Journal of Geomatics Science and Technology

fa مدلسازی توأم میدان ثقل و توزیع دانسیته جرمی بر روی سطح زمین با استفاده از تلفیق دو نوع مسائل معکوس گراویمتری ژئودزی و ژئوفیزیک: مطالعه موردی منطقه فارس ساحلی A Unique Solution to Two Classic Gravimetry Inverse Problems for Simultaneously Doing ‘Local Gravity Field Modeling’ and ‘Earth’s Mass-Density Determination’, a Case Study of the Coastal Fars of Iran ژئودزی و هیدروگرافی Geo&Hydro پژوهشي Research هدف اصلی این تحقیق آن می&lrm;باشد که با تلفیق مسائل معکوس گراویمتری ژئودزی و ژئوفیزیک امکان مدلسازی همزمان "میدان ثقل" و "توزیع دانسیته جرمی" برروی سطح زمین را فراهم آورد. در فیزیکال ژئودزی هدف از طرح مسئله معکوس گراویمتری، مدلسازی دقیق تابعک پتانسیل ثقل و مشتقات آن می&lrm;باشد و در آن طرف در مباحث ژئوفیزیک اکتشافی، عموماً مسئله معکوس گراویمتری با هدف تعیین توزیع دانسیته جرمی زمین طرح می&lrm;گردد؛ در این تحقیق می&lrm;بینیم که از دو منظر ریاضی و فیزیک، وصول به این دو هدف می&lrm;تواند در حل یک مسئله معکوس گراویمتری تلفیقی، خلاصه و محقق گردد. بدین منظور، تئوری&lrm;، مفاهیم و تشابه این دو نوع مسئله معکوس گراویمتری مورد بررسی قرار گرفته و ارتباط بین پارامترهای مجهول این دو نوع مسئله معکوس گراویمتری مشخص خواهد گردید. سپس روشی برای مدلسازی تابعک پتانسیل میدان جاذبه برروی سطح زمین، بر مبنای توابع پایه شعاعی غیرهارمونیک و پایدارسازی تیخونوف تعمیم&lrm;یافته پیشنهاد می&lrm;گردد و با استفاده از روش پیشنهادی و تلفیق معادله دیفرانسیلی پواسن و معادله انتگرالی نیوتن امکان استخراج تابعک دانسیته از مشاهدات ثقلی زمینی فراهم می&lrm;آید. در انتها نیز در یک مطالعه موردی، روش پیشنهادشده با استفاده از 6350 ایستگاه ثقلی در منطقه فارس ساحلی مورد آزمایش عددی قرارگرفته می&lrm;شود. نتایج عددی حکایت از آن دارد که روش پیشنهادشده امکان مدلسازی توأم میدان جاذبه و توزیع دانسیته جرمی در منطقه مورد مطالعه را به ترتیب با دقتهای 0.54 میلی&lrm;گال و 0.25 گرم&lrm;بر&lrm;سانتیمترمکعب فرآهم نموده است.   This paper aims at combining the geodetic gravimetric inverse problem with the geophysical one to do ‘local gravity field modeling’ and ‘topographic mass-density anomaly determination’ simultaneously.  For this purpose, it recalls the basic theories of the two types of gravimetric inverse problems and finds certain relations between their corresponding unknown parameters. The current research proposes a method, which is based on certain anharmonic Radial Basis Functions (RBFs) as well as Generalized Tikhonov regularization method, for modeling the gravity field at the Earth’s surface. Then, it shows that the proposed method can provide certain formulae, based on a combination of the Newton’s IE and the Poisson’s PDE, for extracting the mass-density anomalies from land-based gravity data. Finally, the proposed method is numerically examined by a case study at which the proposed method has been applied to 6350 gravity stations scattered over the coastal Fars of Iran in the north of the Persian Gulf. The case study numerically indicates the possibility of the main idea of the research for simultaneously solving the two types of gravimetry inverse problem. <ol> <li>Gravimetric inverse problems</li> </ol> From point of view of mathematics, a geophysical gravimetric inverse problem can be defined by the following equation: where the mass-density anomaly is considered as the unknown and defined by subtracting the mass-density functional  from its  prior knowledge ;  is the Newton integral operator;   is the gravity anomaly yielded by applying the gravity reduction to land-based gravity observations. In a local coordinate system which z-axis is a tangent to the plumb line and outwards from the Earth’s surface, the above geophysical gravimetric inverse problem (i.e. Eq.) can be re-written as follows: The above equation is a type of the Newton’s IE at which  is the position vector,  is the topographic space of the region of the interest, and  is the differential volume element at the position .  On the other hand, the above definition is so similar to the definition of the gravimetric inverse problem associated with the Earth’s gravity field modeling in physical Geodesy. The Earth’s gravity field modeling in physical Geodesy is often involved with various types of gravimetric inverse problem such as downward continuation, at which the incremental gravity observations such as gravity anomaly are also placed at the right side of corresponding equations and the unknowns are also placed under the mathematical operators. Land-based, airborne, ship borne and satellite gravity data, as well as astronomical,  leveling and tidal observations are counted as common observations of physical Geodesy, which are usually used for modeling the Earth’s gravity field. In this way, the unknowns are usually constituted by coefficients of certain algebraic expansions which enable us to model the Earth’s gravity potential as well as its derivatives such as the gravity vector and gravity gradient tensor. These algebraic expansions are commonly linear and obtained by discretizing linear integral equations achieved by Geodetic Boundary Value Problems (GBVP). Furthermore, these expansions can also be obtained by certain basis functions such as spherical harmonic functions, radial basis functions (RBFs), Slepian and wavelet basis functions. Hence, the current research considers the following equation as the general form of the gravimetric inverse problem associated with the subject of the gravity field modeling in physical Geodesy:        where is an operator which depends on either the corresponding integral equations or the expansions of the basis functions; and the vector  comprises of the unknowns which model the Earth’s gravity potential and its derivatives. Now, if the Earth’ gravity potential is able to be expanded by a set of basis functions  such as <a name="OLE_LINK6"> </a> , the above general form of the gravimetric inverse problem (i.e. Eq.) will have the following form in the above-mentioned local coordinate system. where  and . This paper comprises of an idea of correlating the concepts of the abovementioned gravimetric inverse problems with each other.  Hence, first it analyzes each one of the abovementioned gravimetric inverse problems separately and then combines them with each other by finding mathematical relations between their unknowns. For this purpose, the concept of the Poisson’s PDE is used for converting the unknowns of Eq. (i.e. ) into the unknown of the Newton’s IE (i.e. the mass-density anomaly ).  Indeed, the theories discussed by the paper will also examined by certain numerical experiments in a real case study based on 6350 gravity stations scattered over a part of the north coasts of the Persian Gulf. ثقل, دانسیته, مدلسازی, گرانی, چگالی, وارونسازی, مسئله معکوس Gravity, Mass Density, Gravity Modeling, Inverse Problem 43 62 http://jgst.issgeac.ir/browse.php?a_code=A-10-345-1&slc_lang=fa&sid=1 Y. AllahTavakoli یحیی الله توکلی ytavakoli@ut.ac.ir 10031947532846003749 10031947532846003749 Yes School of Surveying and Geospatial Engineering, Faculty of Engineering, University of Tehran دانشکده مهندسی نقشه برداری و اطلاعات مکانی دانشگاه تهران A. Safari عبدالرضا صفری asafari@ut.ac.ir 10031947532846003750 10031947532846003750 No School of Surveying and Geospatial Engineering, Faculty of Engineering, University of Tehran دانشکده مهندسی نقشه برداری و اطلاعات مکانی دانشگاه تهران A. A. Ardalan علیرضا آزموده ‎اردلان ardalan@ut.ac.ir 10031947532846003751 10031947532846003751 No School of Surveying and Geospatial Engineering, Faculty of Engineering, University of Tehran دانشکده مهندسی نقشه برداری و اطلاعات مکانی دانشگاه تهران A. Bahrodi عباس بحرودی bahroudi@ut.ac.ir 10031947532846003752 10031947532846003752 No School of Mining Engineering, Faculty of Engineering, University of Tehran دانشکده مهندسی معدن دانشگاه تهران