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SATELLITE GRAVIMETRY AND THE SOLID EARTH Mathematical Foundations MEHDI ESHAGH Elsevier Radarweg29,POBox211,1000AEAmsterdam,Netherlands TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates Copyright©2021Elsevierinc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans, electronicormechanical,includingphotocopying,recording,oranyinformationstorage andretrievalsystem,withoutpermissioninwritingfromthepublisher.Detailsonhowto seekpermission,furtherinformationaboutthePublisher’spermissionspoliciesandour arrangementswithorganizationssuchastheCopyrightClearanceCenterandtheCopyright LicensingAgency,canbefoundatourwebsite:www.elsevier.com/permissions. Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightby thePublisher(otherthanasmaybenotedherein). 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LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN:978-0-12-816936-0 ForinformationonallElsevierpublicationsvisitourwebsite athttps://www.elsevier.com/books-and-journals Publisher:CandiceJanco AcquisitionsEditor:AmyShapiro EditorialProjectManager:LizHeijkoop ProductionProjectManager:OmerMukthar CoverDesigner:MilesHitchen TypesetbyTNQTechnologies Dedicated to my wife Elsa and our beautiful daughters Vendela and Helen Preface Satellite gravimetry and its geophysical applications have been my main subjects of research, teaching and supervising students for over 15 years. I felt that a mathematically oriented book covering all aspects of such a subject is needed for higher education students and young researchers in geodesyandgeophysics.Ihadinmindtowritesuchabookandexplainthe mathematicalfoundationsbehindvarious satellitegravimetryobservablesin relation to the Earth’s gravity field and the solid Earth’s geophysics until I talked to one of my close friends, Professor Robert Tenzer, about it. He highly encouraged me to write such a book; also, Dr Martin Pitonak, who hadbeenwithmeforashortpostdoctoralresearchsabbatical,motivatedme towardsthisgoal.However,duetomyhugeamountofacademictasksand duties, taking such an extra and hard job was difficult, until 2years ago, whenMrsMarisaLaFleurfromElsevierproposedthatIwriteabookabout the Earth’s gravity field for publication. Elsevier was one of my favourite publishers, and I had published the majority of my scientific articles in its journals. This was a serious step towards my idea, and after writing a book proposal and its approval after the reviewing process and revisions, my writing was started in June 2018. Mymaingoalwasthatthebookshouldbesimple,complete,didacticand pedagogic and suitable for students and researchers in the related fields. However, I have also developed new ideas in the book for the readers’ benefit. In my opinion, showing how satellite gravimetry data, which are collectedfromplatformsfarfromtheEarth’ssurface,arerelatedtothegravity field and geophysical phenomena inside the Earth is an interesting subject. This is what I have tried to present and explain in this book step by step. I have done all I could during these 2years to present this book in the best form, and checked and rederived formulae several times. In addition, I restructuredthebookacoupleoftimesforitsbetterpresentation,whichled to extra work on the part of the Elsevier Production Division. I would be indeed appreciative if all colleagues, students and readers assist me in improvingthebookthroughtheirconstructiveandusefulcomments,sothat the next edition of the book will be clearer and free of any probable errors. Mehdi Eshagh Summer 2020 xi Acknowledgements I would like to express my gratitude to those without whose support and help my idea could never come true. I am grateful to Professor Robert Tenzer and Dr Martin Pitonak for inspiring me with their positive words and encouragement about the idea of this book. I appreciate all support frommyclosestfriendandcolleague,DrMajidAbrehdary,especiallyallthe kind and scientific discussions about satellite altimetry and the Moho and density contrast determination. My special thanks go to Mr Andenet Ashagrie Gedamuand Mr Farzam Fatolazadeh for help in writing Chapters 8 and 10. Professor Bernhard Steinberger is acknowledged for his help by reading the draft version of Chapter 8 and commenting on it. Dr S.M. Ali Noori Rahimabadi is appreciated for the scientific discussion and advice regarding mantle flow theory. I am indeed appreciative to Dr Michal Sprlak, who acted as the tech- nical editor and reviewer of this book, for his careful reading and detailed andconstructivecommentsonthewholebook.IthankMrsMarisaLaFleur from Elsevier, who contacted me about proposing the book idea, handling administrative processes with Mr Michael Lutz for up to half of the book. Mrs Amy Shapiro and Mrs Liz Heijkoop are cordially acknowledged for their support and help in handling all the administrative processes of the book and for help in the design of the cover page of the book. Mr Omer Mukthar, the production project manager and the Production Division of Elsevier are sincerely thanked for their excellent cooperation in producing and revising the whole book. Aheartfeltwordofthanksgoestomywife,Elsa,forprovidingasuitable atmosphere at home and taking care of the family during my writing from the start to the end, especially during the pandemic times. Mehdi Eshagh Summer 2020 xiii CHAPTER 1 Spherical harmonics and potential theory 1.1 General solution of Laplace equation in spherical coordinates It is known in physics, geophysics and physical geodesy that the earth’s gravitational potential fulfils the Poisson partial differential equation. Since satellite gravimetry data are collected outside the earth’s surface, the gravitational potential becomes harmonic there, meaning that it satisfies the Laplace equation instead.Such a harmonic potential, which we show byV in this book, has the following mathematical form (Heiskanen and Moritz, 1967, p. 12): DV ¼0 for r >¼ R; (1.1) where,asmentioned,Visthepotential,Rstandsfortheradiusofthespher- ical earth and r the geocentric distance, the distance from the earth’s centre ofmass andanypointoutside it.Dstandsfor theLaplaceoperatorwiththe following form in the spherical coordinate system (Heiskanen and Moritz, 1967, p. 19): v2 2 v 1 v2 cotq v 1 v2 D¼ þ þ þ þ ; (1.2) vr2 r vr r2 vq2 r2 vq r2 sin2 q vl2 where q and l are, respectively, the co-latitude and longitude of any point withageocentricdistanceofroutsidetheearth.Byassumingthattheearth isspherical,thesolutionofEq.(1.1)willbe(e.g.,cf.HeiskanenandMoritz, 1967, p. 21): (cid:2) (cid:3) (cid:2) (cid:3) XN Xn R nþ1 XN R nþ1 Vðr;q;lÞ¼ r vnmYnmðq;lÞ ¼ r vnðq;lÞ forr > R; n¼0 m¼(cid:2)n n¼0 (1.3) SatelliteGravimetryandtheSolidEarth ISBN978-0-12-816936-0 ©2021ElsevierInc. https://doi.org/10.1016/B978-0-12-816936-0.00001-3 Allrightsreserved. 1 2 SatelliteGravimetryandtheSolidEarth where Ynmðq;lÞ is the fully-normalised spherical harmonic function of degree n and order m with the following expressions: ( Ynmðq;lÞ¼ scionsmmllPPnnmjmðjcðocossqÞqÞ ffoorrmm>(cid:3)00; (1.4) and PnmðcosqÞ is the fully-normalised associated Legendre functions of degree n and order m and argument of cosq, and vnm the fully-normalised spherical harmonic coefficients (SHCs) of the gravitational potential. The term vn is called the Laplace coefficient with the following relation to vnm: Xn vnðq;lÞ¼ vnmYnmðq;lÞ. (1.5) m¼(cid:2)n As observed, unlike vnm, the Laplace coefficient vn is position dependent. The coefficient ðR=rÞnþ1 in Eq. (1.3) is called the upward continuation factor. The ratio R/r is always smaller than 1 for r > R; this means that it becomes even smaller at the power of nþ1. When n increases, this factor reducesthevaluesofvnm,andfinallywhenngoestoinfinity,theratiogoes to zero and significantly reduces the value of vnm. Obviously, when r is muchlargerthanR,orinotherwords,thepointsarefartherawayfromthe spherical earth, this factor diminishes faster and reduces more of vnm. This means that by increasing the geocentric distance r, the potential contains fewer frequencies and it will be smoother. The spherical harmonic functions presented in Eq. (1.4) have the following orthogonality property: ZZ (cid:4) 1 i ¼ j Ynmðq;lÞYn0m0ðq;lÞds¼4pdnn0dmm0 anddij ¼ 0 isj ; (1.6) s where s stands for the sphere on which the integration is performed, ds is the surface integration element at the sphere and d is the Kronecker delta. Now, consider r¼R in Eq. (1.3), multiply both sides by Yn0m0ðq0;l0Þ andtakethesurfaceintegrationoverthesphere.AccordingtoEq.(1.6),the result will be: ZZ ZZ XN Xn VðR;q0;l0ÞYn0m0ðq0;l0Þds¼ vnm Ynmðq;lÞYn0m0ðq0;l0Þds s n¼0 m¼(cid:2)n s ¼ 4pvnm. (1.7) Sphericalharmonicsandpotentialtheory 3 Solution of Eq. (1.7) for vnm yields: ZZ 1 vnm¼4p VðR;q0;l0ÞYnmðq0;l0Þds. (1.8) s Now, let us introduce another important property of the spherical harmonics, which is known as the addition theorem. This theorem relates the product of two spherical harmonic functions of the same degree and order, but at two different points, to the spherical geocentric distance between them as an argument for the Legendre function. This addition theorem is (Heiskanen and Moritz, 1967, p. 33): Xn Ynmðq0;l0ÞYnmðq;lÞ¼ð2nþ1ÞPnðcosjÞ; (1.9) m¼(cid:2)n wherePnðcosjÞistheLegendrepolynomialwithargumentcosj,wherej is the spherical geocentric angle between the computation and integration points ðq;lÞ and ðq0;l0Þ, which can be computed from the spherical coordinates of both points by (see Fig. 1.5): cosj¼cosqcosq0þsinqsinq0 cosðl0(cid:2)lÞ. (1.10) By inserting Eq. (1.8) into Eq. (1.5) and after applying the addition theorem of spherical harmonics (Eq. 1.9), we obtain another formula for the Laplace coefficient of the potential: ZZ 2nþ1 vnðq;lÞ¼ 4p VðR;q0;l0ÞPnðcosjÞds. (1.11) s Eq. (1.11) is very useful for transforming the integral formula to spherical harmonic expansions and vice versa. Eq. (1.3) is the solution of the gravitational potential outside the spherical earth. When vnm are available, the gravitational potential can be computed by summing up the series. This process is known forward computation or spherical harmonic synthesis. Different methods have been presented by researchers for performing this synthesis effectively and fast. However, in gravimetry the goal is to determine vnm, carrying physical properties of the gravitational potential. This process is known as spherical harmonicanalysis;seeEq.(1.8).TherestoftheparametersinEq.(1.3),i.e., the spherical harmonics and the upward continuation factor, are mathematical functions depending on the position of the computation points. 4 SatelliteGravimetryandtheSolidEarth 1.2 Solving potential from potential outside the earth Here, the mathematical foundation of recovering potential from potentials athigherlevels thantheearth’ssurfaceis presented.Itisexplainedhowthe spherical harmonics and their properties can be used for determining gravitational potential. A boundary-value problem can be organised and solved for recovering the potential outside the sphere from some boundary values of potential at its surface, Dirichlet’s problem. However, in satellite gravimetry the goal is to recover the potential at the surface from the measuredpotentialoutsidethesphere.ThisisthereversecaseofDirichlet’s problem and is called the inverse problem. In this section, and the rest of this chapter, we discuss such subjects in spectral and spatial forms as well as integral equations. 1.2.1 Spectral solution Suppose that Vðr;q;lÞ is known in Eq. (1.3) outside a sphere with the radius R. If both sides of Eq. (1.3) are multiplied by Yn0m0ðq0;l0Þ and a surface integration is performed all over this sphere, we obtain: ZZ (cid:2) (cid:3) XN Xn R nþ1 Vðr;q0;l0ÞY ðq0;l0Þds¼ n0m0 r s n¼0ZmZ¼(cid:2)n (1.12) vnm Ynmðq;lÞYn0m0ðq0;l0Þds. s According to Eq. (1.6), Eq. (1.12), will change to: ZZ (cid:2) (cid:3) XN Xn R nþ1 Vðr;q0;l0ÞYn0m0ðq0;l0Þds¼4p r vnmdnn0dmm0 s (cid:2)n¼0(cid:3)m¼(cid:2)n (1.13) R nþ1 ¼ 4p r vn0m0. Solving Eq. (1.13) for vnm reads: (cid:5) (cid:6) ZZ 1 r nþ1 vnm¼4p R Vðr;q0;l0ÞYnmðq0;l0Þds. (1.14) s This integral (Eq. 1.14) is a direct mathematical formula connecting Vðr;q;lÞ, over a sphere with radius r and vnm at the sphere. The term