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Tectonic implied results from geoid and topography signals
Ply
Chem .Earth (A). Vol.
25, No
I, pp
0 2000 Elsewer
Pergrmon 1464-I
101-106.2000 Scmce
Ltd
All rights reserved 895/00/$ - see front matter
PII: S1464-lS95(00)00017-X
Tectonic Implied Results from Geoid and Topography Signals M. Doufexopoulou
and V. N. Pagounis
National Technical University of Athens, Department of Rural and Surveying Engineering, Higher Geodesy Laboratory, 9 Heroon Polytechniou, 157 73 Zografou, Athens, Greece Received 23 April 1999: revised 27 September
1999; accepted 111October I999
of the lithosphere
within limited geographical regions Fig.( 1). The causal “sources” in both signals are common in the sense of static position. In geodesy, the effect of massdensity distribution beneath topography is not of direct interest and only topographic density information is considered as “known” in the simplified mass models for the computation of the so called Helmert geoid of Physical Geodesy. In any other geodetic application the density information may be introduced either in numerical form or as an indirectly estimable parameter that has minor physical importance, depending on how one views the relation of geodesy to geophysics. In this work we consider the density distribution within lithosphere as unknown prior information input of a partly known system represented by the lithosphere. The term partly known system defines that simple linear methods are used in the study. By the two geodetic quantities, N derived from GPS points through Eq. (1) and orthometric heights H, as outputs. we examine the fulfillment of linear relation: N=a+bH+&N (2) In the test area we compute their spectra by FFT and Maximum Entropy Method (MEM) for locally observed and global scale data. Equation (2) is expected fulfilled not only in regions that have a smooth variation of geoid but also in those regions where the geoid and topography vary strongly, provided that the considered area lies above a single lithospheric “unit”. When an area extends above more than one lithospheric or tectonic “unit”, this relation will be disturbed most due to the geophysical system than due to the implicit existing relation after Eq. (1) for N and the known dependence of geoid on topography. The linearity of the geophysical system beneath topography is examined by these tests from local and ETOPOS topography H, to local GPS undulations N as well as to truncated undulations by the coefficients of EGM96 model. If the linear Eq. (2) is weak, then the corresponding individual FFT and MEM spectra for these data will also reflect that the linearity of the natural system implying therefore existence of is not fulfilled discontinuity.
The topographic surface is a measure of static equilibrium from the actual density distribution within the outmost Earth’s lithosphere. The natural height reference of this surface, known as geoid reflects the mixed massdensity effects, caused by the same sources, without the contribution of topographic mass. Geoid undulation and topography are output signals. which carry in common a large part of the contribution from the causal “sources”. This contribution appears in both types of signal. Comparisons between the signals depict the geographical location and an estimation of the depth occurrence of areas with geophysical and tectonic formations depending on their correlation rate. We present results from the Greek region, known for its complex diversity in topography, tectonics and dynamics. The tests are in point and “surface” concept, from local and global signals of geoid and topography. Local geoid is represented at 91 GPS points and EGM 96 coefficients compute its global representation. The topography is point values within the area, and the ETOPOS 5x5 data within the geographical frame. 0 2000 Elsevier Science Ltd. All rights reserved
Abstract.
1. Introduction.
Geodesy is characterized by duality between geometry and physics. This is mostly illustrated in the definition of a height system, that can be used in engineering. In this system the observed orthometric heights H relate to the geodetic heights h by the geoid undulation N Eq. (1) if the small tilt E (called deflection of the vertical) between the two “vertical” directions is ignored (Torge, 1988) : h=H+N (1) In this study the two height references, the geoid and MSL are considered coincident, by neglecting the discrepancies that may occur between their definition. The leveled or triangulated height H and the geoid undulation N are considered as signals of a certain mass-density distribution beneath a region. Both quantities are expected to reflect mostly the effects of mass-density within the upper layers Correspondence
to: V. N. Pagounis 101
102
M. Doufexopoulou and V. N. Pagounis: Tectonic Implied Results from Geoid and Topography Signals
are treated as signals. The test area extends about S”X9”. The limited extend is a disadvantage that might introduce uncertainties in parametric evaluations by least square algorithms in data fit. However since tests are restricted only to linear fit, the increasing of uncertainty in results is expected low. On the other hand the diversity of the existing geophysical signals in the particular test area, the scattered depths of their occurrence, known from previous geophysical studies (Doufexopoulou, 1986) have been already depicted by studying in detail local GPS undulations (Doufexopoulou et al., 1997, 1998; Pagounis, 1999). 2. Topography, Geoid and Mass-density input existing (pm)
Figure 1 The reference ofheight definition as input - output system
The local undulations N at 91 GPS points after Eq. (1) imply that the GPS geoid does not depend explicitly on topographic density. They consist the local geoid. The global geoid in this area is the one computed by the coefficients C_, S, of EGM96 model truncated at various degrees of expansion at a regular 5X5’ grid over the test area. The truncation degrees are selected after results of previous tests (Pagotmis, 1999). Possible implication of strong changes of lithospheric mass-density in lateral or horizontal sense within the test area, that might be apparent in coefficients C,, , S, of the geopotential model is expected in principle to be mostly attenuated beyond degree n > 30 of expansion, due to the least squares principle applied in the computation of coefficients. Uncertainties in values of N and H that come from observational or analytical errors are supposed to be below the accuracy rates of the present investigation. The local topography comes from trigonometric leveling at the GPS points, enriched by contribution of 180 point heights of the national triangulation network and a large number of heights along coastal lines. This local information is converted into a mean SX5’ file by interpolation, so that it is comparable to ETOPOS values. Testing the relation between topography H and undulation N provides a first insight to the test region that enables to conclude objectively, whether it is possible to depict the existence of possible geophysical formations without physical or analytical assumptions concerning to the massdensity in an area, which - if used - might cause biased conclusions on the goal. The simplicity, the objectivity and the low cost of the method that uses only geodetic data are a main advantage. Actually without using any tectonic model or analytical assumptions on the local gravity field “signatures” N and H
MSL is the traditional reference of topography. The geoid, is a theoretically equivalent reference, when interpolation problems connected to the coincidence of MSL and geoid are neglected (Sjoberg, 1995). The exact coincidence of MSL and geoid depends on how well sea surface topography has been removed. Both references are related to the Earth’s gravity field. Obviously by referring a height system to the gravity field, the effect of mass-density configuration within the Earth is apparent: a) in the “perpendicular distances” of topography (H) and of the geometric height system (h), b) in the vertical “separation” between two height datum (N). Therefore, undulation N and topography H are indeed interrelated not only through Eq. (1) but mainly through common effects of the mass-density distribution within the upper lithosphere in a test area. The question that arises is how this fact, may be used in connection to geophysical inversion Especially, how is possible to depict information about the existence, the location and the depth, of major density “sources” within lithosphere in a region without to use: a) Assumptions on the tectonic structure and on the gravity field within the area h) With data produced without direct implication of density. The detection of location of possible causal sources is thought in geographical distribution as well as depth rate. The requirement not to involve density is fulfilled by the GPS undulations (Doufexopoulou et al., 1998). ETOPOS and local heights do not carry explicitly density information. The Earth’s lithosphere is considered as the unknown system of which topography H and undulations N are at good approximation two collinear directional outputs. Both signals are formed by common causal “sources” (pm), where p is density and m expresses the volume of mass. These outputs carry a common frequency causal band. The density distribution within topographic masses certainly affects mostly the height H at short wavelengths. But this effect is not expected to introduce important distortion of information on the system’s main structure we seek for, due to the following reason: Topography within limited geographical regions does not introduce signal variability at the geoid wavelengths, which could be significantly
M. Doufexopoulou
and V. N. Pagoums: Tectonic Implied Results from Geoid and Topography
recorded in the local geographical scale. On the other hand a part of the information content from the near to surface causal effects in heights H is suppressed by averaging. The GPS undulations do not have these effects, and they are expected to originate by only deeper causal sources. Thus, the main topographic features of mean topography are expected to be apparent in undulations. Therefore topography and the GPS geoid produced without explicit use of the local gravity field are expected to relate linearly as outputs of mostly a common physical system. Measures of the linear relation classify the information system as not linear, weakly linear and linear. The classification certainly depends on the extension of the area and on data density and accuracy. It must be emphasized that the main importance of this experiment exists in that no explicit causative reasons upon the topography and geoid are used. 3. Testing the relation between surfaces and between profiles. The description of systematic or unexpected variations of a system’s output is done by various methods using 2 dimensional arrays, or along selected profiles (sections). Traditional geophysics uses deterministic in concept relevant functions, following functional stochastic approach (Backus et al.: 1970). In these methods the basic structure configuration is of the mass-density supposed approximately known and one seeks for a data fit to this configuration. A second concept, sharing familiarity in geophysical inversion and in geodesy is the statistical one (Jackson. 1972). In the statistical concept one assumes that the variables are stationary in space, so that any deviation of data from this assumption are to be pre-smoothed or filtered so that the parametric inversion becomes stable for a pre-described physical system. In the present tests we assume that: geoid and topography result as stationary effects of an unknown lithosphere system. The relation between the deterministic” [due to the physical system we seek for to trace], but stochastic in the analytical approach, signals, is checked for the local and global data. The data are: a) The 9 1 GPS points within the test region as well as the trigonometric heights H at these points. b) About 180 point heights from the local triangulation network and a large number of point heights along coastal line. These data were converted with analytic interpolation into local 5x5 mean heights for the present tests. c) The ETOPOS elevations within this area. d) The EGM96 undulations computed at a regular grid SXS at truncations n= 60, 90, 120, 360. e) Sections of topography and geoid, along 2 parallels and 2 meridians for the local and global data. The tests include also a comparison between local 5’X5’ and ETOPO5 topography in the area and a preliminary test by which the least square correlation coefficient of scattered individual GPS points is estimated. The spectral power is
Signals.
103
evaluated along the 4 sections in E-W and N-S directions for EGM96 undulations computed at SX5’ grid and for global and local mean topography by FFT and maximum entropy (MEM) spectra evaluated by the PC academic software of Chaos data analysis (Sprott, 1992). 4. Description of tests At first the feasibility of these tests is examined in the particular area. For this purpose the correlation between 5 samples of GPS undulations and the corresponding local heights is evahtated by Eq. (2). For each configuration of GPS points the least square correlation coefficient r is computed. The coefficient between biased outputs due to Eq. (1) is found for the overall test area, r=-O,90 with FUUSof H estimated +/- 2.8 1 m. For 4 subdivisions of data within W, E, NE, SE, the coefficient varies from ~-0.929 to r=-O.878 (Milios, 1998). For the accuracy level of local heights (i.e. RMS=+/2.8 lm), the correlation is expected much higher. Moreover residuals 6N in the fit for all 5 samples, exceed the 20 m with extreme values 16.6 m in the overall area and -2.28m, -1O.O8m, 20.03m, -11.27 m for the 4 subdivisions. The values of correlation show dependence of the coefficient r on the geographical location of the tested sample and the size of residuals 6N indicates that the causative “sources” may exist in lithosphere. This is in agreement with independent investigations of Pagounis (1999). The comparison between ETOPO5 and local topography 5’X5’ is through the slope rates along N-S and E-W azimuth sections. The slope is computed at grid points and the maximal and minimal slope along N-S and E-W is compared between the local and ETOPO5 files Fig. (2). This test aims to evaluate the overall fit of local mean topography in respect to the ETOPO5. The slope rates between ETOPO5 and local heights show that ETOPO5 appears rougher in maximal slope within the test area. Absolute diflerences between these files vary between 1706 - -2072 m, indicating that a direct comparison would be meaningless.
Figure 2 Di&mces
behwn
ETOPOS and local topography
(units in m).
104
M. Doufexopoulou
and V. N. Pagounis:
Tectonic Implied Results from Geoid and Topography
However, it must be pointed out that there were not available studies on the relation between the two reference datum systems. The results are depicted in Table 1. Both tests support the feasibility in testing the relation of EGM96 geoid undulations to both height sources. Max. slope N-S 0,073 0,053
ETOPGS Locsl Heights Tabk
l.cOmpti
Max slope E-W 0,080 0,046
Min. slope N-S -0,049 -0,049
Min. slope E-W -0,056 -0,052
ox N-S
Dif E-W
0,123 0,102
0,136 0,098
of S-N and E-W slopes for ETOPGS
and local
heighta within thetestregion (units in m).
The relation between topography - geoid is tested then for undulations by the EGM96 model truncated at degrees n 60,90, 120 and at full expansion s360 in a 5x5’ grid in respect to global and local mean topography by investigating the spectra along 4 continental sections Fig. (3). To the undulations along the four sections and for the tested truncations, to ETOPOS and to local 5x5 heights the FFf and MEM spectra were computed (E-W and N-S). Thus each section is characterized by 5 spectra per spectral method: a) the spectmm of ETOPO5, b) of 5’X5’local topography and c) the truncated EGM96 geoid at degrees 60,90,120. Through the location and azimuth of each section one expects to draw conclusions on the lateral lithosphere information. Through the truncation degree of EGM96 geoid one hopes to depict indications on the dominating harmonic degrees that are affected by the local conditions of the lithosphere.
1
I
2ooooo.w
Figure 3 The 4 se&as
Mmo0.00
4c8oooo.00 wwo.00
6wooo.w
7ooorm.w
aooow.00
within thetest area in a local projection.
A previous test of least squares polynomial fit along EGM96 geoid sections (Prodromou, 1999) showed that beyond the fit to 4* degree polynomials, there is no more
Signals
added information either by increasing the degree of the polynomial or by increasing the degree of truncation till full model expansion (n=360). Thus the present investigation of the EGM96 undulation spectra is limited to degree n=120. This degree is possibly the highest one within the overall test area, for which the coefficients of two models OSU9 1A and EGM96 carry local lithosphere signal (Pagotis, 1999). By the polynomial fit, only the zero or&r coefficients of each fit were correlated in all sections among the tnmcation degrees of undulation, in respect to the undulation of the full expansion (n=360). The FFT spectra of EGM96 truncated undulations along sections 1,3 (N-S) and 2,4 (WE) were investigated then to depict the range of expansion degree that shows signal concentration Fig. (4). This was done by following the classic method of spectral interpretation through the location of strongest spectral signal (Jenkins et al., 1968). The values of the found maximum slopes are shown in Table 2: Se&m ofgeoid Degree60 N-S 1 (W) 0,281 N-S 3 (E) 0,397 W-E 4(N) 0,521 W-E 2 (S) 0,395 Table Z.The values of spedral undulations.
Degree90. 0,275 0,382 0,365 0,3s 1 slopes of PFT
Deglee120 0,279 0,372 0,371 0,281 spe&a in EGM96
5. Discussion of results The spectral power from the FFT undulation spectra for the N-S sections (1,3) is at 3-5 times higher level in respect to the power in both W-E sections (2,4). Along N-S, the spectra show concentration of spectral power between expansion degrees 60-90. This is not the case for spectra along W-E sections, which is spread out towards n=120 and beyond. This result may be interpreted objectively by any of the ways: a) The solution for EGM96 coefficients Cm, S, contributes more to the longitude 3Lthan to the latitude cp. If this would be the case with the very tested or any other model, then the spectra of geoid are sensitive to the coefficients of the geopotential model. However in the very test area EGM96 and OSU91A models perform practically the same (Doufexopoulou et al., 1998) and this possibility must be rejected. Local conditions in lithosphere affect strongly the b) values of coefficients Cm, S, at those degrees, implying that the EGM96 geoid indicates a more general natural situation in the test area, that n=60-90 between degrees models carry lithospheric signal which is apparent along the N-S azimuth. In all spectra of geoid the power decreases fast towards higher frequencies, implying that the causal source in lithosphere is rather deep. By taking into consideration the higher spectral power level along N-S sections than those of W-E, one may conclude that the lateral geographical
M. Doufexopoulou
and V. N. Pagounis: Tectonic Implied Results from Geoid and Topography
variation of causal sources of the tested EGM96 geoid appears stronger along N-S.
Signals.
105
frequencies than the spectra of geoid undulations, which appear “noisy” beyond the degree 120. 6. Conclusions.
Fiylp 60,90
4 Indicative
FFI’ spedra
for Section 1 in EGM96 for tnmeati~s
,120(power is in arbitrary units).
This result may be supported by the large variation of depths of crust along N-S after Makris (1977) that are between 15-43 Km within a distance of 4 deg. (-440 km). It must be pointed out that the record length of the 4 sections varies from 109 km (section 2, W-E) to about 240 km (section 1, N-S) so that results might alter slightly in case of comparing equal record lengths. However the comparison of equal record lengths is impossible due to the presence of continental and sea parts. The MEM spectra for all sections of undulations at all truncations show only one prominent peak, that occurs at the same range of wavelengths of the particular section. For the W-E sections the peak is at n=120, although for the N-S sections the peak is at n=60-90. This result may be considered objectively as indicative that either the model’s coefficients are sensitive to longitude 1, that seems not to be the case [see first paragraph] or that the lithospheric signal is deeper along both N-S sections at the continental western part. An attempt to define the prospecting depth for section 1 (N-S, maximal power at n=60) and section 2 (W-E, maximal power at n= 120) by the method of Hahn et al., (1976) gives about depth of “source” after EGM96 undulations 120 km along N-S section and about 50 km along W-E section. The spectra of topography do not show important level of spectral power, especially those for ETOPOS with FFf method. This is also apparent for ETOPOS when comparing the maximum entropy spectra along the 4 sections between ETOPOS and local topography. However, MEM spectra are more detailed in the power distribution, as expected by the very spectral method. A possible reason for the overall poor spectral information of topography can be the rather short record length of all 4 sections. The spectra of local topography show a slight shift of power towards higher frequencies than ETOPOS, keeping though the same form in the distribution of spectral power. This observation agrees with the comparison between the two topographic files in space domain by the slope rates and the observed geographical shift of slopes. The magnitude level of spectral power is higher in W-E sections than along N-S, implying that topographic effects on geoid can be more prominent along W-E. Also the dominating topographic frequencies along section 1 (N-S) are lower than along section 3 (N-S), which is shorter in record length. In general the spectra of topography have power distributed at higher
At present it is not possible to carry detailed correlation or admittance studies from only 91 GPS points and the by the local topographic information that were available in the study of Pagounis (1999). The scattered points do not permit computation of reliable spectra for the local GPS geoid Therefore conclusions of this present work may be based only upon the comparison between ETOPOS and local topography within the test area and on the FFT and maximum entropy spectra of topography and of EGM96 truncated undulations. However, the correlation coefficient between the 5 samples of GPS undulations and local topography varies considerably so that the basic idea on the prospecting capacity of local geodetic data is considered confirmed and new tests using more dense geodetic data must be repeated. The comparison between ETOPOS and local topography shows that: a) ETOPOS in the test area gives higher slopes than local mean topography b) There is a geographical shift of slopes between global and local file towards East. The spectra of topography show that ETOPOS gives very smooth signal along the 4 sections and that the information content of local topography appears slightly more detailed. With both spectral methods (FFT and MEM) despite the limited record length of all 4 sections, one may conclude that the topographic signal gives higher frequencies along the W-E sections that along N-S. The spectra of EGM96 undulations along the same sections permit similar conclusions for both spectral methods: The spectml power between degrees 60-90 is more apparent along N-S, while along W-E sections power exists beyond these degrees, at n=120. In combination to the strongly varying power level between the two azimuths, one may conclude that the causal sources of geoid appear stronger and deeper along N-S than along W-E. The attempt to estimate the dominating prospecting depth along the 4 sections from the MEM spectra of EGM96 geoid by the Hahn et al., (1976) method gives: for section 1 (N-S to West) about 115 km, for 3 (N-S towards East) 84 km for 2 (W-E to North) 50 km and for section 4 (W-E to South) 98 Km. These depths must be considered mostly in sense of their variation among profiles than as absolute values. The indicative prospecting depth values depict that the S-W part of the test area shows definitely deeper lithosphere “sources” than the N-E part. Acknowledgement. We thank Ms Ioanna Prodromou, Diploma student in N.T.U.k, for contributing witb her effedive work a large part of the carried out almputatials.
106
M. Doufexopoulou
and V. N. Pagounis: Tectonic Implied Results from Geoid and Topography Signals.
References Backus, G. and Gilbe&, J. Uniqueness iu the Inversion of Inaccurate Gross E&h data, Publ.Trans.Roy.Soc.L.cmd, 4 266: pp.123-192, 1970. Cam&as de Wit, Adaptive control of partially known systems, Theory and Applications, Stud& in Autom&ion and Control, Elsevia; Co.,1987. Chew&, S.J, Monta~er,P., Snieder, R.K., ‘Ihe spedrum of tomographic Earth Models, Geophysical Journal In&national, 133, pp. 783-788,199s. Jadcson,D.D., Jntespolation of Inaccumte, ~k.&lioient and inconsistent data, Geqh. Joumal ofRoyal A&. Sot., 28: pp 97-110,1972. J&is, G.and Watts, D., Spectral Analysis and its Applications, Holdm Day Publ., 1968. Doutkqoulou, M. and Pagounis, V., First numerical results in deeding heterogeneities iu regional gravity field using GPS uodulatiens, XXII EGS Meeting, Viama, 1997. Doufexopoulou, M. and Pagounis, V., Investigation of tectonic traces upccl tbe q&al geoid without the use of gravity, Reports of the Finnish Geockiic In&u&, 98:4 Second Contiaerdal Workshop on the Geoid in Europe, ISBN 951-711-222-X, pp. 153-160,199s. Hahu, A, Kind, E.G., Mishm, D.C., Depth calculatiou of magnetic sourcea by means of Fourier amplitude spe&a, Geophysical Prosped. 24, pp 287308,1976.
h4akris, J., The aust and upper mantle of Aegean region obtained fkom deep seismic soundings. Publ.Jnst. Gecph. Pol. A&c. A-4 (115), 1977. More& L, and Parsons, B., Interpreting gravity, geoid and topography for amve&mwith temperature depad& viscosity. Applicatims to surface features of Venus, Journal of geophysieal Research 100 (ElO) pp 2115521171,1995. Milig K, Student’s work ia t+ame of lesson “Introduction to the Earth’s gravity field”(unpublished), Dep. of Rural Bc Surveying Eng, NnTA, Athens, 1998. Pagounis, V.N Processing methods of geometrical and physics1 geodata for signal detection. Application to the vertical datum. Doctorial thesis submitted at NTu& 1999. Prodromou, I., Managemart of gravity field information from geopotmtial models. Studmt’s work in frame of lgson “Introductkm to the Earth’s gravity field” (unpublished) Dep. of Rural & Surveying E& NTUA, Athens, 1998. Strykowski, G., and Dahl, O.C, The geoid as an equipdartial surface in a sense of N&m’s intergral Ideas and examples, Reports of the Finnish Geodetic Institute, 9814 Second Continental Workshop on the Geoid in Europe, ISBN 951-711-222-X, pp. 113-116,199s. Woodside, J, Regional vertical te&mics in the Eastern Mediterranean, J.G.R A&. Sot No 47,1976.
Chem .Earth (A). Vol.
25, No
I, pp
0 2000 Elsewer
Pergrmon 1464-I
101-106.2000 Scmce
Ltd
All rights reserved 895/00/$ - see front matter
PII: S1464-lS95(00)00017-X
Tectonic Implied Results from Geoid and Topography Signals M. Doufexopoulou
and V. N. Pagounis
National Technical University of Athens, Department of Rural and Surveying Engineering, Higher Geodesy Laboratory, 9 Heroon Polytechniou, 157 73 Zografou, Athens, Greece Received 23 April 1999: revised 27 September
1999; accepted 111October I999
of the lithosphere
within limited geographical regions Fig.( 1). The causal “sources” in both signals are common in the sense of static position. In geodesy, the effect of massdensity distribution beneath topography is not of direct interest and only topographic density information is considered as “known” in the simplified mass models for the computation of the so called Helmert geoid of Physical Geodesy. In any other geodetic application the density information may be introduced either in numerical form or as an indirectly estimable parameter that has minor physical importance, depending on how one views the relation of geodesy to geophysics. In this work we consider the density distribution within lithosphere as unknown prior information input of a partly known system represented by the lithosphere. The term partly known system defines that simple linear methods are used in the study. By the two geodetic quantities, N derived from GPS points through Eq. (1) and orthometric heights H, as outputs. we examine the fulfillment of linear relation: N=a+bH+&N (2) In the test area we compute their spectra by FFT and Maximum Entropy Method (MEM) for locally observed and global scale data. Equation (2) is expected fulfilled not only in regions that have a smooth variation of geoid but also in those regions where the geoid and topography vary strongly, provided that the considered area lies above a single lithospheric “unit”. When an area extends above more than one lithospheric or tectonic “unit”, this relation will be disturbed most due to the geophysical system than due to the implicit existing relation after Eq. (1) for N and the known dependence of geoid on topography. The linearity of the geophysical system beneath topography is examined by these tests from local and ETOPOS topography H, to local GPS undulations N as well as to truncated undulations by the coefficients of EGM96 model. If the linear Eq. (2) is weak, then the corresponding individual FFT and MEM spectra for these data will also reflect that the linearity of the natural system implying therefore existence of is not fulfilled discontinuity.
The topographic surface is a measure of static equilibrium from the actual density distribution within the outmost Earth’s lithosphere. The natural height reference of this surface, known as geoid reflects the mixed massdensity effects, caused by the same sources, without the contribution of topographic mass. Geoid undulation and topography are output signals. which carry in common a large part of the contribution from the causal “sources”. This contribution appears in both types of signal. Comparisons between the signals depict the geographical location and an estimation of the depth occurrence of areas with geophysical and tectonic formations depending on their correlation rate. We present results from the Greek region, known for its complex diversity in topography, tectonics and dynamics. The tests are in point and “surface” concept, from local and global signals of geoid and topography. Local geoid is represented at 91 GPS points and EGM 96 coefficients compute its global representation. The topography is point values within the area, and the ETOPOS 5x5 data within the geographical frame. 0 2000 Elsevier Science Ltd. All rights reserved
Abstract.
1. Introduction.
Geodesy is characterized by duality between geometry and physics. This is mostly illustrated in the definition of a height system, that can be used in engineering. In this system the observed orthometric heights H relate to the geodetic heights h by the geoid undulation N Eq. (1) if the small tilt E (called deflection of the vertical) between the two “vertical” directions is ignored (Torge, 1988) : h=H+N (1) In this study the two height references, the geoid and MSL are considered coincident, by neglecting the discrepancies that may occur between their definition. The leveled or triangulated height H and the geoid undulation N are considered as signals of a certain mass-density distribution beneath a region. Both quantities are expected to reflect mostly the effects of mass-density within the upper layers Correspondence
to: V. N. Pagounis 101
102
M. Doufexopoulou and V. N. Pagounis: Tectonic Implied Results from Geoid and Topography Signals
are treated as signals. The test area extends about S”X9”. The limited extend is a disadvantage that might introduce uncertainties in parametric evaluations by least square algorithms in data fit. However since tests are restricted only to linear fit, the increasing of uncertainty in results is expected low. On the other hand the diversity of the existing geophysical signals in the particular test area, the scattered depths of their occurrence, known from previous geophysical studies (Doufexopoulou, 1986) have been already depicted by studying in detail local GPS undulations (Doufexopoulou et al., 1997, 1998; Pagounis, 1999). 2. Topography, Geoid and Mass-density input existing (pm)
Figure 1 The reference ofheight definition as input - output system
The local undulations N at 91 GPS points after Eq. (1) imply that the GPS geoid does not depend explicitly on topographic density. They consist the local geoid. The global geoid in this area is the one computed by the coefficients C_, S, of EGM96 model truncated at various degrees of expansion at a regular 5X5’ grid over the test area. The truncation degrees are selected after results of previous tests (Pagotmis, 1999). Possible implication of strong changes of lithospheric mass-density in lateral or horizontal sense within the test area, that might be apparent in coefficients C,, , S, of the geopotential model is expected in principle to be mostly attenuated beyond degree n > 30 of expansion, due to the least squares principle applied in the computation of coefficients. Uncertainties in values of N and H that come from observational or analytical errors are supposed to be below the accuracy rates of the present investigation. The local topography comes from trigonometric leveling at the GPS points, enriched by contribution of 180 point heights of the national triangulation network and a large number of heights along coastal lines. This local information is converted into a mean SX5’ file by interpolation, so that it is comparable to ETOPOS values. Testing the relation between topography H and undulation N provides a first insight to the test region that enables to conclude objectively, whether it is possible to depict the existence of possible geophysical formations without physical or analytical assumptions concerning to the massdensity in an area, which - if used - might cause biased conclusions on the goal. The simplicity, the objectivity and the low cost of the method that uses only geodetic data are a main advantage. Actually without using any tectonic model or analytical assumptions on the local gravity field “signatures” N and H
MSL is the traditional reference of topography. The geoid, is a theoretically equivalent reference, when interpolation problems connected to the coincidence of MSL and geoid are neglected (Sjoberg, 1995). The exact coincidence of MSL and geoid depends on how well sea surface topography has been removed. Both references are related to the Earth’s gravity field. Obviously by referring a height system to the gravity field, the effect of mass-density configuration within the Earth is apparent: a) in the “perpendicular distances” of topography (H) and of the geometric height system (h), b) in the vertical “separation” between two height datum (N). Therefore, undulation N and topography H are indeed interrelated not only through Eq. (1) but mainly through common effects of the mass-density distribution within the upper lithosphere in a test area. The question that arises is how this fact, may be used in connection to geophysical inversion Especially, how is possible to depict information about the existence, the location and the depth, of major density “sources” within lithosphere in a region without to use: a) Assumptions on the tectonic structure and on the gravity field within the area h) With data produced without direct implication of density. The detection of location of possible causal sources is thought in geographical distribution as well as depth rate. The requirement not to involve density is fulfilled by the GPS undulations (Doufexopoulou et al., 1998). ETOPOS and local heights do not carry explicitly density information. The Earth’s lithosphere is considered as the unknown system of which topography H and undulations N are at good approximation two collinear directional outputs. Both signals are formed by common causal “sources” (pm), where p is density and m expresses the volume of mass. These outputs carry a common frequency causal band. The density distribution within topographic masses certainly affects mostly the height H at short wavelengths. But this effect is not expected to introduce important distortion of information on the system’s main structure we seek for, due to the following reason: Topography within limited geographical regions does not introduce signal variability at the geoid wavelengths, which could be significantly
M. Doufexopoulou
and V. N. Pagoums: Tectonic Implied Results from Geoid and Topography
recorded in the local geographical scale. On the other hand a part of the information content from the near to surface causal effects in heights H is suppressed by averaging. The GPS undulations do not have these effects, and they are expected to originate by only deeper causal sources. Thus, the main topographic features of mean topography are expected to be apparent in undulations. Therefore topography and the GPS geoid produced without explicit use of the local gravity field are expected to relate linearly as outputs of mostly a common physical system. Measures of the linear relation classify the information system as not linear, weakly linear and linear. The classification certainly depends on the extension of the area and on data density and accuracy. It must be emphasized that the main importance of this experiment exists in that no explicit causative reasons upon the topography and geoid are used. 3. Testing the relation between surfaces and between profiles. The description of systematic or unexpected variations of a system’s output is done by various methods using 2 dimensional arrays, or along selected profiles (sections). Traditional geophysics uses deterministic in concept relevant functions, following functional stochastic approach (Backus et al.: 1970). In these methods the basic structure configuration is of the mass-density supposed approximately known and one seeks for a data fit to this configuration. A second concept, sharing familiarity in geophysical inversion and in geodesy is the statistical one (Jackson. 1972). In the statistical concept one assumes that the variables are stationary in space, so that any deviation of data from this assumption are to be pre-smoothed or filtered so that the parametric inversion becomes stable for a pre-described physical system. In the present tests we assume that: geoid and topography result as stationary effects of an unknown lithosphere system. The relation between the deterministic” [due to the physical system we seek for to trace], but stochastic in the analytical approach, signals, is checked for the local and global data. The data are: a) The 9 1 GPS points within the test region as well as the trigonometric heights H at these points. b) About 180 point heights from the local triangulation network and a large number of point heights along coastal line. These data were converted with analytic interpolation into local 5x5 mean heights for the present tests. c) The ETOPOS elevations within this area. d) The EGM96 undulations computed at a regular grid SXS at truncations n= 60, 90, 120, 360. e) Sections of topography and geoid, along 2 parallels and 2 meridians for the local and global data. The tests include also a comparison between local 5’X5’ and ETOPO5 topography in the area and a preliminary test by which the least square correlation coefficient of scattered individual GPS points is estimated. The spectral power is
Signals.
103
evaluated along the 4 sections in E-W and N-S directions for EGM96 undulations computed at SX5’ grid and for global and local mean topography by FFT and maximum entropy (MEM) spectra evaluated by the PC academic software of Chaos data analysis (Sprott, 1992). 4. Description of tests At first the feasibility of these tests is examined in the particular area. For this purpose the correlation between 5 samples of GPS undulations and the corresponding local heights is evahtated by Eq. (2). For each configuration of GPS points the least square correlation coefficient r is computed. The coefficient between biased outputs due to Eq. (1) is found for the overall test area, r=-O,90 with FUUSof H estimated +/- 2.8 1 m. For 4 subdivisions of data within W, E, NE, SE, the coefficient varies from ~-0.929 to r=-O.878 (Milios, 1998). For the accuracy level of local heights (i.e. RMS=+/2.8 lm), the correlation is expected much higher. Moreover residuals 6N in the fit for all 5 samples, exceed the 20 m with extreme values 16.6 m in the overall area and -2.28m, -1O.O8m, 20.03m, -11.27 m for the 4 subdivisions. The values of correlation show dependence of the coefficient r on the geographical location of the tested sample and the size of residuals 6N indicates that the causative “sources” may exist in lithosphere. This is in agreement with independent investigations of Pagounis (1999). The comparison between ETOPO5 and local topography 5’X5’ is through the slope rates along N-S and E-W azimuth sections. The slope is computed at grid points and the maximal and minimal slope along N-S and E-W is compared between the local and ETOPO5 files Fig. (2). This test aims to evaluate the overall fit of local mean topography in respect to the ETOPO5. The slope rates between ETOPO5 and local heights show that ETOPO5 appears rougher in maximal slope within the test area. Absolute diflerences between these files vary between 1706 - -2072 m, indicating that a direct comparison would be meaningless.
Figure 2 Di&mces
behwn
ETOPOS and local topography
(units in m).
104
M. Doufexopoulou
and V. N. Pagounis:
Tectonic Implied Results from Geoid and Topography
However, it must be pointed out that there were not available studies on the relation between the two reference datum systems. The results are depicted in Table 1. Both tests support the feasibility in testing the relation of EGM96 geoid undulations to both height sources. Max. slope N-S 0,073 0,053
ETOPGS Locsl Heights Tabk
l.cOmpti
Max slope E-W 0,080 0,046
Min. slope N-S -0,049 -0,049
Min. slope E-W -0,056 -0,052
ox N-S
Dif E-W
0,123 0,102
0,136 0,098
of S-N and E-W slopes for ETOPGS
and local
heighta within thetestregion (units in m).
The relation between topography - geoid is tested then for undulations by the EGM96 model truncated at degrees n 60,90, 120 and at full expansion s360 in a 5x5’ grid in respect to global and local mean topography by investigating the spectra along 4 continental sections Fig. (3). To the undulations along the four sections and for the tested truncations, to ETOPOS and to local 5x5 heights the FFf and MEM spectra were computed (E-W and N-S). Thus each section is characterized by 5 spectra per spectral method: a) the spectmm of ETOPO5, b) of 5’X5’local topography and c) the truncated EGM96 geoid at degrees 60,90,120. Through the location and azimuth of each section one expects to draw conclusions on the lateral lithosphere information. Through the truncation degree of EGM96 geoid one hopes to depict indications on the dominating harmonic degrees that are affected by the local conditions of the lithosphere.
1
I
2ooooo.w
Figure 3 The 4 se&as
Mmo0.00
4c8oooo.00 wwo.00
6wooo.w
7ooorm.w
aooow.00
within thetest area in a local projection.
A previous test of least squares polynomial fit along EGM96 geoid sections (Prodromou, 1999) showed that beyond the fit to 4* degree polynomials, there is no more
Signals
added information either by increasing the degree of the polynomial or by increasing the degree of truncation till full model expansion (n=360). Thus the present investigation of the EGM96 undulation spectra is limited to degree n=120. This degree is possibly the highest one within the overall test area, for which the coefficients of two models OSU9 1A and EGM96 carry local lithosphere signal (Pagotis, 1999). By the polynomial fit, only the zero or&r coefficients of each fit were correlated in all sections among the tnmcation degrees of undulation, in respect to the undulation of the full expansion (n=360). The FFT spectra of EGM96 truncated undulations along sections 1,3 (N-S) and 2,4 (WE) were investigated then to depict the range of expansion degree that shows signal concentration Fig. (4). This was done by following the classic method of spectral interpretation through the location of strongest spectral signal (Jenkins et al., 1968). The values of the found maximum slopes are shown in Table 2: Se&m ofgeoid Degree60 N-S 1 (W) 0,281 N-S 3 (E) 0,397 W-E 4(N) 0,521 W-E 2 (S) 0,395 Table Z.The values of spedral undulations.
Degree90. 0,275 0,382 0,365 0,3s 1 slopes of PFT
Deglee120 0,279 0,372 0,371 0,281 spe&a in EGM96
5. Discussion of results The spectral power from the FFT undulation spectra for the N-S sections (1,3) is at 3-5 times higher level in respect to the power in both W-E sections (2,4). Along N-S, the spectra show concentration of spectral power between expansion degrees 60-90. This is not the case for spectra along W-E sections, which is spread out towards n=120 and beyond. This result may be interpreted objectively by any of the ways: a) The solution for EGM96 coefficients Cm, S, contributes more to the longitude 3Lthan to the latitude cp. If this would be the case with the very tested or any other model, then the spectra of geoid are sensitive to the coefficients of the geopotential model. However in the very test area EGM96 and OSU91A models perform practically the same (Doufexopoulou et al., 1998) and this possibility must be rejected. Local conditions in lithosphere affect strongly the b) values of coefficients Cm, S, at those degrees, implying that the EGM96 geoid indicates a more general natural situation in the test area, that n=60-90 between degrees models carry lithospheric signal which is apparent along the N-S azimuth. In all spectra of geoid the power decreases fast towards higher frequencies, implying that the causal source in lithosphere is rather deep. By taking into consideration the higher spectral power level along N-S sections than those of W-E, one may conclude that the lateral geographical
M. Doufexopoulou
and V. N. Pagounis: Tectonic Implied Results from Geoid and Topography
variation of causal sources of the tested EGM96 geoid appears stronger along N-S.
Signals.
105
frequencies than the spectra of geoid undulations, which appear “noisy” beyond the degree 120. 6. Conclusions.
Fiylp 60,90
4 Indicative
FFI’ spedra
for Section 1 in EGM96 for tnmeati~s
,120(power is in arbitrary units).
This result may be supported by the large variation of depths of crust along N-S after Makris (1977) that are between 15-43 Km within a distance of 4 deg. (-440 km). It must be pointed out that the record length of the 4 sections varies from 109 km (section 2, W-E) to about 240 km (section 1, N-S) so that results might alter slightly in case of comparing equal record lengths. However the comparison of equal record lengths is impossible due to the presence of continental and sea parts. The MEM spectra for all sections of undulations at all truncations show only one prominent peak, that occurs at the same range of wavelengths of the particular section. For the W-E sections the peak is at n=120, although for the N-S sections the peak is at n=60-90. This result may be considered objectively as indicative that either the model’s coefficients are sensitive to longitude 1, that seems not to be the case [see first paragraph] or that the lithospheric signal is deeper along both N-S sections at the continental western part. An attempt to define the prospecting depth for section 1 (N-S, maximal power at n=60) and section 2 (W-E, maximal power at n= 120) by the method of Hahn et al., (1976) gives about depth of “source” after EGM96 undulations 120 km along N-S section and about 50 km along W-E section. The spectra of topography do not show important level of spectral power, especially those for ETOPOS with FFf method. This is also apparent for ETOPOS when comparing the maximum entropy spectra along the 4 sections between ETOPOS and local topography. However, MEM spectra are more detailed in the power distribution, as expected by the very spectral method. A possible reason for the overall poor spectral information of topography can be the rather short record length of all 4 sections. The spectra of local topography show a slight shift of power towards higher frequencies than ETOPOS, keeping though the same form in the distribution of spectral power. This observation agrees with the comparison between the two topographic files in space domain by the slope rates and the observed geographical shift of slopes. The magnitude level of spectral power is higher in W-E sections than along N-S, implying that topographic effects on geoid can be more prominent along W-E. Also the dominating topographic frequencies along section 1 (N-S) are lower than along section 3 (N-S), which is shorter in record length. In general the spectra of topography have power distributed at higher
At present it is not possible to carry detailed correlation or admittance studies from only 91 GPS points and the by the local topographic information that were available in the study of Pagounis (1999). The scattered points do not permit computation of reliable spectra for the local GPS geoid Therefore conclusions of this present work may be based only upon the comparison between ETOPOS and local topography within the test area and on the FFT and maximum entropy spectra of topography and of EGM96 truncated undulations. However, the correlation coefficient between the 5 samples of GPS undulations and local topography varies considerably so that the basic idea on the prospecting capacity of local geodetic data is considered confirmed and new tests using more dense geodetic data must be repeated. The comparison between ETOPOS and local topography shows that: a) ETOPOS in the test area gives higher slopes than local mean topography b) There is a geographical shift of slopes between global and local file towards East. The spectra of topography show that ETOPOS gives very smooth signal along the 4 sections and that the information content of local topography appears slightly more detailed. With both spectral methods (FFT and MEM) despite the limited record length of all 4 sections, one may conclude that the topographic signal gives higher frequencies along the W-E sections that along N-S. The spectra of EGM96 undulations along the same sections permit similar conclusions for both spectral methods: The spectml power between degrees 60-90 is more apparent along N-S, while along W-E sections power exists beyond these degrees, at n=120. In combination to the strongly varying power level between the two azimuths, one may conclude that the causal sources of geoid appear stronger and deeper along N-S than along W-E. The attempt to estimate the dominating prospecting depth along the 4 sections from the MEM spectra of EGM96 geoid by the Hahn et al., (1976) method gives: for section 1 (N-S to West) about 115 km, for 3 (N-S towards East) 84 km for 2 (W-E to North) 50 km and for section 4 (W-E to South) 98 Km. These depths must be considered mostly in sense of their variation among profiles than as absolute values. The indicative prospecting depth values depict that the S-W part of the test area shows definitely deeper lithosphere “sources” than the N-E part. Acknowledgement. We thank Ms Ioanna Prodromou, Diploma student in N.T.U.k, for contributing witb her effedive work a large part of the carried out almputatials.
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and V. N. Pagounis: Tectonic Implied Results from Geoid and Topography Signals.
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h4akris, J., The aust and upper mantle of Aegean region obtained fkom deep seismic soundings. Publ.Jnst. Gecph. Pol. A&c. A-4 (115), 1977. More& L, and Parsons, B., Interpreting gravity, geoid and topography for amve&mwith temperature depad& viscosity. Applicatims to surface features of Venus, Journal of geophysieal Research 100 (ElO) pp 2115521171,1995. Milig K, Student’s work ia t+ame of lesson “Introduction to the Earth’s gravity field”(unpublished), Dep. of Rural Bc Surveying Eng, NnTA, Athens, 1998. Pagounis, V.N Processing methods of geometrical and physics1 geodata for signal detection. Application to the vertical datum. Doctorial thesis submitted at NTu& 1999. Prodromou, I., Managemart of gravity field information from geopotmtial models. Studmt’s work in frame of lgson “Introductkm to the Earth’s gravity field” (unpublished) Dep. of Rural & Surveying E& NTUA, Athens, 1998. Strykowski, G., and Dahl, O.C, The geoid as an equipdartial surface in a sense of N&m’s intergral Ideas and examples, Reports of the Finnish Geodetic Institute, 9814 Second Continental Workshop on the Geoid in Europe, ISBN 951-711-222-X, pp. 113-116,199s. Woodside, J, Regional vertical te&mics in the Eastern Mediterranean, J.G.R A&. Sot No 47,1976.