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AD-A252 283Twelfth Moving Base

Gravity/Gradiometer ConferenceUnited States Air Force Academy

Colorado Springs, Colorado

Agenda

Tuesday 14 February

0800 - Buses depart USAFA Officers' Club for Fairchild Hall

0815 - Registration - 3rd floor Fairchild Hail, South End

0900 - Welcome/Introduction - 2Lt Warner

0910 - Opening Remark; - Dr. Martin (DMA)

0920 - "AFGL/DMA GGSS Program Review" - Mr. Borgeson (AFGL'

0935 - "Review of 1983 Moving Base Gravity GradiometerActivities at Ikell Aerospace Textron" - Mr. Metzger(Bell Aerospace Textron)

1010 - "Comparison of at Sea Gradiometer Test Resultswith an Independent Gravity Gradient Reference" -Mr. Zorn, Mr. Henson (Dynamics Research Inc.)

1040 - Break

1055 - "Current GGSS ,lerformance Expectation and ErrorAllocation" - )r. Heller (TASC)

1145 - Buses depart Fairchild Hall for USAFA Officers' Club

1200 - Luncheon - Officers' Club

1300 - Buses depart Officers' Club for Fairchild Hall

1315 - "Efficient Prozessing of Gradiometer Output Usinga Kalman Filter" - Dr. Hutcheson (Bell AerospaceTextron)

1345 - "On-Going Work in Post-Mission Data ProcessingAlgorithm Development" - Dr. Heller (TASC)

1410 - "Three-axis Super Conducting Gravity Gradiometer" -

Dr. Paik (University of Maryland)

1500 - Break

1515 - "Overview of Airborne Gravity and Comparison ofGravity and Gradient Anomolies" - Dr. Hammer(University of Wisconsin)

Approved for public release; distribution unlimited.

1535 - "WGS /2 Gravity Gradient Reference Ellipsoidand Earth Gravitational Model" - Mr. May (NADC)

1600 - Buses depatrt Fairchild Hall for USAFA Officers' Club

1615 - Reception - USAFA Officers' Club

Wednesday 15 February

0800 - Buses depart USAFA Officers' Club for Fairchild Hall

0815 - "Relooking at HASCONS for Representing GravitySurvey Data" - Dr. Breakwell (Stanford University)

0840 - "Integral Formulas Relating Gravity Gradients toVertical Deflection" - Mr. Zorn (Dynamics ResearchInc.)

0910 - "Real-time Vertical Deflection Estimation UsingGradient Data" - Dr. Feldman (Snc.rry Corp)(Mr. B. Epstein, Co-Author, U, Navy SSPO)

0940 - Break

1000 - "Gravity Gradiometer Application to Tunnel Detection" -

Mr. Jircitano (Bell Aerospace Textron)

1040 - ***Time Permitting***"A Simplification of the Least Squares Determinationof the Gravity Field..." - Dr. Reinhardt (Bendix)

1050 - DoD Executive Session

1200 - Buses leave

0

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MINUTES OF THE TWELFTH ANNUALMOVING BASE GRAVITY GRADIOMETER CONFERENCE

United States Air Force AcademyColorado Springs, Colorado

14-15 February 1984

Tuesday Morning Session

Welcome/Introduction - 2Lt D.L. Warner (AFGL)As coordinator of the two-day conference, Lt. Warner opened the mietingon behalf of the co-s-ponsoring agencies, the Air Force GeophysicsLaboratory (AFGL) and the Defense Mapping Agency (DMA). She then introducedDr. C. Martin, Director of the Advanced Technology Division, DMA.

Opening remarks - Dr. C. MartinDr. Martin welcomed participants and attendees. He made referenc? to theagenda noting his enthusiasm over topics of data processing and algorithmdevelopment, and the breaking from topics of hardware development. Heexpressed the urgency of the need to develop the necessary means coanalyze the data. Software development has a long lead time, thereforeDr. Martin urged the gravity community to share insights, applica')lesoftware, anything that might keep individual software developers from"reinventing the wheel".

AFGL/DMA GGSS Program Review - Mr. R. Borgeson (AFGL)Mr. Borgeson, prograi manager of the Gravity Gradiometer Survey S/stem,presented a review oi: those milestones that have been accomplishej, andthose soon to be accomplished. He gave a brief description of the twomodes of the system: the land based and the airborne modes, inclulingmockup models and viewgraph sketches. He stated the accuracy reqjirementsof the system and showed diagrams and pictures of the actual three-axisgravity gradiometer that Bell Aerospace Textron is under contract todevelop and build. Mr. Borgeson stated that AFGL and DMA are in theprocess of determining which aircraft would best suit the needs of theGGSS program. The present choices are the P3, the Convair 580, and the C130.

Questions:Dr. S. Hammer (University of Wisconsin) - To what accuracy can we estimategravity using the gradiometer? Is it a prediction or demonstratei performance?

Mr. Borgeson - .18 arc seconds, .9 milligals and it is a specificationand predicted accuracy.

Mr. M. May (NADC) - What navigation system would be used on the P3, andwill a gravimeter be used?

*Mr. Borgeson - GPS navigation and no gravimeter.

-2-

Review of 1983 Moving Base Gravity Gradiometer Activities at bell Aerospacelextron - Mr. E. Metzger (Bell Aerospace Textron)Mr. Metzger described the two programs Bell is working on: the NavyGravity Survey System (GSS) and the AFGL/DMA GGSS Program. He describedthe performance of the Advanced Development Model (ADM) #1 (Navy) aboardthe USNS Vanguard. Having been in operation since February 1982 theinstrument failed less often than expected and met the feasibilityrequirements in all tests. The ADM #2 has been in operation sinceSeptember 1983. Mr. Metzger outlined the status and future plans of ADM#2 stating that the objective was to provide a test bed for verifyingmodifications considered for operational systems. since Mr. Borgeson hadalready covered the physical or hardware portions of the GGSS, Mr. Metzgerfocused his attention on a discussion of error analysis and track spacing.

Questions:Or. H.J. Paik (University of Maryland) - What is the origin of the red noise?

Mr. Metzger - Drift in the even order calibration coefficients of theaccelerometers.

Dr. Paik - Is the noise proportional to I/f?

Mr. Metzger - It is proportional to D tween 1/f and (I/f) 1 . 6.

Comparison of At-Sea Gradiometer Test Results With An Independent Gravity

Gradient Reference - Mr. D. Benson, Jr. and Mr. A. Zorn (Dynamics ResearchCorporation)Mr. B. Regenauer, Mr. Benson, and Mr. Zorn are co-authors of this paperand it was presented by the latter two. Mr. Benson discussed the generationof a gravity gradient reference off the north coast of Puerto Rico andsubsequent comparison of Gravity Gradiometer gradients with the reference.The gradient reference was derived using position differences between anaccurate marine inertial system and an accurate geodetic position reference,Autotape. Similar techniques have been used for vertical deflectionsover land but, in contrast, require frequent stopping for gyro recalibration.Mr. Zorn presented part two, the feasibility of the at-sea survey techniquewhich was established using covariance error analysis. Subsequently atest plan was developed to acquire data for map development, and thesurvey was performed with equipment onboard the USNS Vanguard. Returnsover the same paths and crossing paths during the survey allow separationof time varying gyro and acclerometer errors from position varyinggradients and deflections when the survey data is processed. The analysisand data processing include local geodetic constraints on the gradientsto satisfy the conditions of the earth's gravitational potential. Mr.Zorn concluded by mentioning that the survey paths were repeated on alater at-sea test with the Bell Gravity Gradiometer onboard. Gravitygradients from the gradiometer compared remarkably well with the gravitygradient reference along the survey paths.

Questions:Mr. Borgeson - Is the progr.im under contract?

Mr. Benson - Yes, tinder contract with Sperry Management Systems, SP24.

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Mr. S. Jordan (Geospace) - Can you give an actual accuracy for test results?

Mr. Benson - No, it is classified.

Mr. P. Fell (DMAHTC) - What is the Autotape system?

Mr. Benson - A radar navigation line of sight positioning system.

Mr. W. Gumert (Carson Geoscience) - What speed were you surveyin( at?

Mr. Zorn - 5 to 8 knots.

Dr. W. Heller (TASC) - Is t'ere an RMS difference between map ani measuredgradients?

Mr. Benson - They exist but are classified.

A discussion followed between Dr. Heller and Mr. Benson concerningpossible RMS differences and how they depend on data processing md use oflow frequency information.

Current GGSS Performance Expectation and Error Analysis Allocati;n -Dr. W. Heller (TASC)Dr. Heller opened with a review of test and survey geometry of tie GGSSand the desired goals. He discussed the error sources as well a'; theGGSS sensitivity to variation in the frequency content of the grivityfield. Mentioned as well were the contribution of each class of errorsand the sensitivity to survey altitude. In his conclusions Dr. Hellerstated that the bottom-line is that the performance of the GGSS is expectedto fall well within prescribed specifications.

Questions:Mr. J. Brozena (NRL) - The downward continuation and truncation ?rrorsrefer to the center of the survey. What happens as you go away from thecenter?

Dr. Heller - We have to be concerned with edge effects. Testing will bedone at the center of the survey region to minimize edge effects.

Mr. Zorn - Was only one gradient, Tzz, used?

Dr. Heller - Yes.

Mr. Zorn - What if other gradients are used, such as those that are moredirectly related to the vertical deflection by integration? The errorcould be overstated because these gradients are not used.S

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Dr. Heller - We used Breakwell's results that the best gradient to use isTzz. We incurred a penalty of a factor of 2 by not using other gradientsand the errors in Tzz were correspondingly scaled.

Mr. M. Molny (Sperry) - Why not put in a wider grid spacing for theoutside areas?

Dr. Heller - That's a good introduction to the next talk!

Dr. Jordan - Is te downward continuation to the physical surface or to areference surface!

Dr. Heller - This has to be resolved. We don't want to continue the databelow the surface. For convenience we chose the physical surface.

Dr. Jordan - I doi't believe we can meet the goals of the program usingthe physical surfice is a reference. We can't downward continue in amountainous area.

Benson, Jordan, aid Holler further dicusse the choicc uf the referencesurface.

Dr. H. Baussus voi Luetzow (ETL) - W" ,eed to consider the shortcomingsof least squares collocation at 1igh frequencies (the use of stationarymodels could introduce errors of 1U-20%). No terrain effects have to beconsidered.

Dr. Heller - We plan to survey over smooth areas to test and demonstratethe gradiometer and should be able to handle mountainous terrain in thefutur'e.

Mr. M. Trageser - Why do we need to downward continue at all? Why notestimate gravity at altitude?

Dr. Heller - We %ant point estimates at the surface.

There was further discussion on this question between Heller, Trageser,and Martin.

Tuesday Afternoon Session

Efficient Procesing of Gradiometer Output Using a Kalman Filter -L7r. J. Hutcheson {BelI Aerospace lextron)Dr. Hutcheson made several points concerning the processing of the datato be collected oy the gravity gradiometer system. First, there are justtoo many data points; they must be compressed. He presented a method toprocess gradient data using a straight line Kalman filter. Dr. Hutchesonshowed comparisons of the STAG model and state space approximations. Hecompared results of fixed point smoothing and least squares collocation.Results were simulated using a MASCON synthetic gravity field.

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Questions:Unidentified questioner - What wavelengths were modeled with the MASCONmode l ?

Dr. Hutcheson - The North Atlantic Gravity Model was used. The vavelengthsare possibly classified.

Or. C. Jekeli (AFGL) - When using the straight line filter, are (orrelationsfrbm track to track taken into account?Dr. Hutcheson - Two dimensional correlations are taken into accoi nt in a

subsequent stage of processing.

Mr. P. Ugincius (NSWC) - What about estimating away from the track?

Dr. Hutcheson - One can always contract the estimation point to ".henearest grid point.

On-goiny Work in Post Mission Data Processing Algorithm Oevelopm-nt -Dr. W. Heller (TASC)Dr. Heller presented some background information on the GGSS projram anddescribed what will be required for post mission analysis of the datacollected. He discussed various survey and data processing issues aswell as prefilter design considerations. On this topic Dr. Hellr concludesthat a 12 second filter time constant will be needed. 0,o the toic ofestimation approacnes to the gravity disturbance vector, Dr. Hellerproposed two alternatives:a) a sequential complementary filter.b) an optimized template algorithm.

Numerical data showing applications of these alternatives were presented.Dr. Heller summarized by stating that on-going development is not a neat"packaged deal" but an active process. He then proceeded to outline thedevelopment of the algorithms.

Questions:Mr. Zorn - How sensitive is the estimation if the aircraft is nct ableto follow the regular grid?

Dr. Heller - The system noise includes navigation error. If thc navigation

error becomes too great, we may cease data collection.

Dr. S. Hammer - How did you simulate the test data?

Dr. Heller - With ocean trench data by computing gradients which werereinverted creating an anomaly field.

Mr. Molny - What is the actual mapping area?

S Dr. Heller 320 X 320 kilometers.

Mr. Horgeson - There are two test sites. Central Florida and Cheyenne,Wyoming, are being considered based on weather data and availab e gravity

-6-

data.

Mr. Molny - When cordensing lata, averaging, etc., do you consider thesurvey design?

or. Heller We must make sure the data are reyular.

Mr. Molny - Lne could enlarge the size of the square by having largertrack intervals in the outside areas to pick up larger wavelengthinformation.

Dr. Heiler - We may do th's ror the test it we are interested only ingravity estimates in the inner zone.

Uyincius, Hutcheson, Fell, Zorn, Benson, and Heller discussed use of thetemplate method.

Three-axis Super Conductiny Gr.vity Gradiometer - Dr. H.J. Paik (Universityof Maryland))r. Paik opened his presentation with a review J his program to devPlopa single axis version of a cryogenic gravi,y giradiometer. Dr. Paik thenwent on to describe plans to develo'r three-axis superconducting gravitygradiometer. He el.iborated ma*hematically on the expected sensitivity ofthe instrument and ,,xplained its physical design with slides and viewgraphs.Dr. Paik described 1he new pendulum suspension ind error compensation. 0For adequate error compensation Dr. Paik proposed the development of asix-axis superconducting accelerometer and described in detail itsconfiguration, desi jn features, and sensitivity as well as its operationwith a gradiometer. He concluded with a proposed time line for thedevelopment of both a three-axis GUM and a six-axis accelerometer. Hisultimate aspirations include space born testing of the GGM in an earthorbit aboard the space shuttle by the year 1990.

Questions:Dr. Heller - At what time will an eror of lu-I E/(Hz)1/ be demonstrated?

Dr. Paik - This is difficult to predict. We may have to do temperaturecontrol experiments to get rid of squid noise. We had hoped to have haddata 2 years ago, but we need to improve the platform. With a littleconfidence and if the platform is OK, maybe next week. We are trying toget better noise levels in the platform. Maybe next year.

Mr. Hastings (Sperry) - If you were to add proportional feed back, wouldit improve the accuracy?

Dr. Paik - This would not improve the accuracy. It would increase thedynamic range.

Dr. Jordan - GRM will get 10J km resolution at a cost of 1/3 billiondollars. Would you care to comment on the resolution and cost of thesatellite gradiometer?

Dr. Paik - For ij-2 E sensitivity we get lUU kin resolution. To improve

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the resolution by a factor of two we need to improve the gradiume:er by afactor of IOU.

Overview of Airborne Gravity and Comparison of Gravity and GradientAnomalies - Dr. S. Hammer (University of Wisconsin)Dr. Hammer introduced his topic with a very brief history of gravitymeasurements in a moving vehicle and a description of the Carson HelicopterGravity Measuring System (HGMS). HGMS has been operational for 6bout fouryears and observes an average of 5CJ0 km of line da.ta per month vith aprobable error of the order of I regal and anomaly resolution limit ofabout 3 km. Survey altitude, as specified, can vary frow low terrainclearance up to about 14,00J ft (43uU m) above sea level and over anytype of terrain including environmentally restricted areas such s theWilnije Reserve in Alaska. Ir. his presentation Dr. Hammer comp( red themagnitudes and rates of attenuation with altitude of gradient an( gravityanomalies. He emphasized that the rate of attenuation of all comiponentsof the gravity gradient is the same for a given type of anomaly. Theqradient attenuates .)h altitude an order of magnitude faster than the

corresponding gravity anomaly. In conclusion Or. Hammer raised 'hequestion: to accomplish the basic objectives oI this conference, what isthe best way to proceed? (a) Concentrate efforts on the developiient oftheory and instrunentation for gravity gradiometry; or (b) Under.aKe aworld-wide gravity mapping program with the rapid and precise HGI1S wni .his now available.

Questions: (deferred to Mr. W. Gumert of Carson Geoscience)Mr. Borgeson - Did you have any problems with civil air traffic?

Mr. Gumert - There were no problems, as long as we filed a fligit plan

and flew low.

Dr. Heller - How far out to sea dicd you fly?

Mr. Gumert - 16U-180 km using line of sight radar.

Mr. Molny - Hoi do you separate vertical acceleration from gravity anomalies?

Ur. Hammer - Using barometric and radar altimeters.

Wednesday Morning Session

k)ue to unexpected circumstances, the order of presentation as shown onthe agenda was slightly rearranged.)

A Simplificatior of the Least Squares Determination of the Gravity Fieldfrom Low-low SaLellite Tracking Data Utilizing An Operator Repr(sentdtionof the Range Observable - Dr. V. Reinhardt (Bendix Field EnginefringCorporation)

* Dr. Reinhardt introduced his paper by explaining that one of th. purposesof NASA/Goddard Space Flight Center's proposed Geopot-ntial ResearchMission is to map the gravity field of the Earth to 1-2 myal in 41,2b3I degree by I degree blocks using range rate tracking data betwoen two

-8-

polar low orbiting satellites in the same orbit. Because there are 041,2b3 blocks in the gravity map, it takes, in general, at least 9.4x1U 1 3

real floating point operations (flop) to solve the least squares fit(LSQF) matrix equation used to determine the gravity potential from thetracking data. Dr. Reinhardt showed that with the circular orbit infinitetrdjectory length approximation, and an iterative method, one obtains anapproximate covariance matrix for the coefficients in less than 1.5x1012

flop. In his presentation Dr. Reinhardt also demonstrated the applicationof this method to airborne gradiometry.

WGS Gravity Gradient Reference Ellipsoid and Earth Gravitational Model -Mr. M. May (NADC)Mr. May described the gravity gradient ellipsoid in terms of geographiclatitude using WGS-72 reference ellipsoid parameters. Additionally, hedescribed a representation of earth's gravity gradients by a Fourierseries for wavelengths down to 120 nautical miles.

Questions:Dr. Hammer - What wavelength ranges were used?

Mr. May - 18U X 180 degree field which cor-x'spor ids to wavelengths down toI2tI nautical miles.

Dr. Heller - Comment: for stochastic modeling, models have been determined

for the degree variances of the gravity field.

Dr. Jekeli - What is the advantage of using gradient maps over gravity maps?

Mr. May - It is advantageous to mix data for vertical deflectiondetermination at the instrument (gradient) level.

Re-looking at MASCONS for Representing Gravity Survey Data -Dr. J. Breakweli (Stanford University)Dr. BrPakwell introduced his presentation with the statement tnat therepresentation of the higher harmonic part of the Earth's gravity fieldby a finite grid of MASCONS in two layers, one at the Earth's surface andone at the bottom of the Earth's crust, leads to an error in the gravitycomponents at altitude (in addition to the error from the estimation ofdensity fluctuations in the two layers). His discussion examined theseerrors for various ratios of grid size to gradiometer altitude.

Integral Formulas Relating Gravity Gradients to Vertical Deflections -Mr. A. Zorn (Dynamics Research Corporation)Mr. Zorn presented a number of integral formulas relating gravitygradients to verticdl deflections. The formulas are derived from standardgeodesy theory and are similar to the Stokes and Vening-Meinesz integralformulas. He explaineii that the deterministic formulas provide a physicalexplanation of certain properties common to various self-consistentstochastic gravity models. For example, the high degree of correlationbetween the gravity gradient, Txz, arid the vertical deflection Tx, inmost gravity models can be physically justified based on a deterministic

-9-

integral formula. The integral formulas may lead to insight into theconstruction of yradiometer filters to estimate vertical deflectionswhich are insensi,;ive to stochastic gravity models. Mr. Zorn discussedimplementation coisiderations involving discretization and truncation ofthe integral formulas using surface gravity data over the Blake Escarpment.He concluded with several statements including that vertical deflectionmaps can be constructed from gradiometer survey data using StokeE' formula.

Questions:

Mr. Ugincius - Going from Tzz one can get any other gradient quantity.Why measure these other components?

Mr. Zorn - To get other components we need Tzz over the entire earth'ssurface. We therefore measure the other components to get more -nformationespecially in limited areas.

Mr. Ugincius - Would the tree diagram hold for spherical formula .ions?

Mr. Zorn - Yes

Dr. D. UeBra (Stanford University) - Are some paths more sensiti/e?

Mr. Zorn indicatcd on his diagram certain paths that would be affected bylow frequency errors.

Mr. May - The tree diagram can be derived from spherical harmonicexpansion. Is there distinction between gravity anomaly and gravitydisturbance?

Mr. Zorn - The difference is negligible.

Mr. A. Rufty (NSWC) - If we have gradient information only, we needintegration constants to get gravity quantities.

Zorn - We would need low frequency information to obtain the corstant ofintegration.

Real-Time Vertical Deflection Estimation Using Gradient Data -Dr. W. Feldman (Sperry; Co-author, Mr. B. Epstein)Dr. Feldman discussed techniques and algorithms for making real-timeestimates of vertical deflections from gravity gradient measurements.First he outlined the techniques and algorithms used to conditi(n sensordata for effects of high-frequency noise, pressure sensitivity, self-gradients, bias/trend, and carouselling. He then showed plots (if typicalgradient data before and after preprocessing to illustrate the (ffectivenessof the preprocessing. Second, a batch filter formulation for e~timatingdeflections from gradient and SEASAT map data was illustrated. Finally,Dr. Feldman demonstrated filter results by plots of filter estiriates ofvertical deflection vs. reference values for several ship track,;.

Questions:Dr. Heller - Why pull out the reference field in the instrument frame

- 10 -

rather than in the NED frame?

Dr. Feldman - It is equivalent.

Dr. Heller - What are the practical considerations?

Dr. Feldman - It was simpler to do in the instrument frame and there wasless chance of error.

Gravity Gradiometer Application to Tunnel Detection - Mr. Jircitano (8ell)Mr. Jircitano described a project being conducted by the Army in thefield of tunnel detection by means of measuring differences in gravitygradients. There are two tunnel detection problems: 1) the detection ofnew tunnels (those which had been dug after a previous survey of thearea) 2) the detection of old tunels (those that were present beforeany previous survey).

Questions:Dr. Reinhardt - What is the feasibility of tunnel dptLtior, if they weremasked by high density fill?

Mr. Jircitano - The possibility exit: but to compensate one would haveto half fill the tunnel with stcel which is impractical.

Dr. DeBra - Seasonal yroundwater has a significant mass. Is this aproblem for new tunnel detection?

Mr. Jircitano - Could be, but water would not collect in a tunnel formation.

Closing Remarks - Lt. D. WarnerLt. Warner thanked everyone for attending this year's conference, especiallythose who presented ptpers. She also offered a special thanks to her AFGLcolleagues who helped with audio/visual equipment and note taking. As adate for next year's conference had not yet been established, she assuredall that they wotld be informed of the date as soon as it was reserved.

0

CONFERENCE ATTENDEES

TWELFTH ANNUAL MOVING BASE GRAVITY GRADIOMETER REVIEW

14, 15 February 1984

AFFLECK, C. Bell

ANDERLE, R. NSWC

BAUSSUS von Luetzow, H. ETL-RI

BENSON, D. Dynamics Research Corporation

BERNSTEIN, R. Navy

BLANTON, J. NSWC

BODIFORD, L. SSPO, Navy

BORGESON, R. AFGL

BRADLEY, L. Geospace

BREAKWELL, J. Stanford University

BROZENA, J. NRL

CHATFIELD, A. Geodynamics

CONFER, M. CIGTF

DEBRA, D. Stanford University

DECKER, L. DMAAC

DELGADO, L. HQAFSC

DEMATTEO, J. NADC

DOWNS, H. CIGTF

ECKHARDT, D. AFGL

FELDMAN, W. Sperry Systems Management

FELL, P. DMAHTC

FLEETWOOD, J. Mobil

FRANSON, J. Johns Hopkins University

FUNDAK, T. AFGL

GOLDSBOROUGH, R. AFGL

GOUKER, R. DMAHTC

GRAHAM, J. DMAAC

GSCHWIND, J. DMAHTC

GUMERT, W. Carson Geoscience

HADFIELD, M. Honeywell

HAMMER, S. University of Wisconsin

HARRIMAN, D. Geodynamics

HASTINGS, R. Sperry Management Systems, U2T20

HEAGOCK, J. ONR

HELLER, W. TASC

HOWLAND, J. Johns Hopkins University

- 11i-

HUTCHESON, J. Bell

JEKELI, C. AFGL

JIRCITANO, A. Bell

JORDAN, S. Geospace

KOWALSKI, M. SOHIO

KURKOWSKI, J. DMAHTC

LAZAREWICZ, A. AFGL

LIST, R. Sperry Systems Management

LUCAS, S. Carson Geoscience

LUMAN, R. Johns Hopkins University

MARTIN, C. DMA

MAY, M. NADC

MAYO, W. DMAHTC

MERTZ, B. DMAHTC

METZGER, E. Bell

MOLNY, M. Sperry Systems m:.&agement

NAVAZIO, F. Carson Oc.-science

PAIK, H.J. University of Maryland

PFEIFER, L. DMAHTC

PFOHL, L. Bell

PINSON, E. DMAHTC

PRICE, C. TASC

REGANAUER, B. Dynamics Research Corporation

REINHARDT, V. Bendix

ROONEY, T. AFGL

RUFTY,A. NSWC

RUTSCHEIDT, E. DMA

RYCHEL, J. NSWC

SENUS, W. DMA

SHAW, G. USAFA

SHULTZ, M. DMAAC

SIMMS, T. NSWC

SUMNER, D. Sperry Systems Management

TANG, W. TASC

TRAGESER, N. C.S. Draper

UGINCIUS, P. NSWC

WARNER, D. AFGL

WIRTANEN, T. AFGL

ZERZAN, J. SOHIO

ZORN, A. Dyn~mics Research Corporation

-12-

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B-507U

COMPARISON OF AT-SEA GRADIOMETER TEST RESULTS

WITH AN INDEPENDENT GRAVITY GRADIENT REFERENCE

Presented At

TWELFTH MOVING BASE GRAVITY GRADIOMETER CONFERENCE

UNITED STATES AIR FORCE ACADEMY S

14-15 February 1984

Dynamics Research Corporation60 Concord Street

Wilmington, Massachusetts 01887

0

-7:2

ABSTRACT

COMPARISON OF AT-SEA GRADIOMETER TEST RESULTSWITH AN INDEPENDENT GRAVITY GRADIENT REFERENCE

Donald 0. Benson, Jr., Alan H. Zorn, and Bernard J. RegenauerDynamics Research Corporation

60 Concord StreetWilmington, Massachusetts 01887

At-sea demonstration and testing of the Bell Gravity Gradiometer nearcoastal regions is desirable to allow access of engineering personnel forequipment grooming and maintenance, and to obtain reasonable signal-to-noise ratios since large deflections and gradients occur near many coastalregions (Ref. 1). Reference gravity maps in these regions either do notexist, do not have sufficient accuracy when derived from satellite altimetryor are extremely time consuming to survey using gravimetric methods.

This presentation discusses the generation of a gravity gradientreference off the north coast of Puerto Rico and subsequent comparison ofGravity Gradiometer gradients with the reference. The gradient referencewas derived using position differences between an accurate marine inertialsystem and an accurate geodetic position reference, Autotape. Similartechniques have been used for vertical deflections over land but, in contrast,require frequent stopping for gyro recalibration (Ref. 2). Feasibility of theat-sea survey technique was established using covariance error analysis.Subsequently a test plan was developed to acquire data for map development,and the survey was performed with equipment onboard the USNS Vanguard.Returns over the same paths and crossing paths during the survey allowseparation of time varying gyro and accelerometer errors from positionvarying gradients and deflections when the survey data is processed. Theanalysis and data processing include local geodetic constraints on the gradientsto satisfy conditions of the earth's gravitational potential (Ref. 3).

The survey paths were repeated on a later at-sea test with the BellGravity Gradiometer onboard. Gravity gradients from the gradiometercompare remarkably well with the independent gravity gradient referencealong the survey paths.

References

1. Dehlinger, P., Marine Gravity, Elsevier Scientific PublishingCo., New York, 1978.

4 2. "Proceedings of the Second International Symposium on InertialTechnology for Surveying and Geodesy, " Banff, Canada,June 1-5, 1981.

3. Heiskanen, W. A. and Moritz, H., Physical Geodesy, W. H.Freeman & Co., San Francisco, 1967.

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THE ANALYTIC SCIENCES CORPORATION

SP-4423-6

CURRENT AIRBORNEGGSS PERFORMANCEEXPECTATIONS AND

ERROR ALLOCAVONS

14-15 February 1984

Prepared for:

TWELFTH MOVING BASE GRAVITY GRADIOMETER REVIEWUnited States Air Force Academy

Colorado

THE ANALYTIC SCIENCES CORPORATIONOne Jacob Way

Reading, Massachusetts 01867

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THE ANALYTIC SCIENCES CORPORATION

SP-4423-7

ONGOING TASC WORK ON GGSSPOST-TEST DATA PROCESSING

ALGORITHM DEVELOPMENT

14-15 February 1984

Prepared for:

TWELFTH MOVING BASE GRAVITY GRADIOMETER REVIEWUnited States Air Force Academy

Colorado

THE ANALYTIC SCIENCES CORPORATIONOne Jacob Way S

Reading, Massachusetts 01867

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OVERVIEW OF AIRBORNE GRAVITY AND COMPARISON OF GRAVITY AND GRADIENT ANOMALIES*

Sigmund Hammer

University of Wisconsin, Madison, WI 53706

William R. Gumert

Carson Geoscience Co., Perkasie, PA 18944

ABSTRACT

The topic is introduced by a very brief history of gravity measurements

in a moving vehicle. The Carson Helicopter Gravity Measuring System (HCMS)

is then described. HGMS has been operational about four years and observes

an average of 5000 km of line data per month with a probable error of the

order of i mgal and anomaly resolution limit of about 3 km. Survey altitude,

as specified, can vary from low terrain clearance up to about 14,000 ft

(4300 m) above sea level and over any type of terrain including environmental-

iy rcs:r1_ areas such as the Wildlife Reserve in Alaska.

The magnitudes and rates of attenuation with altitude of gradient and

gravity anomalies are then compared. It is pertinent to emphasize that the

rate of attenuation of all components of the gravity gradient is the same for

a given type of anomaly. The gradient attenuates with altitude an order of

magnitude faster than the corresponding gravity anomaly.

As a conclusion the question is raised: to accomplish the basic objec-

tives of this Conference, what is the best way to proceed? (a) Concentrate

efforts on the development of theory and instrumentation for gravity gradi-

ometry; or (b) Undertake a world-wide gravity mapping program with the rapid

and precise HGMS which is now available.

*Prepared for presentation at: Twelfth Moving Base Gravity Gradiometry Re-

view, U.S. Air Force Academy, Colorado Springs, CO, February 14-15., 1984

OVERVIEW OF AIRBORNE GRAVITY AND COMPARISON OF GRAVITY AND GRADIENT ANOMALIES

Sigmund Hammer and William R. Gumert

INTRODUCTION

So far as I know there are, as yet, no published field data from func-

tional gravity gradiometers. Our overview, therefore, will assess the pres-

ent state of gravity observations leading up to the present Helicopter Gravity

Measuring System (HGMS). This gravity overview may provide some useful back-

ground for evaluating the comparative potential usefulness of airborne gravity

versus airborne gradiometer measurements for this symposium.

First a very brief history of gravity measurements in a moving vehicle.

The first measurements of gravity in a moving vehicle were accomplished 55

years ago by Vening-Meinesz in a submerged submarine using special three-

pcnduiut a-paratus. Pis spectacular results over the Indonesian Islands Arc-

Trench system were early precursors of the subduction phenomenon which is a

major element in today's plate tectonics theory. Shipborne gravity surveys

with modern high-precision gravimeters began 40 years ago and continue today

on a commercial scale.

Airborne gravity measurements have been under investigation since about

1958. Tests in fixed-wing aircraft have yielded enough encouragement to jus-

tify continuation but satisfactory results have not been achieved up to now.

Successful commercial gravity surveys, using specially stabilized helicop-

ters, have been operational since 1979. High frequency anomalies, applicable

to petroleum exploration, are being mapped at requested altitudes from low

terrain clearance to more than 14,000 feet (4300 m). Since HGMS is quite new

it may be useful to present a very brief explanation of the operation.

3

HELICOPTER GRAVITY MEASURING SYSTEM (HGMS)

The helicopter measurements are made in a large especially stabilized

Sikorski S-61. The first slide shows the helicopter in flight. Gravity is

SLIDE #1: Carson helicopter in flight

measured by a LaCoste-Romberg sea-air meter on a greatly improved stable plat-

form at the center of gravity in the helicopter. It may be of interest to

point out that the helicopter has been modified so that the center of gravity

does not shift as thousands of pounds of fuel are consumed. Total field mag-

netic field data are recorded simultaneously by a trailed proton precession

magnetometer. Terrain clearance is recorded by narrow bean laser. All data

are recorded digitally each second. Flying is done at night to minimize air

turbulence. Flight speed is normally 50 knots but can be specified for the

purpose at hand up to 100 knots (185 km/hr).

The procedure which brought success was to fly a more or less square

grid of intersecting lines as illustrated on slide 2. In this project the

SLIDE #2: Flight pattern

line spacing was 3,300 feet (1 kan) and flight altitude 1000 feet (305 m). The

area is a marshy jungle of extremely difficult access on the ground. Note

that north-south flights are avoided to improve the Etv~s corrections.

0

THE ATTENUATION PROBLEM

The principal elements in the operation are well illustrated in a demon-

stration survey over two well-known salt domes in the Texas Gulf Coast (Car-

SLIDE #3: Gulf Coast survey

son, I98]). The survey consisted of 20 lines, spaced one-half statute mile

(0.805 kI , within the area bordered by dashed lines as shown on Figure 3.

The contoured results clearly define the airborne gravity anomalies on the

two salt domes in the shaded areas. The contour interval is 1/2 mgal.

A comparison with ground data is illustrated on Figure 4. The salt domes

SLIDE #4: Profile results

are sketched at the bottom of the figure. Note the exaggerated vertical

scalp on the domes - the caprock is very shallow. Conventional ground grav-

ity data along the location of the flight line across the centers of the dome

are shown by the central profile. These data were kindly provided by inter-

ested oil companies, but not until after completion of the HGMS project. The

gravity profile measured in flight is the solid curve at the top of the fig-

ure. Upward continuation of the ground data, without filtering, are shown by

the dotted curve (. ). For comparison with the helicopter measurements

the upward continued ground data were then smoothed by the 60-second filter

which removes high frequency motional acceleration effects in the HGMS obser-

vations--------). The differences between the airborne and filtered ground

data nowhere exceed i mgal. However, it is obvious that the sharp, positive

caprock anomalies are strongly attenuated. This will be examined below.

A simulated, quantitative analysis of the anomaly attenuation is illus-

trated in Figure 5. This salt dome is comparable to the Big Hill and High

SLIDE #5: Simulated analysis

/ i

Island domes and is generally typical of salt domes with shallow caprock in

the Gulf Coast, U.S.A. The density contrasts were taken from Nettleton (1976;

Figure 8-14). The calculated gravity anomalies, with and without caprock, at

ground level and at "flight elevation" of 1000 feet (305 m), are plotted in

solid lines. The attenuation of the caprock anomaly is moderate. However,

for comparison with HGMS gravity measurements the "upward continued" ground

data must also be filtered. The filter used by Carson had a time-wid:h of

60 seconds and its width is directly proportional to flight speed. if the

filter width is equal to or larger than the width of the anomaly the smooth-

ing will be great. On the other hand, if the filter width is less than the

width of the anomaly the smoothing will be small. Thus, we see on slide 5

that an assumed flight speed of 50 knots* greatly attenuates the sharp cap-

rock anomaly and spreads it over almost the entire ancmnaly (xxxxx). For an

assumed flight speed of 25 knots, however, the filter widtb is only half as

wide and the attenuation effect for a caprock anomaly of this magnitude is

very much less (.

If high frequency. sharp anomalies are of interest, slow flight speed is

indicated. The flight speed for the 1981 demonstration survey was too fast

to record the complete caprock anomalies but the evidence of shallow positive

influence within the well-defined salt dome minima in Figure 3 are clearly

recognizable to an experienced observer.

Gravity gradient measurements, as mentioned at the beginning. are beyond

the scope of this study. This completes the overview portion of our paper.

*Filter width approximately equal to the width of the caprock anomaly.

0

6

COMPARISON OF GRAVITY AND GRADIENT ANOMALIES

We turn now to the comparative study of gravity and gradient anomalies.

Gravity and gradient anomalies are quite different in both magnitude and be-

havior in space. Figure 6 compares gravity and several gradient components

SLIDE #6: Gravity gradient comparison

from a horizontal cylinder of infinite length at increasing altitudes. In

the present context, the point of interest on this figure is that the gradi-

ents, which are seen to have comparable magnitude at ground level, attenuate

with altitude an order of magnitude faster than the gravity anomalies. This

figure is fr.r an early paper (Harmmer, 1971) which analyzed the behavior of

gradients from a theoretical point of view.

Figure 7 is plotted on a semi-log scale and compares the rate of attenua-

tion with altitude of the gravity and vertical gradient anomalies for a spheri-

ca" ma.as. Gravity is shown by solid curves and right ordinate scale; vertical

SLIDE #7: Gravity and vertical gradient for a spherical mass

gradient by dashed curves and left ordinate. Three sets of curves represent

different depths (D) of the anomalous mass. For all cases the gravity anom-

aly at ground level ig(D) is 10 mgal.

If we assign resolution limits to the gradient and gravity measurements,

thesc curves define the maximum altitude at which the anomalies can be detect-

ed. For exanple, on the first set of curves (with 100 m depth to the center

of mass) the magnitude of the gradient anomaly drops below one (1) Ebtv8s

unit above altitude 1200 m; the gravity anomaly drops below one (1) milligal

abovE altitude 300 m. Thc other two sets of curves show data for greater

depths and indicate higher limiting altitudes.

S

As mentioned, this represents a gravity anomaly at ground level of

10 mgal. For larger (or smaller) gravity anomalies the ordinate magnitudes

are to be multiplied by the ratio.

Figure 8 shows a similar analysis for a two-dimensional gravity anomaly

SLIDE #8: Gravity and vertical gradient for a horizontal cylinder

for a horizontal cylinder of infinite length. The limiting altitudes of

detection for this case are considerably larger. The maximum altitudes at

which the gradient and gravity anomalies in Figures 7 and 8 are detectable

(ig > I mgal, Vzz > iE') are listed in Table 1.

TABLE 1: Maximum Altitudes for Anomaly Detection

Sphere Cylinder

D Gravity Gradient Gravity Cr.;dientmeters meters meters meters meters

100 316 1260 1000 3162

500 1581 3684 5000 7071

2000 6325 9283 20000 14145

This analysis suggests that lower flight altitude is advantageous for

gravity measurements. This may be unfavorable for fixed-wing gravity opera-

tions. Jordan (1978) has shown that low flight altitudes have other advan-

tages as well.

This completes the technical portion of our paper.

0

PRESENT STATUS

Finally, it may be of interest to point out that very extensive coverage

of HCMS surveys has been accomplished in the past four years (Hammer, 1983).

Nearly 200,000 km of gravity line data have been observed in 20 survey areas,

16 of which werE overseas and 4 were in the United States and Alaska. The

normal production speed is 50 knots, but any speed up to 100 knots (185 km!hr)

is possible. The probable error of the data is of the order of 1 mgal and the

anomaiv resolution limit is of the order of 3 km. The helicopter is capable of

flights at altitudes up to 14,000 feet (4300 m) above any type of terrain and

environmentally sensitive access. Duration of flights is up to 7 hours. Nor-

mal output per month is about 3,000 miles (5000 km) of line data.

A very large project, which has been in progress for some time now, is

outlined on Figure 9. If you would like to escape this cold winter weather

SLIDE #9: Bahama,. project

in the sunny Bahamas you might like to get involved with this project.

S

I I ' h !

9

CONCLUSION

In conclusion: the basic purpose of this paper is to emphasize to this

group that the anomaly attenuation of gravity gradient components is an order

of magnitude greater than of the corresponding gravity anomalies.

Finally, I would like to pose a fundamental di e.mra. The question is:

To achieve the basic objectives of this symposium what is the best way to

proceed9

(a) Should we continue the long-range development of the theory and instru-

mentation of the gradient measurements; or

(b) Undertake a world-wide gravity mapping program with the precise and

rapid HGMS which is now available?

0

*****FI S*****

0

REFERENCES

Carson Geoscience Co., 1981, "Airborne Gravity - Gulf Coast 1981". Company

brochure, Perkasie, Pennsylvania 18944.

Hammer, S., 1971, "Vertical attenuation of anomalies in airborne gravimetry".

Geophysics, vol. 36, p. 867-877.

Hafnmmer, S., 1983, "Airborne gravity is here!". Geophyiscs, vol. 48, p. 213-

223.

jordan, S.K., 1978, 'Noving-base gravity gradiometer surveys and interpreta-

tion". Geophysics, vol. 43, p. 94-101.

Nettleton, L.L., 1976, Gravity and Magnetics in Oil Prospecting. McGraw-Hill

Book Co.

.

S

21

FIGURE CAPTIONS

Figure 1

Photograph of Carson HGMS in flight. Note trailing magnetometer at

lower right.

Figure 2

Flight pattern of HGMS survey. Line spacing I km, flight altitude

1000 ft (305 m), and flight speed 50 knots.

Figure 3

Demonstration gravity survey in 1981 over known salt domes in Texas Gulf

Coast. Flight elevation 1000 ft (305 m), flight speed 45 knots, and contour

4 nterval I1/2 mgal.

Figure 4

Profile comparison of HGMS and conventional ground gravity data from

demonstration survey in Figure 3.

Figure 5

Analysis and comparison of simulated HGMS and conventional ground grav-

.tv data relative to data in Figure 4.

Figure 6

Theoretical comparison of magnitude and attenuation of all gradient com-

ponents and gravity anomaly for a horizontal cylinder of infinite length.0

%6 1

12

FigurE 7

Comparative attenuation of gravity and vertical gradient anomalies over

a spherical mass. Note semilog ordinate which indicates that gradients at-

tenuate an order of magnitude faster than gravity. See also Table 1 in the

text.

Figure 8

SamE as Figure 7 but for a two-dimensional mass - a horizontal cylinder

of infinite length.

Figure 9

Layout of HGMS gravity survey now in progress in the Bahamas.

Figure 10 (extra)

Attenuation of airborne terrain correction simulated by an assembly of

:wo-dimtEnsonal mass eIements.

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B-506U

INTEGRAL FORMULAS RELATING GRAVITY GRADIENTS

TO VERTICAL DEFLECTIONS

Presented At

TWELFTH MOVING BASE GRAVITY GRADIOMETER CONFERENCE

UNITE D STATES AIR FORCE ACADEMY

14-15 February 1984

Dynamics Research Corporation60 Concord Street


IABSTRACT

INTEGRAL FORMULAS RELATING GRAVITYGRADIENTS TO VERTICAL DEFLECTIONS

ALAN H. ZORNDynamics Research Corporation

60 Concord Street


A number of integral formulas relating gravity gradients to vertical

deflections are presented. The formulas are derived from standard geodesy

theory and are similar to the Stokes' and Vening-Meinesz integral formulas.

The deterministic formulas provide a physical explanation of certain

properties common to various self-consistent stochastic gravity models.

For example, the high degree of correlation between the gravity gradient,

Txz8 and the vertical deflection, Tx, in most gravity models can be

physically justified based on a deterministic integral formula. The

integral formulas may lead to insight into the construction of gradiometer

filters to estimate vertical deflections which are insensitive to stochastic

gravity models. Implementation considerations involving discretization

and truncation of the integral formulas are explored using surface gravity

data over the Blake Escarpment.

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REAL-TIME VERTICAL DEFLECTION ESTIMATIONUSING GRADIENT DATA

Presented at

TWELFTH MOVING BASE GRAVITY GRADIOMETER CONFERENCEUNITED STATES AIR FORCE ACADEMY

14 - 15 February 1984

Presented by 0

W. FeldmanSperry CorporationElectronic Systems

Great Neck, New York 11020

and

B. EpsteinStrategic Systems Project Office

Department of the Navy


0

• ' I I I I II i

SYNOPSISp

REAL-TIME VERTICAL DEFLECTION ESTIMATION

USING GRADIENT DATA

WALTER K. FELDMANSperry Corporation/Electronic Systems


and

BERNARD EPSTEINStrategic Systems Project Office

Department of the Navyp Great Neck, New York 11020

Techniques/algorithms for making real-time estimates of verticaldeflections from gravity gradient measurements are discussed. First,techniques/algorithms used to condition sensor data for effects ofhigh-frequency noise, pressure sensitivity, self-gradients, bias/trend,and carouselling are outlined. Plots of typical gradient data before andafter preprocessing are shown to illustrate the effectiveness of thepreprocessing. Second, a batch filter formulation for estimatingdeflections from gradient and SEASAT map data is illustrated. Finally,filter results are demonstrated by plots of filter estimates of verticaldeflection vs. reference values for several ship tracks.

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PREPRINT

A SIMPLIFICATION OF THE LEAST SQUARES DETERMINATION OF THE GRAVITY FIELD FROMLOW-LOW SATELLITE TRACKING DATA UTILIZING AN OPERATOR REPRESENTATION OF THE

RANGE OBSERVABLE

V. ReinhardtBendix Field Engineering Corporation, Columbia, Maryland 21045

February 1984

ABSTRACT

One of the purposes of NASA/Goddard Space Flight Center's proposedGeopotential Research Mission is to map the gravity field of the Earth to 1-2rgal in 41,253 1 degree x 1 degree blocks using range rate tracking databetween two polar low orbiting satellites in the same orbit. Because there are41,253 blocks in the gravity map, it takes, in general, at least 9.4x10 13

real floating point operations (flop) to solve the least square fit (LSQF)matrix equation used to determine the gravity potential from the trackingdata. In this paper, by studying the symmetry of an operator which generatesthe range rate acceleration between the satellites from the along trackgravity field, it is shown that the LSQF cross correlation matrix (T-matrix)formed from a complex spherical harmonic (Yzm) expansion of the gravitynotential is, in the circular orbit infinite trajectory length approximation,iagonal in n. Further, for the actual mission, an iterative method which

utilizes the inverse of the m-diagonal portion of the T-matrix, is shown toproduce the 32758 coefficients of the Yzm model of the gravity potential to 64bit precision in less than 1x101

2 flop. A second iterative method is shown tooroduce an approximate covariance matrix for the coetficients in less thananother 5x10 11 flop. This paper also demonstrates a method which eliminatesthe problem of aliasing when attempting to solve the full problemsequentially, first, by solving for the approximate trajectories of thesatellites with a reduced model of the gravity fiId, and then, by solving fora full c,odel of the gravity field using the approximate trajectories.

1. INTRODUCTION

One of the purposes of Goddard Space Flight Center's (GSFC) proposedGeopotential Research Mission (GRM) (Smith, et. al., 1982) is to map thegravity field of the Earth to 1-2 mgal in 41,253 1 degree x 1 degree blocksusing range rate tracking data between t*3 polar low orbiting satellites inthe same orbit (low-low range rate tracking). Using worst case analysis, bothGSFC and the Johns Hopkins University Applied Physics Laboratory (APL), whichhas Derformpd much of the design and study work for the proposed mission, haveconcluded that the data analysis required to turn the range rate data into the*jravity map will consume very large amounts of computer time even on thefastest computers that exist today (Weiffenbach, 1983; vonBun, 1983). One ofthe "nst time consuming parts of the data analysis is in solving the leastsquare it (LSQF) natrix equation used to determine the gravity potential from

2

the tracking data because of the 41,253 blocks in the gravity map. It islater shown that the solution of this equation using Gaussian eliminationtechniques (See Section 5 and Appendix B.) will take, in general, at least9.4x1O 13real floating point operations (flop).

By examining the symmetry of an operator which generates the rangeacceleration from the along track gravity field, this paper demonstrates thatthe LSQF cross correlation matrix (T-matrix) formed from a complex sphericalharmonic (Y,,) expansion of the gravity potential is, in the circular orbitinfinite trajectory length approximation, diagonal in m Further, for theactual mission, an iterative method which utilizes the inverse of them-diagonal portion of the T-matrix, is shown to produce the 32758 coefficientsof the YZm model of the gravity potential in less than ix 1O'2 flop. A seconditerative method is shown to produce an approximate covariance matrix for thecoefficients in less than another 5x101' flop.

This paper also demonstrates a method which eliminates the problem ofaliasing when attempting to solve the full problem sequentially by solving forthe approximate trajectories of the satellites, and then, first, by solvingfor the full gravity potential using the approximate trajectories.

2. GRM GRAVITY FIELD NORMAL EQUATIONS

As shown in Figure 1, the 2 GRM satellites will be launched in thesame 160 Km polar nearly circular orbit with an along track separation ofabout 300 Km (Smith, et. al., 1982). On board navigation and control systemseffectively make the motion of both satellites drag free so changes insatellite motion for times longer than a few seconds are solely due to thegravity field of the Earth (Ibid.). Thus, the motion of each satellite can bedescribed by:

S= - V(r ,t) (2.1)

where r, satellite i's center of mass position, and V, the gradient, are in

units of Rh, the average orbital radius. The coordinate system for thisequation is an approximately inertial geocentric system, so the gravitypotential of the Earth,V, is a function of time due to the Earth's rotation.From this point on, the time dependence of V will not be explicitly written.

To determine the Earth's gravity field, GRM will use the observable p,the magnitude of the range vector between the center of mass of the satellitesgiven by:

r 2 1 -(2.2)2 1

Since the satellites are in the same nearly circular orbit and p is smallcompared with the circumference of the orbit, p is essentially the along trackcomponent of - . Thus we can write:

p(t) - vsV(r 2 ) + :sV(r 1 (2.3)

where 7s is the component of the gradient along dg, the differential alongtrack vector fr'rn satellite I to 2 in units of Rh. We can rewrite this

3

in a more compact form using a position difference operator, P( ), defined by:

P()f(j=) = f(jr+) - f(jt). (2.4)

Using this operator, (2.3) becomes a normal equation for * in terms of V andthe operator P:

p(t)= - P VsV(r) (2.5)

where P will be implicitly P( ) unless the functional dependence is otherwisespecified. A normal equation for , the range rate observable, can begenerated from (2.5) by integrating the equation.

3. CIRCULAR ORBIT APPROXIMATION

If we assume that the orbit is circular, (2.5) can be written in aform which will prove extremely useful later.

In a polar coordinate system, (r, 0, fl, whose north polar axiscoincides with the North Pole of the Earth, the along track diferentialcoordinate, ds, is parallel to the 9 coordinate. If the descending node of theorbit, n is 0 deg - 180 deg, ds = do. If the descending node of the orbit is180 deg - 360 deg, ds = -do. Thus, since r =1 at the orbit height:

± a (3.1)

where the plus or minus depends on the value of 0n as given above.

Using (3.1) and integrating (2.5) with respect to t, we obtain anexplicit form for the normal equation for in the circular orbitapproximation given by:

p(t) = - PV/v (3.2)

where v is the satellite orbital speed. To obtain (3.2), we have used the factthat:

dt = ds/v = *de/v (3.3)

and that P commutes with the integration.

We can also derive an explicit operator expression for P byconsidering the properties of the displacement operator, 0, defined by:

D( f(+ r) = f(i+p). (3.4)

To derive the properties of D, first consider the operator for a differentialdisplacement. (This is a standard derivation found in many texts, i.e.,Tinkham, 1964; Messiah, 1966.) For a differential displacement in the along

* track direction, di, D is given by:

(ds)f(r) = d -s.V f *) + f (r (3.5)

4

so:

D(d) = 1 + dsV s . (3.6)

If we can assume all the dt are along the direction of ' (This true for p<<1,the case here.), then we can write p = nds. Thus D(p) can be written as(D(ds))n yielding:

D(p) : lim (1 + pVs/n)n : epv (3.7)n-w

Using (3.1), we obtain:D(-P) = e - p - - (3.8)

(If p were not small, in the above expression, p would have to be replaced by

the arc length between the end points of p.)

But P can be written as:

P(:) = D( ) - 1. (3.9)

Therefore: 3

P( ) = e±-1. (3.10)

4. SEQUENTIAL LEAST SQUARES FITTING

The standard technique of deriving V from (2.5) or (3.2) using therange rate data is co represent V by a set of known basis functions timesunknown coefficiencs and to use least squares fitting techniques to determinethe coefficients and the trajectory, r(t). Since our primary concern is V andnot F(t), one can attemot to solve the full problem sequentially, first, bysolving for the approximate trajectories of the satellites with a reducedmodel of the gravity field, and second, by solving for a full model of thegravity field using the approximate trajectories. Let us consider theconsequences of using this approach.

For this discussion, let our normal equation be represented as:

p(t) = Ea iPf i(r) + (b i +c i)Pg i(r) (4.1)i i

where we want to determine the cofficients ai and bi+c i given range rate datataken over some unknown trajectory (t) and the known sets of functions fi andgi. Assume also that, through least squares fitting over the same trajectory,possibly using ancilliary data, we have found a solution:

Po(t) = EbiPgi( ) (4.2)i

which has yielded the values bi and the approximate orbit vector ro(t). (Thefit may involve each satellite individually and may use a priori b i's.)Defining p = - po and subtracting (4.2) from (4.1), yields:

5

O :aiPfi(O) + E(bi+ci)Pgi( ) - biPg i(ro). (4.3)1 1 1

Further assume that the functions f.(r') and gi(r) can be approximated byf.(-o) and g'(i'0). (f and g don't ci~ange appreciably over the distance

This turns (4.3) into:0

A (t) = aiPfi( r) + zriPg i(j) (4.4)i i

It is necessary to include the set of c-coefficients in the problembecause the coefficients of the g-fuctions derived from a least square fit of(4.1) will not, in general, be the same as the the coefficients derived from aleast squares fit which doesn't include the f-functions; to obtain the leastsquares condition without the f-functions, the g-function must alias theeffects described by the f-function through alteration of the b-cofficientsfrom the "true" value obtained if the full least squares fit were performed.(This is also true if a priori b-coefficients used in the 1st fit, weredetermined from past fits.) Thus by including the the c- €oeff icied+s A4Heg-functions in a second least squares fit on (4.4), we can correct ouroriginal b-coefficients to obtain better values and we can ensure that theprevious fit does not cause aliasing problems in the a-coefficients. Since thecoefficients of the g-functions are being refit, the only importance of thefirst fit is to produce a good r (t).

If we were interested in a further refinement in the trajactory, wecould apply the approach used to obtain (4.4) to generate an expression for acorrection to the trajectory given by:

" (t) E a aivf( -Z~b i+C i) V(vg i Cd 4.5)1 1

This expression could be integrated twice to generate a better estimate ofr(t) after the a;'s , bi's, and ci's were determined. This better r(t) couldthen be used to obtain an improvea data set for generating Ap(t) which in turncould be used to obtain improved a. and ci values, etc. Thus (4.4) and (4.5)can be used together in an iterative chain to refine the results obtainedbefore iteration.

4.2 USING LEAST SQUARE FITTING TO DETERMINE THE GRAVITY FIELD

s If we now assume the g-functions are to be part of the f-functions and

set Ap = y, we can write (4.4) in the general form:

y(t) = z.aiPfi(r o ) (4.6)

Let us allow the fit functions and the coefficients to be complex even thoughy is real. (This will lead to simpler notation later.) The sum of the squaresor x2 is then given by:

S fsw'(r °y-(aiPf( - i )) (4.7)

r 10 i i 0

where w'(ro) is a weighting factor tq be determined later and where theintegration is over the trajectory, rO.

6

If the elapsed time for the trajectory is long compared with 24hours, the rotation period of the Earth, the trajectory will pass over ornearly over every point on Earth with a probability which is only function ofE (infinite trajectory limit). (The orbit period of the satellite must alsonot be an exact harmonic of the Earth's rotation period, the case here.) Inthe infinite trajectory limit, the orbit also passes over each point an equalnumber of times with ds=+dO and ds=-d. In this limit and assuming thecircular orbit approximation, (4.7) thus becomes:

X2 E f dow( )(y-Ea*Pf*(Q))(y-Za iPfi(n))._82... ~ 2 ~ .~ .(4.8)

-+ 1 1where we have replaced the integral over the trajectory with a sum over thetwo possible directions of ds in P of an integral over a geostationaryspherical surface of radius Rh times an orbit densitV function absorbed intow("4. In (4.8), notice that y is now a function of r, not t, and the implicitt dependence in V has disappeared. y must now be interepreted as the averagevalue of j) over the earth fixed point (eq).The new weighting function w(o) isgiven in terms of the old one by:

w ( M):n( e)w' ('r) (4.9)0

where n(6) is the density of trajectory points over the Earth region (e,p) givenby:

n(e) = Nrev/(nsin8) (4.10)

where Nrev is the number of revolutions in the trajectory. (4.8) is not apractical x2 for performing the actual least square fit, but will be usefulfor the theoretical analysis of the least square fit problem to follow.

From (4.8), least square fit equations can be developed in the usualway (Wolberg, 1967) by taking the derivatives of x with respect to thea-coefficients. This results in the following matrix equation:

L = TA (4.11)

where:

A. = a. (4.12)

Li = (I/2)E+id~wyPf.(6,,p) (4.13)

Tij = (1/2)ZdP2wP fi Pfi (4.14)

L and A are column vectors and T is a hermetian matrix whose inverse providesthe solution:

A = T-1 L (4.15)

The covariance matrix Cij = <SAi A*> (<..> is an ensemble average) isqivyen by: 1ebyC

= TI x/F (4.16)

¢oA1 " ~~~~~~~ i Pk ,;, , ,fA .. , 4-I, k. ,v" * I,,,,,

& PA 1 C...4I. fu

7

where F is the number of degrees of freedom, the number of independentmeasurements minus the number of a-coefficients. We can determine x2 byexpanding (4.8) and using (4.15) to yield:

X2 = _A-A L = y? - L+A (4.17)

where AtL = EA*Li , etc, and where:1

y2 = j£fW(e) y2 do (4.18)+

If the x2 of (4.7) is used instead of that of (4.8), the equationsremain the same except that the sum plus integral over the spherical surfacegoes back to an integral over the trajectory and w is replaced by w'

5. SOLVING THE MATRIX EQUATIONS

THE SCALE OF THE OVERALL PROBLEM

The basic GRM gravity data analysis problem is to find the orbitapproximation, r+(t), to solve the matrix equation (4.11) in terms offunctions at the orbit height, and to analytically extend the solution atorbit height to the geoid. The analytic extension problem can be solvedwithout large computations by using basis functions for V which are solutionsto Laplace's equation. We will not concern ourselves with the orbitcomputation problem here. We will concern ourselves here with the remainingproblem, the solution of the matrix equation.

There are two aspects of the matrix problem, forming the matrixelements and inverting T. The size of these tasks are determined by the numberof basis functions, N, and the number of range rate data points collected, n.N is determined by the number of blocks in the gravity map. In a coordinatelattice which minimizes the number of blocks (regular square spherical shelllattice), the 1 degree resolution requirement of GRM (Smith, et. al., 1983)translates into 41,253 such blocks. If localized functions are used to makethe map, one function to a map block, then there will be 41,253 basisfunctions. If complex spherical harmonics,Yzm, are used to the point where onehalf wavelength of the function (n/t) equals the block size? there will bespherical harmonics to order =180 and there will be ((941)'-3)=32758 realbasis functions. For a 6 month mission with 1 data point per 4 seconds(Ibid.), n is 3.9x106.

Inverting an NxN matrix using Gaussian elimination techniques requiresapproximately (4/3)N3 real floating point operations (flop) (See Appendix B.)and requires access to the matrix elements N times. Use of caching techniquesas planned by GSFC in the proposed mission (Weiffenbach, 1983) and asdescribed by others (Duff, 1981) can reduce the number of of disk accesses tothe point where a large processor can run at 10 to 20 percent of fullprocessing speed (Briggs, 1984). This means that with a Cyborg type computer,

* one can plan on having 10-20 Mflop/s of effective speed (Ibid.). Using thisnumber for the processing speed and 41,253 for N, we obtain that 9.4x10 13

flops and 1300-2600 hrs of computing time required for inversion of T.

.(, ;

Forming the matrix elements for the matrix equation is dominated bythe calculation of T. Naively approaching the problem, it would seem thatthere are on the order of nxN = 7xO flop required to calculate theelements of T, based on calculating N"/2 basis functions for n data points.Even at 80 Mflop, this would require 23,000 hrs to compute the elements.However, since the basis functions only change for lengths on the order of ablock size, one does not have to calculate the basis functions for each point.APL has developed a scheme to do this in approximately 5000 hrs of computertime (Weiffenbach, 1980). The problem of calculating the matrix elements,though clearly the dominant problem, will not be addressed here.

One approach to finding a way of reducing the computation timeassociated with inverting T is to investigate the symmetries and nearsymmetries associated with the P operator and various sets of basis functionsto see if the matrix can be broken down or approximately broken down intoseveral irreduceable submatrices (diagonal blocks of submatrices) of lowerdimensionality (Tinkham, 1964). These can be inverted with less computationtime because the number of operations goes as such a high power (N3 ) of thedi mensionali ty.

If a symmetry is only approximate, it will create small terms insteadof zeros off the irreduceable submatrices. If a residual matrix equation iscreated by operating on the original matrix equation, TA = L, with the inverseof the approximately irreproduceable submatrices of T, it is shown in AppendixB that the residual equation can be solved iteratively if the residualT-matrix is diagonally dominant. If this is the case, Appendix B also showsthat the residual equation can be solved for A iteratively in less than x1O 2

flop (Appendix B.4) and that an aproximate T-1 for determining the covariancecan be generated in another 5x10 flop (Appendix B.6). 0

Investigations by the author using the form of the P operator derivedin Section 3 were carried out using point anomaly basis functions andspherical harmonic basis functions. The point anomaly functions were generatedfrom the gravity field of point masses at the center of square spherical shelllattice blocks at a depth below the surface equal to the width of the block.The T-matrix generated from these anomaly functions was highly diagonal butunfortunately was not diagonally dominant. Also unfortunate was the lack ofcorrelation in the large off diagonal terms (nearest neighbors) which may causeback-filling problems (See Appendix B.I.). (This is an essential feature oftrying to index a two-dimensional surface with a one dimensional matrix index- most nearest neighbors are not nearest in index number.) To show that thematrix does or does not have back-filling problems will require furtherinvestigation.

The investigation using spherical harmonic basis functions proved sosuccessful in demonstrating that there is an approximate symmetry which willgreatly reduce the computation time required for inverting T that only thisapproach will be presented here. The next section presents the approach basedon these functions.

0

9

6. CONSEQUENCES OF USING SPHERICAL HARMONIC BASIS FUNCTIONS

Let us consider the consequences of using spherical harmonics as thebasis functions for V. To simplify writing the equations and for other reasonswhich shall become apparent later in this section, let us use the complexspherical harmonics, Ym(e,), with complex coefficients a... rather than thereal spherical harmoni f, R (0,) and S m(e,@) used normally in Geodesy(Heisman and Moritz, 1967)."Tn terms of im a solution to Laplace's equationvalid in all regions outside a sphere of any radius is given by (Jackson,1962):

0, z aim (6.1)V(r,fl = vE E_ i- i Y Zm(e,f)

Z=o m=-k r

where we have normalized the coefficients so:

V 9 (6.2)

f(1,64) E a Y m(e, )Z=0 Mn=-Z

Yem is given by (Messiah, 1966, Vol. 1, Appendix B):

Y= (-)m [ (2z+QI- 'm)! (cose) eim (6.3)km 4 (+m)! PM(63

where P.im is the associated Legendre polynomial of order (Zm) given by thefollowing for positive m (Ibid.):

P m(x) (1-x3) m di+m (xa-1)k (6.4)dx

For negative m, the PIm are defined by (Ibid.):

Y (-I) m * (6.5)

As defined here, the Y0,are orthonormal with respect to integrations on aunit sphere (Ibid.).

For a spherical harmonic basis, in the circular orbit approximation,the T-matrix elements become:

Tm m = A E f Y M P Y m(e) dQ (6.6)

where we have put a restriction on w(Q) that it be only a function of e.Normally, the weighting function in terms of the trajectory integral, w'(rd,will be a constant; since p is approximately the same everywhere along thetrajectory, we expect the range rate data collected all along the trajectoryto have about the same noise content. This means that normally w(sQ) isproportional to n(e), so the restriction is normally satisfied. In the nextpiraqraph, we will show that this restriction, along with the properties of Rin the circular orbit approximation, are instrumental in simplifying theinversion of the T-matrix.

10

The expression for the P operator in the circular orbit approximation,

(3.10), is written in terms of the operator a/aQ. Since P has no 0 dependence,it does not effect the functional dependence of the Y m' s with respect to(e Im€). Thus, since we assumed w(s) is only a function of e, the matriSelements will be zero if m is not equal to m' because the integral of eimo

over 360 degrees is zero unless m=m', regardless of the 3 behavior. In otherwords, the T-matrix is diagonal in m. For a given m, there are only Y m's withwith 0>=1ml, so each T mem is non-zero only for Z > im: and z'> ml.

The fact that the T-matrix is diagonal in m greatly reduces the numberof computer operations required to invert the T-matrix. If we subgroup theindices by m first rather than by 9. first, we can form the T-matrix into2Z +1 diagonal m submatrices. (zm =180 is the maximum value Iml can have.)Eaci m submatrix for m>=1 has ( Z_-tmf+1)2 complex elements ( (2Z +1)elements for m=O). Summing for m = - m to Zm the approximatelym

(16/3)(Z -lml+l) ' real operations required to invert each submatrix, we findthat only approximately (8/3) Z4 =2.8x10 9 flop are needed to invert the fullmmatrix. Also of impact on the computation time is the fact that the largestsubmatrix (m=O) only requires 260642 real words for storing the matrixelements (and only 64800 words for the next largest, etc.), so we can loadeach submatrix to be inverted wholly into core memory on a large machine. Thismeans the machine can run at virtually full speed to invert the matrix. Soassuming an 80 Mflop/s computation rate and a 1 Mbyte/s disc transfer rate,one can invert the matrix in 66 seconds.

To aid in further investigations of this approach, it would be usefulto be able to calculate the matrix elements of T in the circular orbitapproximation. Appendix A shows how to generate closed formulas for the matrixelements without actually performing the integrations. It shows that theoperator P in the circular orbit approximation can be written in terms of thequantum mechanical angular momentum operators, L+and L_ (Messiah, 1966), whichhave well known properties when acting on Y and yield linear combinations ofother Y m's. It also shows that the orbit density function in the circularorbit approximation times Y, can be expressed as linear combinations of otherY ms. Together these properiies, along with the orthonormal property of they s, allow T , to be written in terms of sums over coefficients whichare functions J ',m',and p.

7. WILL AN ITERATIVE INVERSION OF THE RESIDUAL T-MATRIX CONVERGE?

We showed for (A) a circular orbit polar orbit, (3) w(r) = w(a), and(C) the infinite trajectory limit that the T-matrix can be inverted fastbecause it is diagonal in m. In Appendix B, it is shown that the actual matrixequation, TA = L, can be solved iteratively for A (Appendix B.4) and for anapproximate covariance (Appendix B.6) if the residual T-matrix formed byoperating on T with the inverse ot the diagonal-m portion of T isdiagonally dominant (Appendices 9.3 and 8.5). Let us examine the violations ofthe assumptions (A), (8), and (C) that will occur in the actual plannedmission to see if the diagonal dominance criterion is satisfied or not.

11

In the infinite trajectory limit, for the diagonalization in m to beviolated there must be some 0 dependence in P, w(i?), or vs (See equation2.5.).A small eccentricity in the orbit would not cause problems because this onlycauses mixing of the radial and ecoordinates. A deviation of the orbit from apolar one causes mixing of the Ocoordinate into the p. argument of P andinto vs, but in a way which doesn't effect the diagonalization ao f.Iows TI.sedependent part of p-' can be written as, and the dependent part of Vs isproportional to:

(p.V)o = tp (6)(1/sine ) . (7.1)

where p ( 0) functionally describes the orbit with 6 as the independentvariable and wheje .the * is for the 2 possible directions of ds at the point, ). Since 4e mO =ime ilm , this term does not change the orthogonality

of the Ym'S, so the matrix elements off diagonal in m remain zero. Also adeviation of the orbit from a polar one does not cause w(J) to become afunction of p since the density remains axially symmetric regardless of theorbit in the infinite trajectory limit. Therefore, a deviation of the orbitfrom a polar one does not produce terms off diagonal in m in the infinitetrajectory limit.

The finiteness of the actual data, however, can introduce p dependenceinto w(i). As stated previously, this effect goes to zero in the limit of aninfinite trajectory length since the phase relationship between the rotationof the Earth and the rotation of the spacecraft around the geocentric fixed

*orbit go through all possible relationships equally. Since the orbital periodis much less than the Earth's rotation period, the effect of the finite datalength in the limit of many orbits, is given, to 1st order, by the residualassymmetry in the trace of the orbit as the Earth turns under the orbit. Sincethe Earth turns under the orbit once in 24 hours, the relative assymmetry inn(2) will be given by a maximum of (12 hrs)/(6 mo) =3xi0 "3. This number is justabove the level of the order of magnitude criterion for diagonal dominancebased on the assumption that all the off diagonal elements in m are equal asgiven in (B.18). (See Appendix B for a more detailed discussion.) We must,thus, examine this effect in more detail.

The assymmetry is a periodi step type function in with a 21r periodand an amplitude 0.003. (For Oa<0<4b, w is less by 0.003 than w for othervalues of P). Since the - dependence of Yom is given by eimrP , the relativesize of 1TZmZ,r,, to IT I will be given by the the absolute value of(m-m')th harmonic amplitfiemof the fourier transform of a periodic step typefunction. The absolute value this harmonic is less than 0.003*2/(7:m-m':)regardless of the width of the step (Selby, 1974). Therefore the criterion forconvergence of the iterative inversion without assuming the off diagonal matrixelements are equal, (B.17) (See section B for more detail), can be written as:

0.5 > MAXz,z' JT~mZmI/JTtmZ m I = 0.002*2 ?n(1/m)

m= 1

=0.004fmdx/x (7.2)

0 0.004 In( m0.5 > 0.02

12

where we are using the worst case conditions off,2!= Z and m=O. Notice thatthe diagonal dominance criterion is more than met. Th% inclination of theorbit an angle z from a polar orbit will not change this result since we didnot assume any particular w(e) for the infinite trajectory length case toobtain the relative size of the step function in 0.

Off diagonal T-matrix terms in m will be produced by higher orderspherical harmonic terms in the gravity potential through their effect on thetrajectory estimate (t). (In our anqular integration model, this willproduce 0 dependence in wCP through ro(t)'s influence on n(ed.) Terms in thegravity field with the index im' will cause coupling terms in T between m andm'. However, all gravity potential spherical harmonic terms with Im',>O areless than 3x10 6 of the 1/r term (Gray, 1972) so we expect the relative sizeof terms off diagonal in m due to this effect to be on the order of 3x10 "6orless. (B.18), as mentioned above, states that the iterative inversion willconverge if the relative size of the off diagonal elements is less than 0.001.Thus, this effect should not prevent the full matrix from convergingiteratively.

Another effect which must be considered is roll and yaw of thesatellites with frequencies equal to nv/(21T) (n= 1,2,3 ..... ) producing Tterms between m and m+_m'. Because the orbit is polar, however, this would notproduce terms coupling different m's but terms coupling different 's. Also,since this effect would produce range rate data which would alias the effectof gravity terms, GSFC and APL have taken great pains to ensure that thiseffect will be measured and taken out of the range rate data before furtheranalysis. 0

Lunar and Solar effects are also too small to effect theconvergence of the iterative inversion of the T-matrix and they will usuallybe removed from the range rate residual by the least squares fit whichdetermines the approximate trajectory.

8. CONCLUSIONS

The approach discussed in this paper of examining the symmetries of anoperator representation of the observable proved very fruitful in simplifyingthe solution of the matrix equation for determining the gravity potential fromlow-low satellite range rate data. This still leaves the even larger problemof determining the matrix elements of the equation. This problem may also besimplified by utilizing the T-matrix in the circular orbit infinite trajectoryapproximation, which can be very quickly calculated (See Appendix A.), as azeroth order approximation for the actual matrix. Further study along theselines will probably prove very promising. In addition, the circular orbitinfinite trajectory approximation should prove very useful in studying theeffects of various system errors and interactions on the errors in thecoefficients of the spherical harmonic expansion of the gravity potential.This aporoach may also nrove very fruitful in simplifying other problems insatellite geodecy and orbit determination.

0

13

APPENDIX A. CALCULATING THE T-MATRIX IN THE CIRCULAR ORBIT APPROXIMATION

The effect of P on YZmcan be calculated by using the followingformulas (Messiah, 1966,V.1, Appendix B):

a/e = (1/2)(e'0L - eiCL ) (A.1)

where:

L. =e-+;€(+ 3 + icot(e)L ) (A.2)

and where:

L Y = (t(t+i) - m(m±1)) Y (A.3)

If we now turn the full formalism of quantum mechanics, we can obtaina representation of the T-operator. In bra-ket notation (Messiah,1966), onecan write the matrix elements of T as:

TiniMe = iZ<m"Ptw(O)P' Vm'> (A.4)

where the superscript t mnans the hermetian adjoint of the operator and wherethe w(6) is put between P and P to indicate that the P operators only operateon the Ym functions. Thus the T operator is:

T = (ee6\6±-)w(e)(e- /o-1) (A.5)4.

where the hermetian adjoint of a/3e is written as 66\6 to emphasize that itoperates in the backward direction.

If we expand the exponentials, collect terms of the same order inp, and oerform the sum over the two directions of the orbit, we obtain:

G 2n-i 6 2n-j 2nT = e (A.6)

n=1 j= (2n-j)! j!

The lowest order term in p, TI is given as:

T I = ( e)6\6)pw(o)( ;/ae). (A.7)

Setting w'(r)-1, we can write w(a) as:

w(e) = Nrev/(wsine). (A.8)

From a recursion relation derived from the fact that Y is proportional toe-l sin@ (Mertzbacher, 1961), one can show that the "uptprator" 1/sineoperating on Y ;m yields:

Yzdslne F2 m [ (A .9)

=0m(2 Y2j..l,m.I-2j+2

nwhere Fm, the number of permutations of m object out of n, is:

nFm = n(n-1)(n-2) ..... (n-m+1) (A.1O)

and where:

Z - 2jm - I . Im+1| (A.11)

Using (A.1), (A.3), (A.6), (A.9), and the orthonormality condition(6.6), a closed formula can be written for the T-matrix elements as a sum ofcoefficients which are calculable functions of ,m,z',m', and p without havingto perform any numerical integrations. This could be set up in recursive formon a cremputer and accomplished without using a great deal of computer time.

S

Is

APPENDIX B. SOLVING THE MATRIX EQUATIONS

8.1 DIRECT METHODS

Before proceeding it is useful set up some simplifying terminology. Thematrix equation to be solved is given by:

TX = B (8.1)

where T and B are known and X is to be solved for. T is a square matrix ofsize NxN, B is an NxM matrix (a column vector when M=1), and the solution X isan NxM matrix. In the case of finding the solution of the inversion of T, L6 isthe NxN identity matrix I, and X=T-'. When perfo,'ming certain operations onthe equation in finding the solution, ,t is useful to introduce the concept ofthe augmented matrix, Ta. Ta is an Nx(N M) matrix formed by augmenting the Ncolumns of T with the M columns of B; that is:

Ta = T (+) B (B.2)

where (+) indicates the dimensional addition of the columns of T and Btogether.

The various direct numerical methods used to solve matrix equationsand invert matrices fall into 3 basic catagories, Triangular Gaussianelimination methods (Doolittle,Crout, etc.), Full Gaussian elimination(Gauss-Jordan), and sequential transformation methods (LU deconpostion,partitioned matrices, Householder) (Riess, 1981; Gastinel, 1970; Detoma andWardrip, 1982; Ralston and Rabinowitz, 1978). The sequential transformationmethods are usually used only for special cases to take advantage of specialsynmetries such as large blocks of zero elements (LU and partitioning) or whenonly a small amount of memory is available (Householder). For the general caseon large computers, usually one of the first two catagories is used.

Both of the first two catagories rely on Gaussian elimination (Riess,198I; Gastinel, 1970). This consists of turning the off diagonal elements of okpivotal column (full), 1, or a section of the pivotal column (triangular) of Tinto zeros by operating on Ta with:

Ta (n+i) = Ta(n) - (T(n) /T (n) )Ta(n) (B.3)i= ij il lj 11

for all j not equal I in the Jordan method and for ,11 j greater than 1 in thetriangular method. This is performed N-i times wits. I = 1 to N-i to produceI(+)X in the Jordan method and T"(+)B' in the triangular method where T" is anupper triangular matrix. In the Jordan method the row j=l of Ta is alsooperated on by dividing the Ith row by T .(The complication of pivoting tominimize the error will not be discussedlBere (Riess, 1981; Gastinel, 1970).)In the trianqular method, X is found by back-solving the modified equation asfollows: Since T" is upper triangular, the lowest row of the equation is ofthe 'orm T" X N=BN.' so it can be solved for X by dividing BNi by T"

1 The value oX can iow be used in the next lowe 4 row of equatioa to find*x and this in turn can be used to find XN2 ,j, etc., until t.,e wholePe t 4 on is solved.N-j

For M=1, and T symmetric, the number of operations to performtriangular Gaussian elimination (The triangular method takes fewer or as manyoperations as the Jordan method for all cases (Riess, 1981).) is approximately(1/3)N 3 real floating point operations (flop) (Riess, 1981; Gastinel, 1970).When M=N and T is symmetric (inversion) it takes approximately (4/3)N 3 flop.

Using this method to invert T, for T complex,it takes 4 times as manyreal operations as for T real, Thus for complex and symmetric,(4/3)N 3 (real) flop are required to find A and (16/3)N 3 (real) flop arerequired to invert T.

It is not enough just to show that large numbers of the matrixelements are zero or near zero; when a matrix is inverted using one of thevarious Guassian elimination techniques zeros and near zeros can beback-filled with non-small values very quickly during the factoring stageunless there is some symmetry to prevent this from happening (Riess, 1981;Duff, 1981). To see how fast the non-zeros turn into zeros, consider thefollowing. Suppose we have a large NxN matrix with a relatively small number,k, of non-zero off diagonal elements in each row with the non-zero elementsoccuring in a random pattern from row to row. In the factoring process ofGuassian elimination each row of the matrix T other than a pivotal row, 1, istras formed by the relation (B.3). If T.(n) were zero oi near zero andT il n) and Tl (n) were non-small, one ca see that T.(n 1)would become anon-small num er. This is what is called back-fillin After operating on themarx elements with 1 pivotal rows and columns, because the probability ofTi) being zero is virtually I when there are only a small number ofn3l-zero off diagonal elements, one can show that the average number ofnon-zero terms in the remaining columns grows approximately as k2l until thenumber of non-zero elements per row in the remaining columns becomescomoarable to N. To keep this from happening, the zeros in one row must becorrelated to the zeros in another row to a very high order and must remaincorrelated during the elimination operatations.

3.2 ITERTIVE METHODS

The two principal iterative methods are the Jacobi method and the

1 muss-Seidel method (Riess, 1981; Gastinel, 1970). The Jacobi method finds then+l)th estimate of X from the(nth estimate by:

X(n+l) = (B. - T X n) )/T. (B.4)im im i J 11

The Gauss-Seidel method uses the most current estimate for X(n) , which for k<i

is X 1ni1). Each iteration takes 2N4 flop for M=1 and 2N fodminverting aniatrix when T is symmetric. Thus, there is no gain in using tnis technique forinverting mitrices, but there is a very substantial gain for M=1 as long astne solution converges in a relatively small number of steps.

B.3 THE COPVEOrGENCE OF ITERATIVE SOLUTIONS

If T is diigonally dominant and non-singular, the Jacobi,]iuss-Seidel, and other similar iterative methods for finding the inverse of amatrix or solving nAitrix equations will converge regardless of the first guess

r

'7

* (Riess, 1981; Gastinel, 1970). Diagonal dominance is defined by:

1 > Maxi IT ij /IT i I (B.5)

where E' means the sum over j with j not equal to i and Max i means the largestvalue for i = I to N. In fact if we define q as:

q = Max i z'ITijj/ITiiI (B.6)

one can show that after n iterations:

IlAX(n) tl/llAX(O)tl S q n (B.7)

where llAX(n),I is the largest error in the nth estimate of the solution, X.This means that, starting with Ti as a first guess for X, and given q =1/2,the matrix elements would converge to 2-n times the largest element in niterations or approximately to a precision of 2 n in n iterations. If thematrices are complex, as stated above each iteration in the Jacobi orGauss-Seidel methods requires 8N 2 (real) flop. Thus, obtaining a result with2-nprecision requires only 8N2xn flop. For 64 bit precision, this becomes512N 2 (real) flop.

Diagonal dominance is a stringent condition for large matrices. If allthe of diagonal elements were the same, (B.5) would become:

ITij I/ITii I = q/(N-1) (B.3)

For N=41,253 and q =1/2, this condition is IT.. /T.. I < 1.2x10-5.i3 11

B.4 ITERATIVE METHOD FOR DETERMINING THE EXACT COEFFICIENTS.

In this section, we derive an iterative method which solves for Ausing X', the inverse of T', where T' is the T-matrix with all elements offdiagonal in m set to zero. The method produces A to 64 bit precision in lessthan ix1IO 2 (real) flop. (See Sections B.5 and 7 for proof of convergence.)It can produce A to any desired precision with corespondingly fewer or moreflop.

If we observe that:

T = T' + AT, (B.9)

we can write:

TA = (T' AT)A : L. (B.1O)

Multiplying both sides of (B. 9) by X', we obtain:

(I X' T)A = X'L. (B.11)

This equation can now be solved by the Gauss-Seidel or Jacobi methods if thematrix elements of T off diagonal in m are small since I+X'AT has no matrixelements diagonal in m which are off diagonal in Z. (See Sections B.5 and 7for proof of convergence.)

Calculating A from (B.11) is dominated by the number of operations inthe multiplication of X' by AT and in the iterative proceedure. Since X' isdiagonal in m, and both Xand AT are symmetric, it takes 8N z' /3= 5.1xi0 11

(real) flop to perform the multiplication. Each iteration of the solutiontakes 8N2 (real) flop. Assuming the solution will converge 64 iterations orless (See Section B.3.), the iterative solution will take 512N 2 =5.5x1011

(real) flop. This means that the total number of operations to find A is1.06x10' 2 (real) flop. This is a factor of 88 fewer flop than our originalestimate based on (4/3)41,253 3=9.4x101 3 flop.

B.5 CONVERGENCE CRITERION FOR MATRIX ELEMENTS OFF DIAGONAL IN m

Based on the iterative method of the previous section, iterativeconvergence criteria for the T-matrix elements off diagonal in m will bederived.

Since T has no diagonal terms in m and X' has only diagonal terms inm, we note that I +X'T is equal to I for diagonal terms in m,and equalto X' AT for off diagonal terms in m. The criterion for convergence of theiterative solution of (8.11) is that I+X' AT be diagonally dominant as given by(B.5) and (B.6), which in this case, with the noted properties of I+X'AT,become:

1 > q Max Z'Z ,, X I I. (B.12)Zm Z m fm Z m m

Noting that:

I ITII = Maxi 'IT .1 (B.13)j 1j

is a matrix norm of T which satisfies the Cauchy-Schwarz inequality (Riess,1982), 'e + .re'+e;

I IX' ATI I < IIX' I IAT1I (B.14)

q < I IX'II I ATI. (8.15)

But IIX'II is on the order of 1IIT'II, so we can write the approximaterelation:

q < MAX E' 'ITQm "m / 'ZITQ'm'm I (B.16)

Assuming IT m d, is on the order or bigger than IT s,,m,m I (For this matrix,we expect tle o F diagonal elements to get smaller as m and m' differ.), wecan "divide out" the sum over ;" in the numerator. This yields the followingaooroximate relation:

(q

q < MAX ,IT , J/IT , I (B.17)

where MAX, ., means the maximum value ranging over allp'k and V. (B.17),though of goiwhat dubious derivation, is probably good for order of magnitudecalculations.

A crude estimate for diagonal dominance can be obtained from (B.17) bysetting all the off diagonal elements equal. Setting q =0.5, for the worstcase of z and Z' equal to 180 and m equal to 0, we obtain:

IT mm, I/IT zm,m < q/(2 1 *1) = 0.001 (B.18)m

where the value given is for q= 0.5.

B.6 PERTURBATION THEORY METHOD FOR DETERMINING THE COVARIANCE MATRIX

From (4.16), the covariance matrix can be determined if we determineT-1 and x2. To determine T -1 for this purpose, we don't need an exactsolution, but only an approximate one good enough for error estimates. Thefollowing method will obtain an approximate solution for T -1 in only 5x10''(real) flop.

We want the solution to:

TX=I (B.19)

where I is the identity matrix.

Suppose we have a solution:

T' X' =I (B.20)

where T is close to T. Then by manipulating (B.19) and (B.20), we cangenerate:

X-X' = (I - X'T)T -i (8.21)

If we use X) the separation T = T' + AT from Section B.4)and make theapproximate substitution T-1 = X', we obtain the following approximate formula(1st order perturbation solution):

T = X - X'ATX' (B.21)

Given both X'AT and X' (from the solution to TA=L in Section B.4), ittakes 8N Z3 /3 (real) flop to produce X'TX' and 4 Z3 /3 (real)flop to

m2Derform_ the remaining subtraction. The actual x can'pe calculated from (4.17)using y defined by (4.18) with the integral in (4.18) changed back to theintegral over the trajectory. For 3.9x10 6 data points, calculatingtakes 8x10 (real) flop and calculating A L takes only 3N=2.6xlO '(real) flop.Thus, it takes approximately 8N Z3 /3 (real) = 5.1xlO 1' (real) flop toproduce this estimate of the covariance.

ACKNOWLEDGEMENT

The author would like to acknowledge the help of the many persons atGoddard Space Flight Center and the Applied Physics Laboratory who cheerfullysupplied a great deal of unpublished material and verbal information on GRMand the matrix inversion problem.

REFERENCES

W. Briggs, Bendix Field Engineering Corporation, Columbia, Maryland, privatecommunication based on experience with the inversion of a 10,000xlO,000matrix, 1984.

E. Detoma and S. C. Wardrip, "Useful Statistical Techniques for Time andFrequency Data Reduction", NASA/Goddard Spaceflight Center internal report(unpublished), 1982.

I. Duff, ed., SPARSE MATRICES AND THEIR USES, Academic Press, 1981.

N. Gastinel, LINEAR NUMERICAL ANALYSIS, Academic Press, 1970.

E. Gray, ed., AMERICAN INSTITUTE OF PHYSICS HANDBOOK, 3rd ed., McGraw-Hill,1972.

W. Heisman and H. Moritz, PHYSICAL GEODESY, Freeman and Co., 1967.

J. Jackson, CLASSICAL ELECTRODYDNAMICS, Wiley and Sons, 1962.

E. Mertzbacher, QUANTUM MECHANICS, Wiley and Sons, 1961.

A. Messiah, QUANTUM MECHANICS, Wiley and Sons, 1966.

A. Ralston and P. Rabinowitz, A FIRST COURSE IN NUMERICAL ANALYSIS,McGraw-Hill, 1978.

J. Riess, NUMERICAL ANALYSIS, Addison-Wesley, 1982.

S. Selby, ed., CRC STANDARD MATHEMATICAL TABLES, 22nd ed., CRC Press, 1974.

D. Smith, R. Langel, and T. Keating, "Geopotential Research Mission (GRM)",NASA/Goddard Space Flight Center document (no I.D. number), August, 1982.

M. Tinkham, GROUP THEORY AND QUANTUM MECHANICS, McGraw-Hill, 1964.

F. 0. vonBun, D. Smith, W. Kahn, S. Bergson-Willis, T. Keating, et. al.,orivate communications, NASA/Goddard Space Filght Center, 1980-1983.

G. Weiffenbach, V. Pisacani, et. al, private communications, Johns Hopkins

University Applied Physics Laboratory, 1980-1983.

J. Wolberg, PREDICTION ANALYSIS, D. Van Nostrand, 1967.

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DEPARTMENT OF THE AIR FORCEAIR FORCE GEOPHYSICS LABORATORY (AFSC)

HANSCOM AIR FORCE BASE, MASSACHUSETTS 01731

REPLY TO

ATTNOF LWG/Lt Warner

SUBJECT Minutes, Gravity Gradiometer Conference, 12-13 February 1985

TO Conference Attendees

1. Please find attached a copy of the meeting minutes, presentations

(if individually requested), and list of conference attendees. The minutes

were compiled by Dr. Chris Jekeli and me from abstracts submitted by

various presenters and notes taken at the conference. In some cases our

notes may have been sketchy resulting in our interpretation of questions

and responses after each presentation. In all cases we attempted to

report each presentation correctly and adequately.

2. 1 have reserved the same facilities at the USAF Academy for next

year's conference to be held 11-12 February 1986. We'll start requesting

papers late this summer. Registration will proceed soon after.

3. Thank you for your interest in gravity gradiometry. We look forward

to your continuing support.

DONNA L. WARNER I Atch: MinutesConference Coordinator

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