SOFTWARE INTERFACE SPECIFICATION

SPHERICAL HARMONICS ASCII DATA RECORD (SHADR)

prepared by

Frank G. Lemoine

Code 698, Planetary Geodynamics Laboratory

NASA Goddard Space Flight Center

Greenbelt, Maryland, 20771 USA

Peggy L. Jester

SGT, Inc. / Code 614.1

NASA GSFC/ Wallops Flight Facility

Wallops Island VA, 23337 USA

Version 2.1

22 May 2013

|====================================================================|

| |

| DOCUMENT CHANGE LOG |

| |

|====================================================================|

|--------+--------+------------+-------------------------------------|

| 1.0 |06/02/20| All |Adapted MGS SHADR SIS to include |

|--------+--------+------------+-------------------------------------|

| 1.0 |06/03/15| All |Miscellaneous edits |

| | | | |

|====================================================================|

| 1.0 |06/06/29| All |Integration of PDS review comments |

|====================================================================|

| 1.1 |07/04/04| 4.2.2.1 |Integration of RS review comments |

| | | 4.2.2.2 | |

|====================================================================|

| 1.2 |08/07/28| 2.3 |Updated file naming convention |

|====================================================================|

| 2.0 |13/04/17| All |Adapted to include GRAIL |

|====================================================================|

| 2.1 |13/05/22| All |Minor edits |

|====================================================================|

Contents

Document Change Log................................................. 2

Contents........................................................... 3

Acronyms and Abbreviations...........................................5

1. General Description...............................................6

1.1. Overview........................................................6

1.2. Scope...........................................................6

1.3. Applicable Documents............................................6

1.4. System Siting...................................................8

1.4.1. Interface Location and Medium.................................8

1.4.2. Data Sources, Transfer Methods, and Destinations..............8

1.4.3. Generation Method and Frequency...............................8

1.5. Assumptions and Constraints.....................................8

1.5.1. Usage Constraints.............................................8

1.5.2. Priority Phasing Constraints..................................8

1.5.3. Explicit and Derived Constraints..............................8

1.5.4. Documentation Conventions.....................................8

1.5.4.1. Data Format Descriptions....................................9

1.5.4.2. Time Standards..............................................9

1.5.4.3. Coordinate Systems..........................................9

1.5.4.4. Limits of This Document.....................................9

1.5.4.5. Typographic Conventions....................................10

2. Interface Characteristics........................................11

2.1. Hardware Characteristics and Limitations.......................11

2.1.1. Special Equipment and Device Interfaces......................11

2.1.2. Special Setup Requirements...................................11

2.2. Volume and Size................................................11

2.3. Labeling and Identification....................................11

2.4. Interface Medium Characteristics...............................12

2.5. Failure Protection, Detection, and Recovery Procedures.........12

2.6. End-of-File Conventions........................................12

3. Access...........................................................13

3.1. Programs Using the Interface...................................13

3.2. Synchronization Considerations.................................13

3.2.1. Timing and Sequencing Considerations.........................13

3.2.2. Effective Duration...........................................13

3.2.3. Priority Interrupts..........................................13

3.3. Input/Output Protocols, Calling Sequences......................13

4. Detailed Interface Specifications................................14

4.1. Structure and Organization Overview............................14

4.2. Detached PDS Label.............................................14

4.2.1. Label Header.................................................14

4.2.2. TABLE Object Definitions.....................................17

4.2.2.1. SHADR Header Object Definition.............................18

4.2.2.2. SHADR Coefficients Object Definition.......................20

4.2.2.3. SHADR Covariance Object Definition.........................22

4.3. Data File......................................................24

4.3.1. SHADR Header Object/Block....................................24

4.3.2. SHADR Coefficients Block.....................................26

4.3.3. SHADR Covariances Block......................................27

Appendix A. Description of Spherical Harmonic Model Normalization...28

A.1 Definition of Model for the Potential...........................28

A.2 Definition of the normalization used............................29

Appendix B. Example Data Products...................................31

B.1. Example Label..................................................31

B.2. Example Data Object............................................37

Tables

4-3-1. SHADR Header Block...........................................25

4-3-2. SHADR Coefficients Block.....................................26

4-3-3. SHADR Covariance Block.......................................27

Figures

4-2-1. SHADR Label Header...........................................15

ACRONYMS AND ABBREVIATIONS

ANSI American National Standards Institute

APL Applied Physics Laboratory

ARC Ames Research Center

ARCDR Altimetry and Radiometry Composite Data Record

ASCII American Standard Code for Information Interchange

CCSDS Consultative Committee for Space Data Systems

CD-WO Compact-disc write-once

CNES Centre National d'Etudes Spatiales

CR Carriage Return

dB Decibel

DSN Deep Space Network

DVD Digital Video Disc or Digital Versatile Disc

EGM96 Earth Gravitational Model 1996

FEA Front End Assembly

GRAIL Gravity Recovery and Interior Laboratory

GSFC Goddard Space Flight Center

IEEE Institute of Electrical and Electronic Engineers

IAU International Astronomical Union

JHU Johns Hopkins University

JPL Jet Propulsion Laboratory

J2000 IAU Official Time Epoch

K Degrees Kelvin

kB Kilobytes

km Kilometers

LAST Laser Altimeter Science Team (MESSENGER)

LF Line Feed

LP Lunar Prospector (mission or spacecraft)

MB Megabytes

MESSENGER MErcury Surface Space ENvironment, GEochemistry, and Ranging

(acronym for mission to Mercury)

MGN Magellan

MGS Mars Global Surveyor

MIT Massachusetts Institute of Technology

MLA MESSENGER Laser Altimeter

MO Mars Observer

MRO Mars Reconnaissance Orbiter

NAIF Navigation and Ancillary Information Facility

NASA National Aeronautics and Space Administration

NAV Navigation Subsystem/Team

ODL Object Definition Language (PDS)

PDB Project Data Base

PDS Planetary Data System

RST Radio Science Team

SCET Space Craft Event Time

SDS Science Data System

SFDU Standard Formatted Data Unit

SHADR Spherical Harmonics ASCII Data Record

SHBDR Spherical Harmonics Binary Data Record

SHM Spherical Harmonics Model

SIS Software Interface Specification

SPARC Sun Scaleable Processor Architecture

SPK Spacecraft and Planet Kernel Format, from NAIF

TBD To Be Determined

UTC Universal Time Coordinated

1. GENERAL DESCRIPTION

1.1. Overview

This Software Interface Specification (SIS) describes Spherical Harmonics

ASCII Data Record (SHADR) files. The SHADR is intended to be general and may

contain coefficients for spherical harmonic expansions of gravity,

topography, magnetic, and other fields.

1.2. Scope

The format and content specifications in this SIS apply to all phases of the

project for which a SHADR is produced.

The SHADR was defined initially for gravity models derived from Magellan

(MGN) and Mars Observer (MO) radio tracking data [1], but the format is more

generally useful. The original SHADR has been adapted for the Mars Global

Surveyor (MGS), Lunar Prospector (LP), Mars Reconnaissance Orbiter (MRO),

and MESSENGER missions. This update of the SIS was made to include the

Gravity Recovery and Interior Laboratory (GRAIL) mission [16]. Some of the

original mission-specific documentation has been omitted, but the file

format descriptions should still be applicable for the GRAIL mission.

Specifics of the various models are included in [2], which will be updated

as data for new spherical harmonic models are incorporated within the SHADR

definition. A Spherical Harmonic Binary Data Record is also defined [3],

which may be more suitable for large models or when all error covariances

will be included in the final product.

The Jet Propulsion Laboratory (JPL), Pasadena, California, manages the Mars

Reconnaissance Orbiter Mission [4], the Mars Global Surveyor Mission, and the

GRAIL Mission for the National Aeronautics and Space Administration (NASA).

The Johns Hopkins University, Laurel, Maryland, USA manages the MESSENGER

Mission [5,6] for NASA.

1.3. Applicable Documents

[1] Tyler, G.L., G. Balmino, D.P. Hinson, W.L. Sjogren, D.E. Smith, R. Woo,

S.W. Asmar, M.J. Connally, C.L. Hamilton, and R.A. Simpson, Radio Science

Investigations with Mars Observer, J. Geophys. Res., 97, 7759-7779, 1992.

[2] Simpson, R.A., Interpretation and Use of Spherical Harmonics ASCII Data

Record (SHADR) and Spherical Harmonics Binary Data Record (SHBDR), Version

1.0, 1993.

[3] Lemoine, F.G., Software Interface Specification: Spherical Harmonics

Binary Data Record (SHBDR), 2006.

[4] Mars Reconnaissance Orbiter Mission Plan, Revision C: July 2005,

prepared by Robert Lock. Document JPL D-22239, MRO-31-201.

[5] McAdams, J. V. (JHU/APL), MESSENGER mission overview and trajectory

design, American Institute of Aeronautics and Astronautics, American

Astronautical Society (AIAA/AAS) Astrodynamics Specialist Conference, Paper

AAS 03-541, 20 pp., Big Sky, MT, August 3-7, 2003.

[6] McAdams, J. V., D. W. Dunham, R. W. Farquhar, A. H. Taylor, and

B. G. Williams, Trajectory design and maneuver strategy for the MESSENGER

mission to Mercury, 15th American Astronautical Society (AAS)/American

Institute of Aeronautics and Astronautics (AIAA) Space Flight Mechanics

Conference, Paper AAS 05-173, 21 pp., Copper Mountain, CO, Jan. 23-27, 2005.

[7] Seidelmann, P.K., V.K. Abalakin, M. Bursa, M. E. Davies, C. de Bergh,

J. H. Lieske, J. Oberst, J. L. Simon, E. M. Standish, P. Stooke, P. C.

Thomas, Report of the IAU/IAG Working Group on Cartographic Coordinates and

Rotational Elements of the Planets and Satellites: 2000, Celes. Mechanics

and Dyn. Astronomy, 82, 83-110, Dec 2002.

[8] MRO-D-22685, Rev B., Planetary Constants and Models, 05-15-2003.

[9] Konopliv, A.S, C.F. Yoder, E. M. Standish, D.-N. Yuan, and

W. L. Sjogren, A global solution for the Mars static and seasonal gravity,

Mars orientation, Phobos, Deimos Masses, and Mars Ephemeris, Icarus, 182(1),

23-50, 2006.

[10] Konopliv A.S., S.W. Asmar, E. Carranza, W.L. Sjogren, and D.N. Yuan,

Recent Gravity models as a results of the Lunar Prospector Mission, Icarus,

150, 1-18, 2001.

[11] Lambeck, Kurt, Geophysical Geodesy, Oxford University Press, Oxford,

UK, 1988.

[12] Kaula, William M., Theory of Satellite Geodesy, Applications of

satellites to geodesy, Dover Publications, Mineola, NY, 2000.

[13] Lemoine, FG, SC Kenyon, JK Factor, RG Trimmer, NK Pavlis, CM Cox,

SM Klosko, SB Luthcke, MH Torrence, YM Wang, RG Williamson, EC Pavlis,

RH Rapp and TR Olson, The Development of the Joint NASA GSFC and the

National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96,

NASA/TP-1998-206861, NASA Goddard Space Flight Center, Greenbelt, Maryland

20771, July 1998.

[14] JPL D-7116, Rev. F, Planetary Science Data Dictionary Document, Jet

Propulsion Laboratory, Pasadena, California, October 20, 2008.

(http://pds.jpl.nasa.gov/documents/psdd/psdd.pdf)

[15] JPL D-7669 Part 2, Planetary Data System Standards Reference, PDS

Version 3.8, Jet Propulsion Laboratory, February 27, 2009.

(http://pds.jpl.nasa.gov/documents/sr/index.html)

[16] Roncoli, R. B., and K. K. Fujii, Mission Design Overview for the

Gravity Recovery and Interior Laboratory (GRAIL) Mission, AIAA/AAS

Astrodynamics Specialist Conference, Toronto, Ontario, Canada, 2010.

http://arc.aiaa.org/doi/pdf/10.2514/6.2010-8383.

[17] Standish, E. M., Jr. (November 1982), Conversion of positions and

proper motions from B1950.0 to the IAU system at J2000.0, Astronomy and

Astrophysics 115 (1): 20-22. Bibcode 1982A&A...115...20S.

http://adsabs.harvard.edu/full/1982A%26A...115...20S

[18] Folkner, William M., The Planetary and Lunar Ephemeris DE 421, 2009.

IPN Progress Report 42-178.

http://ipnpr.jpl.nasa.gov/progress_report/42-178/178C.pdf

1.4. System Siting

1.4.1. Interface Location and Medium

SHADR files are created at the institution conducting the science analysis.

SHADR files can be electronic files or can be stored on compact-disc

write-once (CD-WO) or DVD type media.

1.4.2. Data Sources, Transfer Methods, and Destinations

SHADR files are created from radio tracking, vertical sounding, in situ,

and/or other measurements at the institution conducting the scientific data

analysis. They are transferred to and deposited in a data system (such as

the PDS) specified by the managing institution.

1.4.3. Generation Method and Frequency

Spherical Harmonic Models are developed separately at each institution

conducting scientific analyses on raw data; each model meets criteria

specified by the investigators conducting the analysis. Each model requires

data with complete sampling (in terms of longitude and latitude coverage on

the planet), so that SHADR files will be issued infrequently and on

schedules which cannot be predicted at this time.

1.5. Assumptions and Constraints

1.5.1. Usage Constraints

None.

1.5.2. Priority Phasing Constraints

None.

1.5.3. Explicit and Derived Constraints

None.

1.5.4. Documentation Conventions

1.5.4.1. Data Format Descriptions

The reference data unit is the byte. Data may be stored in fields with

various sizes and formats, viz. one-, two-, and four-byte binary integers,

four- and eight-byte binary floating-point numbers, and character strings.

Data are identified throughout this document as:

char 8 bits character

uchar 8 bits integer

short 16 bits integer

long 32 bits integer

float 32 bits floating point (sign, exponent, and mantissa)

double 64 bits floating point (sign, exponent, and mantissa)

u (prefix) unsigned (as with ulong for unsigned 32-bit integer)

other special data structures such as time, date, etc. which are described

within this document

If a field is described as containing n bytes of ASCII character string

data, this implies that the leftmost (lowest numbered) byte contains the

first character, the next lowest byte contains the second character, and so

forth.

An array of n elements is written as array[n]; the first element is

array[0], and the last is array[n-1]. Array[n][m] describes an n x m element

array, with first element array[0][0], second element array[0][1], and so

forth.

Floating point (real) numbers are represented as double precision character

strings in the FORTRAN 1P1E23.16 format. Fixed point (integer) numbers are

represented using the FORTRAN I5 format.

1.5.4.2. Time Standards

SHADR files use the January 1.5, 2000 epoch as the standard time. Within the

data files, all times are reported in Universal Coordinated Time (UTC) as

strings of 23 ASCII characters. The time format is

"YYYY-MM-DDThh:mm:ss.fff", where "-", "T", ":", and "." are fixed

delimiters; "YYYY" is the year "19nn" or "20nn"; "MM" is a two-digit month

of year; "DD" is a two-digit day of month; "T" separates the date and time

segments of the string; "hh" is hour of day; "mm" is the minutes of hour

(00-59); "ss" is the seconds of minute (00-59); and "fff" is fractional

seconds in milliseconds.

The date format is "YYYY-MM-DD", where the components are defined

as above.

1.5.4.3. Coordinate Systems

The SHADR uses the appropriate planetocentric fixed body coordinate system

[7,8]. This may be an IAU system (e.g. IAU2000 [7]) or the new

body-fixed Mars reference frame defined by Konopliv et al. [9]. At present,

the MESSENGER mission has adopted the IAU2000 model for Mercury [7].

The coordinate system for lunar geopotential models will be a body figure

axis system defined by the lunar librations which are resolved by lunar

laser ranging [10], or a more coarse frame defined by the IAU [7].

GRAIL uses the DE 421 Lunar Body-Fixed Frame [17] as defined in the DE 421

planetary ephemeris [18].

1.5.4.4. Limits of This Document

This document applies only to SHADR data files.

1.5.4.5. Typographic Conventions

This document has been formatted for simple electronic file transfer and

display. Line lengths are limited to approximately 80 ASCII characters,

including line delimiters. No special fonts or structures are included

within the file. Constant width characters are assumed for display.

2. INTERFACE CHARACTERISTICS

2.1. Hardware Characteristics and Limitations

2.1.1. Special Equipment and Device Interfaces

Users of the SHADR product must have access to the data system (or media)

on which the SHADR files are stored.

2.1.2. Special Setup Requirements

None.

2.2. Volume and Size

SHADR products have variable length, depending on the degree and order of

the model and the number of tables included. A model of degree and order N

will require approximately (N*(N+1)/2)*137 bytes for storage of spherical

harmonic coefficients and associated uncertainties. A SHADR file for the

geopotential that contains coefficients and coefficient standard deviations

through degree 90 will be 510 kB in size. Vector quantities (e.g., magnetic

field) may be described by a single SHADR (in which all components are

represented) or by a separate SHADR for each field component. If the single

SHADR includes covariances, the file size will be approximately 27 times

larger than the combined volumes of the three component files because of the

inter-component covariance terms. In general, the SHBDR [3] is recommended

when the data include error covariances because of the smaller data volume

associated with binary formats.

2.3. Labeling and Identification

The length of file names is limited to 27 or fewer characters before the

period delimiter and 3 characters after the period delimiter. Each file has

a name which describes its contents. The name includes the following

structure which uniquely identifies it among SHADR products. Beginning with

the MRO gravity products the following naming convention is used:

GTsss_nnnnvv_SHA.TAB

where

"G" denotes the generating institution

"J" for the Jet Propulsion Laboratory

"G" for Goddard Space Flight Center

"M" for Massachusetts Institute of Technology

"T" indicates the type of data represented

"G" for gravity field

"sss" is a 3-character modifier specified by the data producer.

This modifier is used to indicate the source spacecraft or project, such as

GRX for the pair of GRAIL spacecraft.

"_" the underscore character is used to delimit modifiers in the file name

for clarity.

"nnnnvv" is a 4- to 6-character modifier specified by the data producer.

Among other things, this modifier may be used to indicate the target body,

whether the SHADR contains primary data values as specified by "T" or

uncertainties/errors, and/or the version number. For GRAIL, this modifier

indicates the degree and order of the solution for the gravity field,

topography or magnetic field.

"SHA" denotes that this is an ASCII file of Spherical Harmonic coefficients

".TAB" indicates the data is stored in tabular form.

Each SHADR file is accompanied by a detached PDS label; that label is a file

in its own right, having the name GTsss_nnnnvv_SHA.LBL.

2.4. Interface Medium Characteristics

SHADR products are electronic files.

2.5. Failure Protection, Detection, and Recovery Procedures

None.

2.6. End-of-File Conventions

End of file labeling complies with standards for the medium on which the

files are stored.

3. ACCESS

3.1. Programs Using the Interface

Data contained in SHADR files will be accessed by programs at the home

institutions of science investigators. Those programs cannot be identified

here.

3.2. Synchronization Considerations

3.2.1. Timing and Sequencing Considerations

N/A

3.2.2. Effective Duration

N/A

3.2.3. Priority Interrupts

None.

3.3. Input/Output Protocols, Calling Sequences

None.

4. DETAILED INTERFACE SPECIFICATIONS

4.1. Structure and Organization Overview

The SHADR is a file generated by software at the institution conducting

scientific data analysis. Each SHADR file is accompanied by a detached PDS

label.

4.2. Detached PDS Label

The detached PDS label is a file with two parts -- a header, and a set of

one, two, or three PDS TABLE object definitions. The header contains

information about the origin of the file and its general characteristics

such as record type and size. The TABLE object definitions describe the

format and content of the tables that make up the SHADR data file. The SHADR

Header Table Object definition is required. The SHADR Coefficients Table

Object definition is required if there is a SHADR Coefficients Table in the

file; the SHADR Covariance Table Object definition is required if there is a

SHADR Covariance Table. Each detached PDS label is constructed of ASCII

records; each record in the label contains exactly 80 characters. The last

two characters in each record are the carriage-return (ASCII 13) and

line-feed (ASCII 10) characters. An example of a complete label and data

object is given in APPENDIX C.

4.2.1 Label Header

The structure of the label header is illustrated in Figure 4-2-1.

Keyword definitions are given below.

PDS_VERSION_ID =

The version of the Planetary Data System for which these data have been

prepared; set to PDS3 by agreement between the mission and PDS.

RECORD_TYPE =

The type of record. Set to "FIXED_LENGTH" to indicate that all logical

records have the same length.

RECORD_BYTES =

The number of bytes per (fixed-length) record. It is usually most convenient

if this has been set equal to the length of records in the

SHADR_COVARIANCE_TABLE.

FILE_RECORDS =

The number of records in the SHADR file: instance dependent.

^SHADR_HEADER_TABLE =

File name and record number at which SHADR_HEADER_TABLE begins. Set to

("GTsss_nnnnvv_SHA.TAB ",1) where " GTsss_nnnnvv_SHA.TAB " is the file name

as described in Section 2.3, and 1 is the record number since this is the

first record in the SHADR file.

|====================================================================|

| |

| Figure 4-2-1 SHADR Label Header |

|====================================================================|

| |

| PDS_VERSION_ID = PDS3 |

| RECORD_TYPE = FIXED_LENGTH |

| RECORD_BYTES = nnn |

| FILE_RECORDS = nnn |

| ^SHADR_HEADER_TABLE = ("GTsss_nnnnvv_SHA.TAB",1) |

| ^SHADR_COEFFICIENTS_TABLE = ("GTsss_nnnnvv_SHA.TAB ",nn) |

| ^SHADR_COVARIANCE_TABLE = ("GTsss_nnnnvv_SHA.TAB ",nnn) |

| INSTRUMENT_HOST_NAME = "cccccccccccccccccccc" |

| TARGET_NAME = "cccc" |

| INSTRUMENT_NAME = "ccccccccccccccccccccccc" |

| DATA_SET_ID = "ccccccccccccccccccccccc" |

| OBSERVATION_TYPE = "ccccccccccccc" |

| ORIGINAL_PRODUCT_ID = "ccccccccccccc" |

| PRODUCT_ID = " GTsss_nnnnvv_SHA.TAB " |

| PRODUCT_RELEASE_DATE = YYYY-MM-DD |

| DESCRIPTION = "cccccccccccccccccc" |

| START_ORBIT_NUMBER = nnnn |

| STOP_ORBIT_NUMBER = nnnn |

| START_TIME = YYYY-MM-DDThh:mm:ss |

| STOP_TIME = YYYY-MM-DDThh:mm:ss |

| PRODUCT_CREATION_TIME = YYYY-MM-DDThh:mm:ss.fff |

| PRODUCER_FULL_NAME = "cccccccccccc" |

| PRODUCER_INSTITUTION_NAME = "ccccccccccc" |

| PRODUCT_VERSION_TYPE = "cccccccccccc" |

| PRODUCER_ID = "ccccccc" |

| SOFTWARE_NAME = "ccccccc;Vn.m" |

|====================================================================|

^SHADR_COEFFICIENTS_TABLE =

File name and record number at which the SHADR_COEFFICIENTS_TABLE begins.

The Coefficients Table is optional; this pointer will not appear in the

SHADR label if there is no Coefficients Table. Set to

("GTsss_nnnnvv_SHA.TAB",nn) where "GTsss_nnnnvv_SHA.TAB" is the file

name as described in Section 2.3, and "nn" is the record number in the

file where the Coefficients Table begins.

^SHADR_COVARIANCE_TABLE=

File name and record number at which SHADR_COVARIANCE_TABLE begins.

The Covariance Table is optional; this pointer will not appear in the SHADR

label if there is no Covariance Table. Set to ("GTsss_nnnnvv_SHA.TAB",nn)

where "GTsss_nnnnvv_SHA.TAB" is the file name as described in Section 2.3,

and "nn" is the record number in the file where the Covariance Table begins.

INSTRUMENT_HOST_NAME =

Name of the spacecraft; acceptable names include "MARS GLOBAL SURVEYOR"

"LUNAR PROSPECTOR", "MARS RECONNAISSANCE ORBITER", "MERCURY SURFACE, SPACE,

ENVIRONMENT, GEOCHEMISTRY, AND RANGING", "MAGELLAN", and "GRAVITY RECOVERY

AND INTERIOR LABORATORY".

TARGET_NAME =

A character string which identifies the target body.

For MRO or MGS SHADR files, the character string "MARS".

For MESSENGER SHADR files the character string will be "MERCURY".

For LP and GRAIL SHADR files, the character string will be "MOON".

For Magellan SHADR files, the character string will be "VENUS".

INSTRUMENT_NAME =

Name of the instrument; set to "RADIO SCIENCE SUBSYSTEM" for products

generated from radio science data, or set to other instrument names as

appropriate. Set to "LUNAR GRAVITY RANGING SYSTEM" for GRAIL.

DATA_SET_ID =

Identifier for the data set of which this SHADR product is a member.

-Set to "MRO-M-RSS-5-SDP-Vn.m" for Mars Reconnaissance Orbiter;

-Set to "MESS-H-RSS-5-SDP-Vn.m" for MESSENGER;

-Set to "MGS-M-RSS-5-SDP-Vn.m" for MGS; and "

-Set to "LP-L-RSS-5-SHGBDR-L2-Vn.m" for Lunar Prospector;

-Set to "GRAIL-L-LGRS-5-RDR-Vn.m" for GRAIL.

The suffix Vn.m indicates the version number of the data set.

OBSERVATION_TYPE=

A character string which identifies the data in the product. For the

spherical harmonic model of a gravity field, the character string

"GRAVITY FIELD". For a model of planet topography, the character string

"TOPOGRAPHY".

ORIGINAL_PRODUCT_ID =

Optional. An identifier for the product provided by the producer.

Generally a file name, different from PRODUCT_ID, which would be recognized

at the producer's home institution.

PRODUCT_ID =

A unique identifier for the product within the collection identified by

DATA_SET_ID. Generally, the file name used in pointers ^SHADR_HEADER_TABLE,

^SHADR_COEFFICIENTS_ TABLE, and/or ^SHADR_COVARIANCE_TABLE. The naming

convention is defined in Section 2.3.

PRODUCT_RELEASE_DATE =

The date on which the product was released to the Planetary Data System;

entered in the format "YYYY-MM-DD", where components are defined in

Section 1.5.4.2.

DESCRIPTION =

A short description of the SHADR product.

START_ORBIT_NUMBER =

Optional. The first orbit represented in the SHADR product. An integer.

STOP_ORBIT_NUMBER =

Optional. The last orbit represented in the SHADR product. An integer.

START_TIME =

Optional. The date/time of the first data included in the model, expressed

in the format "YYYY-MM-DDThh:mm:ss" where the components are defined in

section 1.5.4.2.

STOP_TIME =

Optional. The date/time of the last data included in the model, expressed

in the format "YYYY-MM-DDThh:mm:ss" where the components are defined

in section 1.5.4.2.

PRODUCT_CREATION_TIME =

The time at which this SHADR was created; expressed in the format "YYYY-MM-

DDThh:mm:ss.fff" where the components are defined in Section 1.5.4.2.

PRODUCER_FULL_NAME =

The name of the person primarily responsible for production of this SHADR

file. Expressed as a character string, for example "JAMES T. KIRK".

PRODUCER_INSTITUTION_NAME =

The name of the institution primarily responsible for production of this

SHADR. Standard values include:

"STANFORD UNIVERSITY"

"GODDARD SPACE FLIGHT CENTER"

"JET PROPULSION LABORATORY"

"CENTRE NATIONAL D'ETUDES SPATIALES"

"MASSACHUSETTS INSTITUTE OF TECHNOLOGY"

PRODUCT_VERSION_TYPE =

The version of this SHADR. Standard values include "PREDICT", "PRELIMINARY",

and "FINAL".

PRODUCER_ID =

The entity responsible for creation of the SHADR product. For products

generated by the Mars Reconnaissance Orbiter Gravity Science Team set to

"MRO GST". For products generated by the MESSENGER Laser Altimeter Science

Team set to "MESS LAST". For products generated by the Mars Global Surveyor

Radio Science Team, set to "MGS RST". For products generated by the GRAIL

Science Data System, set to "SDS".

SOFTWARE_NAME =

The name and version number of the program creating this SHADR file;

expressed as a character string in the format "PROGRAM_NAME;n.mm"

where "PROGRAM_NAME" is the name of the software and "n.mm" is the version

number. (e.g. "SOLVE;200201.02")

4.2.2 TABLE Object Definitions

TABLE object definitions completely define the TABLE objects for each SHADR

file. Minor tailoring of the definitions for different OBSERVATION_TYPES

precludes specification of exact definitions here. DESCRIPTION values, for

example, will likely be tailored for each product type. In no case should

there be a need to change the structure of the file, however. Entries "*"

are provided by the label generating program based on information supplied

elsewhere.

4.2.2.1 SHADR Header Object Definition

Each SHADR Header Object is completely defined by the Header Object

Definition in its Label. The Definition which follows gives the structure of

the Header Object; some of the DESCRIPTION values may vary from product to

product. The SHADR Header Object Definition is a required part of the SHADR

label file. It immediately follows the label header.

OBJECT = SHADR_HEADER_TABLE

ROWS = 1

COLUMNS = 8

ROW_BYTES = 137

ROW_SUFFIX_BYTES = 107

INTERCHANGE_FORMAT = ASCII

DESCRIPTION = "The SHADR header includes descriptive

information about the spherical harmonic coefficients which follow

in SHADR_COEFFICIENTS_TABLE. The header consists of a single record

of eight (delimited) data columns requiring 137 bytes, a pad of

105 ASCII blank characters, an ASCII carriage-return, and an ASCII

line-feed."

OBJECT = COLUMN

NAME = "REFERENCE RADIUS"

DATA_TYPE = "ASCII REAL"

START_BYTE = 1

BYTES = 23

FORMAT = "E23.16"

UNIT = "KILOMETER"

DESCRIPTION = "The assumed reference radius of the

spherical body."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "CONSTANT"

DATA_TYPE = "ASCII REAL"

START_BYTE = 25

BYTES = 23

FORMAT = "E23.16"

UNIT = "KM^3/SEC^2"

DESCRIPTION = "For a gravity field model the assumed

gravitational constant GM in kilometers cubed per seconds squared

for the body. For a topography model, set to 1."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "UNCERTAINTY IN CONSTANT"

DATA_TYPE = "ASCII REAL"

START_BYTE = 49

BYTES = 23

FORMAT = "E23.16"

UNIT = "KM^3/SEC^2"

DESCRIPTION = "For a gravity field model the

uncertainty in the gravitational constant GM in kilometers cubed per

seconds squared for the planet. For a topography, set to 0."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "DEGREE OF FIELD"

DATA_TYPE = "ASCII INTEGER"

START_BYTE = 73

BYTES = 5

FORMAT = "I5"

UNIT = "N/A"

DESCRIPTION = "The degree of model field."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "ORDER OF FIELD"

DATA_TYPE = "ASCII INTEGER"

START_BYTE = 79

BYTES = 5

FORMAT = "I5"

UNIT = "N/A"

DESCRIPTION = "The order of the model field."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "NORMALIZATION STATE"

DATA_TYPE = "ASCII INTEGER"

START_BYTE = 85

BYTES = 5

FORMAT = "I5"

UNIT = "N/A"

DESCRIPTION = "The normalization indicator.

For gravity fields:

0 coefficients are unnormalized

1 coefficients are normalized

2 other."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "REFERENCE LONGITUDE"

POSITIVE_LONGITUDE_DIRECTION = "EAST"

DATA_TYPE = "ASCII REAL"

START_BYTE = 91

BYTES = 23

FORMAT = "E23.16"

UNIT = "DEGREE"

DESCRIPTION = "The reference longitude for

the spherical harmonic expansion; normally 0."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "REFERENCE LATITUDE"

DATA_TYPE = "ASCII REAL"

START_BYTE = 115

BYTES = 23

FORMAT = "E23.16"

UNIT = "DEGREE"

DESCRIPTION = "The reference latitude for

the spherical harmonic expansion; normally 0."

END_OBJECT = COLUMN

END_OBJECT = SHADR_HEADER_TABLE

4.2.2.2 SHADR Coefficients Object Definition

The SHADR Coefficients Object is completely defined by the Coefficients

Object Definition in the label. Small differences in DESCRIPTION values

should be expected from product to product. The structure outlined in the

Definition below should not vary, however.

The SHADR Coefficients Object is an optional part of the SHADR data file.

This allows the SHADR to be used for targets which are too small or too

remote to have easily discerned coefficients, but for which estimates of

mass have been obtained (e.g., satellites Phobos and Deimos). If the

Coefficients Object is not included in the SHADR file, either the SHADR

Coefficients Object Definition will be omitted or the number of rows will be

set to zero (ROWS = 0). If the SHADR Coefficients Object is not included,

the pointer ^SHADR_COEFFICIENTS_TABLE will not appear in the label header.

If the SHADR Coefficients Object Definition is included in the label, it

immediately follows the SHADR Header Object Definition.

No requirements are placed on the order in which coefficient values appear

in the table or that all possible combinations of the pairs {m,n} be

included. The coefficients are defined by their COEFFICIENT DEGREE and

COEFFICIENT ORDER; see [2] for interpretation.

OBJECT = SHADR_COEFFICIENTS_TABLE

ROWS = *

COLUMNS = 6

ROW_BYTES = 107

ROW_SUFFIX_BYTES = 15

INTERCHANGE_FORMAT = ASCII

DESCRIPTION = "The SHADR coefficients table contains the

coefficients for the spherical harmonic model. Each row in the table

contains the degree index n, the order index m, the coefficients Cnm

and Snm, and the uncertainties in Cnm and Snm. The (comma delimited)

data require 107 ASCII characters; these are followed by a pad of 13

ASCII blank characters, an ASCII carriage-return, and an ASCII

line-feed."

OBJECT = COLUMN

NAME = "COEFFICIENT DEGREE"

DATA_TYPE = "ASCII INTEGER"

START_BYTE = 1

BYTES = 5

FORMAT = "I5"

UNIT = "N/A"

DESCRIPTION = " The degree index n of the C and S

coefficients in this record."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "COEFFICIENT ORDER"

DATA_TYPE = "ASCII INTEGER"

START_BYTE = 7

BYTES = 5

FORMAT = "I5"

UNIT = "N/A"

DESCRIPTION = "The order index m of the C and S

coefficients in this record."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "C"

DATA_TYPE = "ASCII REAL"

START_BYTE = 13

BYTES = 23

FORMAT = "E23.17"

UNIT = "N/A"

DESCRIPTION = "The coefficient Cnm for this

spherical harmonic model."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "S"

DATA_TYPE = "ASCII REAL"

START_BYTE = 37

BYTES = 23

FORMAT = "E23.17"

UNIT = "N/A"

DESCRIPTION = "The coefficient Snm for this spherical

harmonic model."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "C UNCERTAINTY"

DATA_TYPE = "ASCII REAL"

START_BYTE = 61

BYTES = 23

FORMAT = "E23.17"

UNIT = "N/A"

DESCRIPTION = "The uncertainty in the coefficient Cnm

for this spherical harmonic model."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "S UNCERTAINTY"

DATA_TYPE = "ASCII REAL"

START_BYTE = 85

BYTES = 23

FORMAT = "E23.17"

UNIT = "N/A"

DESCRIPTION = "The uncertainty in the coefficient Snm

for this spherical harmonic model."

END_OBJECT = COLUMN

END_OBJECT = SHADR_COEFFICIENTS_TABLE

4.2.2.3 SHADR Covariance Object Definition

The SHADR Covariance Object is completely defined by the Covariance Object

Definition in the label. Small differences in DESCRIPTION values should be

expected from product to product. The structure established by the

Definition below should not change, however. The SHADR Covariance Object is

an optional part of the SHADR data file. If the Covariance Object is not

included, either the Covariance Object Definition will be omitted or the

number of rows will be set to zero (ROWS = 0). If the SHADR Covariance

Object is not included, the pointer ^SHADR_COVARIANCE_TABLE will not appear

in the label header. If the SHADR Covariance Object Definition is included

in the label, it immediately follows the SHADR Coefficients Object

Definition.

No requirements are placed on the order in which covariance values appear in

the table. Nor is there a requirement that all possible combinations of the

quadruplet values {i,j,n,m} be included. By careful editing and use of

symmetry arguments, it may be possible to define all covariances with fewer

than the maximum number of rows in the table.

OBJECT = SHADR_COVARIANCE_TABLE

ROWS = *

COLUMNS = 8

ROW_BYTES = 119

ROW_SUFFIX_BYTES = 3

INTERCHANGE_FORMAT = ASCII

DESCRIPTION = "The SHADR covariance table contains

the covariances for the spherical harmonic model coefficients.

For each index quadruplet {i,j,n,m} the covariances of CijCnm,

SijSnm, CijSnm, and SijCnm are given. In each row of the table the

(comma delimited) indices occupy 24 ASCII characters and the (comma

delimited) covariances occupy 95 ASCII characters. These are followed

by an ASCII blank, an ASCII carriage-return and an ASCII line-feed."

OBJECT = COLUMN

NAME = "COEFFICIENT DEGREE I"

DATA_TYPE = "ASCII INTEGER"

START_BYTE = 1

BYTES = 5

FORMAT = "I5"

UNIT = "N/A"

DESCRIPTION = "The degree index i of the C and S terms

in this record."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "COEFFICIENT ORDER J"

DATA_TYPE = "ASCII INTEGER"

START_BYTE = 7

BYTES = 5

FORMAT = "I5"

UNIT = "N/A"

DESCRIPTION = "The order index j of the C and S terms

in this record."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "COEFFICIENT DEGREE N"

DATA_TYPE = "ASCII INTEGER"

START_BYTE = 13

BYTES = 5

FORMAT = "I5"

UNIT = "N/A"

DESCRIPTION = "The degree index n of the C and S terms

in this record."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "COEFFICIENT ORDER M"

DATA_TYPE = "ASCII INTEGER"

START_BYTE = 19

BYTES = 5

FORMAT = "I5"

UNIT = "N/A"

DESCRIPTION = "The order index m of the C and S terms

in this record."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "COVARIANCE (C_IJ,C_NM)"

DATA_TYPE = "ASCII REAL"

START_BYTE = 25

BYTES = 23

FORMAT = "E23.16"

UNIT = "N/A"

DESCRIPTION = "Covariance (C_IJ,C_NM) the coefficients

of this spherical harmonic model."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "COVARIANCE (S_IJ,S_NM)"

DATA_TYPE = "ASCII REAL"

START_BYTE = 49

BYTES = 23

FORMAT = "E23.16"

UNIT = "N/A"

DESCRIPTION = "Covariance (S_IJ,S_NM) for the

coefficients of this spherical harmonic model."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "COVARIANCE (C_IJ,S_NM)"

DATA_TYPE = "ASCII REAL"

START_BYTE = 73

BYTES = 23

FORMAT = "E23.16"

UNIT = "N/A"

DESCRIPTION = "Covariance (C_IJ,S_NM) for the

coefficient of this spherical harmonic model."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "COVARIANCE (S_IJ,C_NM)"

DATA_TYPE = "ASCII REAL"

START_BYTE = 97

BYTES = 23

FORMAT = "E23.16"

UNIT = "N/A"

DESCRIPTION = "Covariance (S_IJ,C_NM) for the

coefficients of this spherical harmonic model."

END_OBJECT = COLUMN

END_OBJECT = SHADR_COVARIANCE_TABLE

4.3. Data File

Each SHADR data file comprises a SHADR Header TABLE Object, an (optional)

Coefficients TABLE Object, and an (optional) Covariances TABLE Object.

Each TABLE Object comprises one or more data blocks. The TABLE Objects were

defined in Section 4.2. The data blocks are illustrated below.

4.3.1. SHADR Header Object/Block

The SHADR Header Object contains the parameters necessary to interpret the

data in the SHADR file. The structure of the SHADR Header Object is defined

in Section 4.2.2.1. The SHADR Header Object is a one-row table; hence the

Header Object and the Header Block are logically synonymous. The structure

of the Header Block is shown in Table 4-3-1. Note that the logical content

of the Header Object is delimited by the ASCII carriage return and line feed

characters. The physical block is padded to 244 bytes (an integral multiple

of RECORD_BYTES).

|====================================================================|

| |

| Table 4-3-1. SHADR Header Block |

| |

|====================================================================|

|--------+--------+--------+--------+--------------------------------|

| 1 | +0 | 23 | E23.16 |Planetary Radius |

|--------+--------+--------+--------+--------------------------------|

| 2 | 24 | 23 | E23.16 |Constant |

|--------+--------+--------+--------+--------------------------------|

| 3 | 48 | 23 | E23.16 |Uncertainty in Constant |

|--------+--------+--------+--------+--------------------------------|

| 4 | 72 | 5 | I5 |Degree of Field |

|--------+--------+--------+--------+--------------------------------|

| 5 | 78 | 5 | I5 |Order of Field |

|--------+--------+--------+--------+--------------------------------|

| 6 | 84 | 5 | I5 |Normalization State |

|--------+--------+--------+--------+--------------------------------|

| 7 | 90 | 23 | E23.16 |Reference Longitude |

|--------+--------+--------+--------+--------------------------------|

| 8 | 114| 23 | E23.16 |Reference Latitude |

|--------+--------+--------+--------+--------------------------------|

| | 137| 105 | | blanks |

|--------+--------+--------+--------+--------------------------------|

| | 242| 1 | | carriage return |

|--------+--------+--------+--------+--------------------------------|

| | 243| 1 | | line feed |

|--------+--------+--------+--------+--------------------------------|

| | +244 | |

|========|========|========|========|================================|

4.3.2. SHADR Coefficients Block

The SHADR Coefficients Object is made up of one or more SHADR Coefficient

Blocks. Each block contains one pair of coefficients and associated

uncertainties for the overall model defined by the SHADR product. The

structure of the SHADR Coefficients Object is defined in Section 4.2.2.2.

The structure of an individual block is shown in Table 4-3-2.

Note that the logical content of the Coefficients Block is delimited by the

ASCII carriage return and line feed characters. The Coefficients Block is,

by definition, an integral multiple of RECORD_BYTES.

|====================================================================|

| |

| Table 4-3-2. SHADR Coefficients Block |

| |

|====================================================================|

|--------+--------+--------+--------+--------------------------------|

| 1 | +0 | 5 | I5 |Coefficient Degree n |

|--------+--------+--------+--------+--------------------------------|

| 2 | 6 | 5 | I5 |Coefficient Order m |

|--------+--------+--------+--------+--------------------------------|

| 3 | 12 | 23 | E23.16 |Cnm |

|--------+--------+--------+--------+--------------------------------|

| 4 | 36 | 23 | E23.16 |Snm |

|--------+--------+--------+--------+--------------------------------|

| 5 | 60 | 23 | E23.16 |Uncertainty in Cnm |

|--------+--------+--------+--------+--------------------------------|

| 6 | 84 | 23 | E23.16 |Uncertainty in Snm |

|--------+--------+--------+--------+--------------------------------|

| | 107 | 13 | | blanks |

|--------+--------+--------+--------+--------------------------------|

| | 120 | 1 | | carriage return |

|--------+--------+--------+--------+--------------------------------|

| | 121 | 1 | | line feed |

|--------+--------+--------+--------+--------------------------------|

| | +122 | |

|========|========|========|========|================================|

4.3.3. SHADR Covariance Block

The SHADR Covariance Object is made up of one or more SHADR Covariance

Blocks. Each block contains the CijCnm, SijSnm, CijSnm, and SijCnm

covariances for the overall model defined by the SHADR product. The

structure of the SHADR Covariance Object is defined in Section 4.2.2.3. The

structure of an individual block is shown in Table 4-3-3. Note that the

logical content of the Covariance Block is delimited by the ASCII carriage

return and line feed characters. The SHADR Covariance Block is, by

definition, an integral multiple of RECORD_BYTES.

|====================================================================|

| |

| Table 4-3-3. SHADR Covariance Block |

| |

|====================================================================|

|--------+--------+--------+--------+--------------------------------|

| 1 | +0 | 5 | I5 |Coefficient Degree i |

|--------+--------+--------+--------+--------------------------------|

| 2 | 6 | 5 | I5 |Coefficient Order j |

|--------+--------+--------+--------+--------------------------------|

| 3 | 12 | 5 | I5 |Coefficient Degree n |

|--------+--------+--------+--------+--------------------------------|

| 4 | 18 | 5 | I5 |Coefficient Order m |

|--------+--------+--------+--------+--------------------------------|

| 5 | 24 | 23 | E23.16 |Covariance {Cij,Cnm} |

|--------+--------+--------+--------+--------------------------------|

| 6 | 48 | 23 | E23.16 |Covariance {Sij,Snm} |

|--------+--------+--------+--------+--------------------------------|

| 7 | 72 | 23 | E23.16 |Covariance {Cij,Snm} |

|--------+--------+--------+--------+--------------------------------|

| 8 | 96 | 23 | E23.16 |Covariance {Sij,Cnm} |

|--------+--------+--------+--------+--------------------------------|

| | 119 | 1 | | blank |

|--------+--------+--------+--------+--------------------------------|

| | 120 | 1 | | carriage return |

|--------+--------+--------+--------+--------------------------------|

| | 121 | 1 | | line feed |

|--------+--------+--------+--------+--------------------------------|

| | +122 | |

|========|========|========|========|================================|

APPENDIX A.

A.1 Definition of Spherical harmonic models for the geopotential.

Spherical harmonics satisfy Laplace's equation in spherical coordinates.

The gravity potential field of the planets and the mathematical

representation of magnetic fields and topographic fields are readily

expressed in terms of spherical harmonics. Useful reviews are by Lambeck

[11] (Section 2.2, Elements of Potential Theory) and Kaula [12] (Section 1.1

Potential Theory, and Section 1.2 Spherical Harmonics).

The expression for the geopotential takes the form

V = (GM/r) + (GM/r)*SUMMATION_n SUMMATION_m (Re/r)**n [Cnm" cos(mL) + Snm"

sin(mL)]* Pnm"(sin(phi))

(Equation A-1-1)

where GM is the gravitational constant of the planet, r is the radial

distance of the test point from the origin, and Re is the assumed reference

radius of the spherical planet for which the coefficients were calculated.

The summations take place from degree n=1 to infinity, and order m=0 to n;

Cnm" and Snm" refer to the normalized spherical harmonic coefficients (see

Section A.2 below); L is the longitude; the Pnm" are the normalized

associated Legendre functions of degree n and order m; and phi is the

latitude of the test point. If we assume the origin is at the center of mass,

the degree one terms vanish, and the summation in degree starts at degree

n=2.

A "solution" for a spherical harmonic model of the geopotential refers to a

solution for these spherical harmonic coefficients and the gravitational

constant, GM, of the body.

In practice the spherical harmonic series is truncated at a maximum degree

nmax. For MRO, the likely degree of truncation will be between n=100 and

n=120. For MESSENGER, gravity solutions for the planet Mercury will likely

be truncated at degree 20. The degree of truncation depends on the quality

of the tracking data and the orbits of the spacecraft in the geopotential

solution. For Lunar Prospector derived gravity solutions, the maximum degree

has ranged from n=100 to n=165 [10]. For GRAIL, which used a different

measurement technique, the gravity signal to noise ratio was very strong and

the truncation was at a high degree. nmax was unprecedented 660 for the

Prime Mission. The field size is expected to exceed 1000 for the combined

Prime and Extended missions solution.

If the origin is placed at the center of mass, the degree 1 terms vanish

from the spherical harmonic expansion, and the first summation above is then

from (n=2) to the maximum degree, nmax.

Figure 1, section 1.2 from Kaula [11] gives examples of spherical harmonics.

The zonal terms, m=0, have n zeros in a distance pi along a north-south

meridian; in other words, they represent only latitudinal variations in the

potential.

Zonal terms may be represented in the literature as Jn = - Cn0.

Aside from GM, C20 is the most significant term in the gravity field (for

planets such as the Earth and Mars), and reflects the dynamical expression

of the planet's polar flattening.

Tesseral harmonics (coefficients where n is not equal to m, and m > 0, have

n-m zeros in a distance pi along a meridian (like the tesserae of a mosaic).

Sectoral harmonics are coefficients where n=m and are constant in sectors of

longitude (N-S) and have n zero crossings in a distance pi along a meridian

of latitude (E-W).

A.2 Definition of the normalization used for geopotential coefficients.

The normalization for spherical harmonic coefficients is given by

Lambeck[11]

Cnm" = Cnm/PI_nm (Equation A-2-1)

where Cnm" is normalized and Cnm is un-normalized, and

[PI_nm]**2 = (2 - delta_0m) * (2n+1) * (n-m)! / (n+m)! (Equation A-2-2)

delta_0m refers to the Kronecker delta function -- unity for coefficients

where m=0 (the zonal terms), zero for order m > 0.

For zonal coefficients (m=0) the relation reduces to

Cnm" = Cnm / sqrt(2n+1)

For example, for the Earth C20 = -1.08262668355E-03 (un-normalized) so

C20" = C20 / sqrt(5) = -4.8416537173572E-04 (normalized)

Working the process backwards for Earth's C22 we have

C22" = .24391435239839D-05

from the Earth Gravitational Model 1996, EGM96, [13].

[PI_nm]**2 = (2-0)*(2n+1) (2-2)! / (4)!

= 2*5*1/(4!) = 5/12

which yields

C22 = sqrt(5/12) * (.24391435239839E-05)

= 1.5744604E-06

closely matching Lambeck's [11] result (page 14).

Likewise for Earth's S22, we have the normalized value [13]

S22" = -.14001668365394E-05

Thus,

S22= sqrt(5/12) * (-.14001668365394E-05)

= -9.038038E-07 (un-normalized)

which matches closely the example given by Lambeck [11].

APPENDIX B EXAMPLE DATA PRODUCTS

APPENDIX B.1 Example Label

The following lists an example SHADR LBL file for a Mars Global Surveyor

derived gravity solution. For GRAIL the "INSTRUMENT_HOST_NAME" would be

listed instead of MARS GLOBAL SURVEYOR. The DESCRIPTION would be changed to

reflect the data content of the GRAIL-derived gravity solutions. Other

fields (e.g., PRODUCT_RELEASE_DATE, PRODUCT_ID, INSTRUMENT NAME, START_TIME,

STOP_TIME, PRODUCT_CREATION TIME) would also be changed as appropriate.

This example can be found in its original electronic form at the URL

http://pds-geosciences.wustl.edu/geodata/mgs-m-rss-5-sdp-v1/mors_1021/sha/

PDS_VERSION_ID = PDS3

RECORD_TYPE = FIXED_LENGTH

RECORD_BYTES = 122

FILE_RECORDS = 4185

^SHADR_HEADER_TABLE = ("GGM1041C.SHA",1)

^SHADR_COEFFICIENTS_TABLE = ("GGM1041C.SHA",3)

INSTRUMENT_HOST_NAME = "MARS RECONNAISSANCE ORBITER"

TARGET_NAME = "MARS"

INSTRUMENT_NAME = "RADIO SCIENCE SUBSYSTEM"

DATA_SET_ID = "MRO-M-RSS-5-SDP-V1.0"

OBSERVATION_TYPE = "GRAVITY FIELD"

ORIGINAL_PRODUCT_ID = "MGM1041C"

PRODUCT_ID = "GGM1041C.SHA"

PRODUCT_RELEASE_DATE = 2003-03-28

DESCRIPTION = "

This file contains coefficients and related data for a spherical

harmonic model of the Mars gravity field. Input data are from

radio tracking of the Mars Global Surveyor spacecraft; no Mariner

9 or Viking data are included. Coordinate system is IAU 2000

(Seidelmann et al., Celestial Mechanics and Dynamical Astronomy,

82(1), 83-110, January 2002).

Constants relevant for Mars in IAU 2000 are:

alpha (right ascension) = 317.68143 deg - 0.1061 deg/century

delta (declination) = 52.88650 deg - 0.0609 deg/century

Wo (prime meridian) = 176.630 deg

Wdot = 350.89198226 deg/day

Gravitational constants obtained from the solution are:

GM (Mars) = 4.282837024529127 E+13 m**3/s**2

sigma = 0.617 E+05 m**3/s**2

GM (Phobos) = 0.68012569 E+06 m**3/s**2

sigma = 0.842 E+04 m**3/s**2

The model was constructed from 2,568,683 observations, summarized

in the table below. MGS data are limited to tracking from the

Aerobraking Hiatus and Science Phasing Orbit (SPO) subphases of

the Orbit Insertion phase of the mission and to February 1999 to

May 2002 after the orbit was circularized.

Time Periods Total

Arcs Observations

------------------------ ---- ------------

Hiatus 2 24119

SPO-1 8 31014

SPO-2 17 144253

Feb-Mar 1999 10 86069

2 Apr 1999 - 22 May 2002 167 2186533

Doppler + Crossovers 5 96875

------------------------ ---- ------------

Total 2568863

Orbit reconstruction was improved using Mars Orbiter Laser

Altimeter (MOLA) data on 5 arcs between March and December 1999.

Inter-arc and intra-arc crossovers at 21679 points were included

in the orbit solutions. Crossovers poleward of 60 degrees north

and south were excluded to avoid possible contamination by any

time-varying signature in the polar caps. The altimeter data were

edited for large off-nadir angles as well as for the roughness and

slope of the terrain. The crossovers supplemented the normal

radiometric tracking for these arcs, which included 75196 Doppler

and range observations.

One-way Doppler data were included once these data started to

become available in March 2000 (just after the start of the first

Beta Supplement operations). Range biases were adjusted on a pass

by pass basis, and frequency biases were adjusted for the one-way

data.

The Gravity Calibration Orbit (GCO, February 1999) data were

weighted at 0.0044 Hz (0.16 mm/s). Data for 2000 and for 2001

through the start of Relay 16 operations were weighted at 0.18

mm/s. Data after the start of Relay 16 operations in 2001 were

weighted at 0.357 mm/s in light of the higher RMS fit for these

data. The one-way data were downweighted with respect to the

two-way data by a factor of 2, and are thus weighted at 0.36 mm/s

for the 2000-2001 data and at 0.71 mm/s for Relay 16.

Compared to the GMM2B model, this model contains about 2.5 times

as much data. The data were also rigorously re-edited to remove

spurious signatures which were particularly apparent in some of

the arcs with one-way Doppler data. Finally, the non-conservative

force model was refined to include the high gain antenna. This

improvement reduced the average solar radiation reflectivity

(Cr)from 1.15 to 1.05.

The gravity model was derived using a Kaula type constraint:

sqrt(2)*13*10**(-5)/L**2 (Kaula, W.M., Theory of Satellite

Geodesy, Blaisdell, Waltham, MA, 1966).

Further improvements to the model are expected as additional MGS

data are incorporated.

The C20 coefficient is given in the zero-tide system, meaning

that the deformation due to the (solar-induced) permanent tide

is included in the coefficient. An apriori K2 Love number of

0.10 was used in the derivation of this model.

This product is a set of two ASCII tables: a header table and a

coefficients table. Definitions of the tables follow.

This Mars gravity model was produced by F.G. Lemoine under the

direction of D.E. Smith of the MGS Radio Science Team."

START_TIME = 1997-10-13T00:00:00

STOP_TIME = 2002-05-27T23:59:59

PRODUCT_CREATION_TIME = 2003-02-05T20:34:50

PRODUCER_FULL_NAME = "FRANK G. LEMOINE"

PRODUCER_INSTITUTION_NAME = "GODDARD SPACE FLIGHT CENTER"

PRODUCT_VERSION_TYPE = "FINAL"

PRODUCER_ID = "MRO GST"

SOFTWARE_NAME = "SOLVE;200201.02"

OBJECT = SHADR_HEADER_TABLE

ROWS = 1

COLUMNS = 8

ROW_BYTES = 137

ROW_SUFFIX_BYTES = 107

INTERCHANGE_FORMAT = ASCII

DESCRIPTION = "The SHADR header includes

descriptive information about the spherical harmonic coefficients

which follow in SHADR_COEFFICIENTS_TABLE. The header consists of

a single record of eight (delimited) data columns requiring 137

bytes, a pad of 105 unspecified ASCII characters, an ASCII

carriage-return, and an ASCII line-feed."

OBJECT = COLUMN

NAME = "REFERENCE RADIUS"

DATA_TYPE = "ASCII REAL"

START_BYTE = 1

BYTES = 23

FORMAT = "E23.16"

UNIT = "KILOMETER"

DESCRIPTION = "The assumed reference radius

of the spherical body."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "CONSTANT"

DATA_TYPE = "ASCII REAL"

START_BYTE = 25

BYTES = 23

FORMAT = "E23.16"

UNIT = "KM^3/S^2"

DESCRIPTION = "For a gravity field model the

assumed gravitational constant GM in km cubed per seconds

squared for the body. For a topography model, set to 1."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "UNCERTAINTY IN CONSTANT"

DATA_TYPE = "ASCII REAL"

START_BYTE = 49

BYTES = 23

FORMAT = "E23.16"

UNIT = "KM^3/S^2"

DESCRIPTION = "For a gravity field model the

uncertainty in the gravitational constant GM in km cubed per

seconds squared for the planet (or, set to 0 if not known).

For a topography model, set to 0."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "DEGREE OF FIELD"

DATA_TYPE = "ASCII INTEGER"

START_BYTE = 73

BYTES = 5

FORMAT = "I5"

UNIT = "N/A"

DESCRIPTION = "Degree of the model field."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "ORDER OF FIELD"

DATA_TYPE = "ASCII INTEGER"

START_BYTE = 79

BYTES = 5

FORMAT = "I5"

UNIT = "N/A"

DESCRIPTION = "Order of the model field."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "NORMALIZATION STATE"

DATA_TYPE = "ASCII INTEGER"

START_BYTE = 85

BYTES = 5

FORMAT = "I5"

UNIT = "N/A"

DESCRIPTION = "The normalization indicator.

For gravity field:

0 coefficients are unnormalized

1 coefficients are normalized

2 other."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "REFERENCE LONGITUDE"

POSITIVE_LONGITUDE_DIRECTION = "EAST"

DATA_TYPE = "ASCII REAL"

START_BYTE = 91

BYTES = 23

FORMAT = "E23.16"

UNIT = "DEGREE"

DESCRIPTION = "The reference longitude for

the spherical harmonic expansion; normally 0."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "REFERENCE LATITUDE"

DATA_TYPE = "ASCII REAL"

START_BYTE = 115

BYTES = 23

FORMAT = "E23.16"

UNIT = "DEGREE"

DESCRIPTION = "The reference latitude for the

spherical harmonic expansion; normally 0."

END_OBJECT = COLUMN

END_OBJECT = SHADR_HEADER_TABLE

OBJECT = SHADR_COEFFICIENTS_TABLE

ROWS = 4183

COLUMNS = 6

ROW_BYTES = 107

ROW_SUFFIX_BYTES = 15

INTERCHANGE_FORMAT = ASCII

DESCRIPTION = "The SHADR coefficients table

contains the coefficients for the spherical harmonic model. Each

row in the table contains the degree index n, the order index m,

the coefficients Cnm and Snm, and the uncertainties in Cnm and

Snm. The (delimited) data require 107 ASCII characters; these are

followed by a pad of 13 unspecified ASCII characters, an ASCII

carriage-return, and an ASCII line-feed."

OBJECT = COLUMN

NAME = "COEFFICIENT DEGREE"

DATA_TYPE = "ASCII INTEGER"

START_BYTE = 1

BYTES = 5

FORMAT = "I5"

UNIT = "N/A"

DESCRIPTION = "The degree index n of the

C and S coefficients in this record."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "COEFFICIENT ORDER"

DATA_TYPE = "ASCII INTEGER"

START_BYTE = 7

BYTES = 5

FORMAT = "I5"

UNIT = "N/A"

DESCRIPTION = "The order index m of the C and S

coefficients in this record."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "C"

DATA_TYPE = "ASCII REAL"

START_BYTE = 13

BYTES = 23

FORMAT = "E23.17"

UNIT = "N/A"

DESCRIPTION = "The coefficient Cnm for this

spherical harmonic model."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "S"

DATA_TYPE = "ASCII REAL"

START_BYTE = 37

BYTES = 23

FORMAT = "E23.17"

UNIT = "N/A"

DESCRIPTION = "The coefficient Snm for this

spherical harmonic model."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "C UNCERTAINTY"

DATA_TYPE = "ASCII REAL"

START_BYTE = 61

BYTES = 23

FORMAT = "E23.17"

UNIT = "N/A"

DESCRIPTION = "The uncertainty in the

coefficient Cnm for this spherical harmonic model."

END_OBJECT = COLUMN

OBJECT = COLUMN

NAME = "S UNCERTAINTY"

DATA_TYPE = "ASCII REAL"

START_BYTE = 85

BYTES = 23

FORMAT = "E23.17"

UNIT = "N/A"

DESCRIPTION = "The uncertainty in the

coefficient Snm for this spherical harmonic model."

END_OBJECT = COLUMN

END_OBJECT = SHADR_COEFFICIENTS_TABLE

END

APPENDIX B.2 Example SHADR Data Object

The following lists the first few lines from an example SHADR file, the

MGM1041C Gravity solution.

Note that the lines here wrap after 70 characters whereas the header record

length is 244 and the coefficient record length is 122.

3.3970000000000000E+03, 4.2828370245291269E+04,

6.1699999999999995E-05, 90, 90, 1, 0.0000000000000000E+00,

0.0000000000000000E+00

2, 0,-8.7450461309664714E-04, 0.0000000000000000E+00,

8.6998585172904000E-11, 0.0000000000000000E+00

2, 1, 3.4361530466444738E-10,-2.6812730136287860E-10,

5.2026417903363999E-11, 5.1856231628722999E-11

2, 2,-8.4585864260034122E-05, 4.8905472151326622E-05,

2.4262638528121999E-11, 2.4711067535925999E-11

3, 0,-1.1889488636438340E-05, 0.0000000000000000E+00,

7.1845677542599005E-11, 0.0000000000000000E+00