65 , computed from these departures at deviations and checkpoints corroborate the fact that even for large continuous x , h values of acceptable accuracy can be computed from torsion balance measurements. If for any point of the investigated area the initial value of geoid height N 0 is known, further if adequately interpolated deflection values are available for this particular area, the detailed map of the geoid can be obtained for the given area. Based on the above, computations were carried out for our experimental area shown in Fig. 1, using the deflection components interpolated previously. Table 1 " -0.69 +0.54 +0.55 0.60 Checking point 27 13 14 Standard deviations " -0.51 +0.96 +0.29 0.65 As on this territory no geoid height related to the same reference surface as the given deflection components was available, the geoid height N 0 for the experimental area in Fig. 1 was chosen arbitrarily to be 40.00 m in the astrogeodetic point 30, to serve as basis for determining geoid heights of further points. The mentioned geoid map is presented in Fig. 6. It is characteristic for the accuracy of the computations that going along the chains 30 10, 10 20, 20 30, and returning to point 30, instead of the initial value N30 = 40.00 m, N30 = 39.93 m was obtained, - i.e. going around the chain length of about 115 km, a misclosure of only 7 cm was obtained. This misclosure is characteristic not only of the accuracy of the geoid height, but is at the same time an excellent possibility to check the confidence of the interpolated and values. Therefore deflection values interpolated from torsion balance observations can be stated to give very economically highly reliable geoid maps which are most suitable to study local details. 4 10872 438876 440424 420 427 423428 432 422 426 316 415419 421 434320 318314288 411 413417 430 728 735 737 20000 726 727 736 732 731 729 730 774 776 775 853 807 797 5955 5951 798 779 5954 5952 5949 10000 14 5933 5953 5947 820 821 5936 5935 5878 5876 705 27 711 694 706 704 698 699 700 636 695 637 696 764 811 819 818 816 755 788 808 789 745 752 786 787 718 666 753 771 598 625 624 631 630 609 611 610 612 613 614 553 608 641 686 574 607 575 20 578 579 596 770 769 767 766 633 640 639 716 746 747 754 751 757 817 748 759 717 719 720 749 13 756 810 815 762 758 802 750 761 638 715 697 701 703 760 692 632 595 597 623 594 577 576 622 621 518 619 620 615 616 570 617 676 785 855 5946 5945 5937 5938 712 714 763 803 812 813 5934 772 804 805 814 5930 713 707 710 709 721 773 765 5956 806 5931 722 796 777 5950 723 852 5957 5958 724 725 5944 30 5939 0 0 5940 10000 20000 30000 Fig. 2. W 5 40000 50000 10872 438876 440424 420 427 423428 432 422 426 316 415419 421 434320 318314288 411 413417 430 723 725 724 728 727 726 735 737 20000 736 732 731 729 730 774 776 775 807 797 5955 5951 798 779 5954 5952 5949 10000 14 5933 5953 5947 820 821 5936 5935 5878 5876 711 704 698 699 700 636 695 811 819 818 816 755 788 808 789 745 752 786 787 666 753 766 598 630 609 611 610 612 613 614 553 608 641 686 574 607 575 20 578 579 596 770 769 767 771 625 624 631 640 639 716 718 746 747 754 751 757 817 748 717 719 720 749 759 13 756 810 815 762 758 802 750 761 638 715 697 701 703 760 633 632 637 696 692 595 597 623 594 577 576 622 621 518 619 620 615 616 570 617 676 785 855 5946 5945 5937 5938 803 812 813 5934 764 804 805 814 5930 5931 712 714 763 765 5956 806 772 796 777 853 5950 713 773 705 27 709 721 694 706 852 5957 5958 722 707 710 5944 30 5939 0 0 5940 10000 20000 30000 Fig. 3. W xy 6 40000 50000 30000 10872 438876 440424 420 427 423428 432 422 426 316 415419 421 434320 318314288 411 413417 430 723 725 724 728 727 726 735 737 730 774 776 775 807 797 5955 5951 798 779 5954 5952 5949 5934 14 5933 5953 5947 820 821 5936 5935 5878 5876 711 704 698 699 700 636 695 811 817 815 819 818 816 789 752 786 787 666 753 767 766 771 769 630 598 609 611 610 612 613 614 553 608 641 686 574 607 575 20 578 579 596 770 625 624 631 640 639 716 718 746 754 755 788 808 745 747 748 751 757 13 756 810 717 719 720 749 759 758 802 750 761 762 638 715 697 701 703 760 633 632 637 696 692 595 597 623 594 577 576 622 621 518 619 620 615 616 570 617 676 785 855 5946 5945 5937 5938 763 803 812 813 10000 712 714 764 804 805 814 5930 5931 772 765 5956 806 5950 713 705 27 709 721 773 796 777 853 5958 722 694 706 852 5957 20000 732 731 729 736 707 710 5944 30 5939 0 0 5940 10000 20000 30000 Fig. 4. Deflection of the vertical ( component) 7 40000 50000 30000 10872 438876 440424 420 427 423428 432 422 426 316 415419 421 434320 318314288 411 413417 430 728 735 737 730 774 776 853 775 807 5955 5951 798 779 5934 5954 5952 5949 14 5933 5953 5947 5876 705 27 711 694 706 704 698 699 700 636 695 637 696 803 811 762 757 810 817 815 819 818 808 752 786 755 787 718 666 753 767 766 771 598 625 624 631 630 609 611 610 612 613 614 553 608 641 686 574 607 575 20 578 579 596 770 769 633 640 639 716 746 754 788 789 816 745 747 748 751 13 756 717 719 720 749 759 758 802 750 761 638 715 697 701 703 760 692 632 595 597 623 594 577 576 622 621 518 619 620 615 616 570 617 676 785 855 5946 5945 5937 5938 764 804 812 820 821 5936 5935 5878 712 714 763 765 805 813 10000 713 707 710 709 721 772 796 797 814 5930 5931 722 773 5956 806 5950 723 852 777 5958 724 725 732 731 729 736 5957 20000 726 727 5944 30 5939 0 0 5940 10000 20000 30000 Fig. 4. Deflection of the vertical ( component) 8 40000 50000 30000 10872 438876 440424 420 427 423428 432 422 426 316 415419 421 434320 318314288 411 413417 430 723 725 724 728 727 726 735 737 20000 732 731 729 736 730 774 776 775 853 807 797 5955 5951 798 779 5954 5952 5949 10000 14 5933 5953 5947 820 821 5936 5935 5878 5876 711 704 698 699 700 636 695 811 819 818 816 755 788 808 789 745 752 786 787 718 666 753 766 598 630 609 611 610 612 613 614 553 608 641 686 574 607 575 20 578 579 596 770 769 767 771 625 624 631 640 639 716 746 747 754 751 757 817 748 759 717 719 720 749 13 756 810 815 762 758 802 750 761 638 715 697 701 703 760 633 632 637 696 692 595 597 623 594 577 576 622 621 518 619 620 615 616 570 617 676 785 855 5946 5945 5937 5938 803 812 813 5934 764 804 805 814 5930 5931 712 714 763 765 5956 806 772 796 777 5950 713 773 705 27 709 721 694 706 852 5957 5958 722 707 710 5944 30 5939 0 0 5940 10000 20000 30000 40000 50000 Fig. 4. Geoid heights Our investigations were supported by the Hungarian National Research Fund (OTKA), contract No. T-015960; and the Geodesy and Geodynamics Research Group of the Hungarian Academy of Sciences. 9 References Biró, P.: Geodesy*. University Lecture Notes. Technical University of Budapest. Tankönyvkiadó, Budapest, pp. 1-196 (1982) Völgyesi, L.: Interpolation of Deflection of the Vertical in Hungary Based on Gravity Gradients. Periodica Polytechnica Ser. Civil Eng. Vol.37. No.2, pp. 137-166 (1993) Völgyesi, L.: Test Interpolation of Deflection of the Vertical in Hungary Based on Gravity Gradients. Periodica Polytechnica Ser. Civil Eng. Vol.39. No.1, pp. 37-75 (1995) * In Hungarian *** Völgyesi L (1998): Geoid Computations Based on Torsion Balance Measurements. Reports of the Finnish Geodetic Institute, 98:4, pp.145-151. Dr. Lajos VÖLGYESI, Department of Geodesy and Surveying, Budapest University of Technology and Economics, H-1521 Budapest, Hungary, Műegyetem rkp. 3. Web: http://sci.fgt.bme.hu/volgyesi E-mail: volgyesi@eik.bme.hu 10
