Geoid.java
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* Licensed to CS GROUP (CS) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* CS licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
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*
* http://www.apache.org/licenses/LICENSE-2.0
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* Unless required by applicable law or agreed to in writing, software
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package org.orekit.models.earth;
import org.hipparchus.CalculusFieldElement;
import org.hipparchus.Field;
import org.hipparchus.analysis.CalculusFieldUnivariateFunction;
import org.hipparchus.analysis.UnivariateFunction;
import org.hipparchus.analysis.solvers.AllowedSolution;
import org.hipparchus.analysis.solvers.BracketingNthOrderBrentSolver;
import org.hipparchus.analysis.solvers.FieldBracketingNthOrderBrentSolver;
import org.hipparchus.analysis.solvers.UnivariateSolver;
import org.hipparchus.exception.MathRuntimeException;
import org.hipparchus.geometry.euclidean.threed.FieldLine;
import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
import org.hipparchus.geometry.euclidean.threed.Line;
import org.hipparchus.geometry.euclidean.threed.Vector3D;
import org.hipparchus.util.FastMath;
import org.orekit.bodies.FieldGeodeticPoint;
import org.orekit.bodies.GeodeticPoint;
import org.orekit.errors.OrekitException;
import org.orekit.forces.gravity.HolmesFeatherstoneAttractionModel;
import org.orekit.forces.gravity.potential.GravityFields;
import org.orekit.forces.gravity.potential.NormalizedSphericalHarmonicsProvider;
import org.orekit.forces.gravity.potential.TideSystem;
import org.orekit.frames.FieldStaticTransform;
import org.orekit.frames.Frame;
import org.orekit.frames.StaticTransform;
import org.orekit.time.AbsoluteDate;
import org.orekit.time.FieldAbsoluteDate;
import org.orekit.utils.TimeStampedPVCoordinates;
/**
* A geoid is a level surface of the gravity potential of a body. The gravity
* potential, W, is split so W = U + T, where U is the normal potential (defined
* by the ellipsoid) and T is the anomalous potential.[3](eq. 2-137)
*
* <p> The {@link #getIntersectionPoint(Line, Vector3D, Frame, AbsoluteDate)}
* method is tailored specifically for Earth's geoid. All of the other methods
* in this class are general and will work for an arbitrary body.
*
* <p> There are several components that are needed to define a geoid[1]:
*
* <ul> <li>Geopotential field. These are the coefficients of the spherical
* harmonics: S<sub>n,m</sub> and C<sub>n,m</sub></li>
*
* <li>Reference Ellipsoid. The ellipsoid is used to define the undulation of
* the geoid (distance between ellipsoid and geoid) and U<sub>0</sub> the value
* of the normal gravity potential at the surface of the ellipsoid.</li>
*
* <li>W<sub>0</sub>, the potential at the geoid. The value of the potential on
* the level surface. This is taken to be U<sub>0</sub>, the normal gravity
* potential at the surface of the {@link ReferenceEllipsoid}.</li>
*
* <li>Permanent Tide System. This implementation assumes that the geopotential
* field and the reference ellipsoid use the same permanent tide system. If the
* assumption is false it will produce errors of about 0.5 m. Conversion between
* tide systems is a possible improvement.[1,2]</li>
*
* <li>Topographic Masses. That is mass outside of the geoid, e.g. mountains.
* This implementation ignores topographic masses, which causes up to 3m error
* in the Himalayas, and ~ 1.5m error in the Rockies. This could be improved
* through the use of DTED and calculating height anomalies or using the
* correction coefficients.[1]</li> </ul>
*
* <p> This implementation also assumes that the normal to the reference
* ellipsoid is the same as the normal to the geoid. This assumption enables the
* equation: (height above geoid) = (height above ellipsoid) - (undulation),
* which is used in {@link #transform(GeodeticPoint)} and {@link
* #transform(Vector3D, Frame, AbsoluteDate)}.
*
* <p> In testing, the error in the undulations calculated by this class were
* off by less than 3 meters, which matches the assumptions outlined above.
*
* <p> References:
*
* <ol> <li>Dru A. Smith. There is no such thing as "The" EGM96 geoid: Subtle
* points on the use of a global geopotential model. IGeS Bulletin No. 8:17-28,
* 1998. <a href= "http://www.ngs.noaa.gov/PUBS_LIB/EGM96_GEOID_PAPER/egm96_geoid_paper.html"
* >http://www.ngs.noaa.gov/PUBS_LIB/EGM96_GEOID_PAPER/egm96_geoid_paper.html</a></li>
*
* <li> Martin Losch, Verena Seufer. How to Compute Geoid Undulations (Geoid
* Height Relative to a Given Reference Ellipsoid) from Spherical Harmonic
* Coefficients for Satellite Altimetry Applications. , 2003. <a
* href="http://mitgcm.org/~mlosch/geoidcookbook.pdf">mitgcm.org/~mlosch/geoidcookbook.pdf</a>
* </li>
*
* <li>Weikko A. Heiskanen, Helmut Moritz. Physical Geodesy. W. H. Freeman and
* Company, 1967. (especially sections 2.13 and equation 2-144 Bruns
* Formula)</li>
*
* <li>S. A. Holmes, W. E. Featherstone. A unified approach to the Clenshaw
* summation and the recursive computation of very high degree and order
* normalised associated Legendre functions. Journal of Geodesy, 76(5):279,
* 2002.</li>
*
* <li>DMA TR 8350.2. 1984.</li>
*
* <li>Department of Defense World Geodetic System 1984. 2000. NIMA TR 8350.2
* Third Edition, Amendment 1.</li> </ol>
*
* @author Evan Ward
*/
public class Geoid implements EarthShape {
/**
* uid is date of last modification.
*/
private static final long serialVersionUID = 20150312L;
/**
* A number larger than the largest undulation. Wikipedia says the geoid
* height is in [-106, 85]. I chose 100 to be safe.
*/
private static final double MAX_UNDULATION = 100;
/**
* A number smaller than the smallest undulation. Wikipedia says the geoid
* height is in [-106, 85]. I chose -150 to be safe.
*/
private static final double MIN_UNDULATION = -150;
/**
* the maximum number of evaluations for the line search in {@link
* #getIntersectionPoint(Line, Vector3D, Frame, AbsoluteDate)}.
*/
private static final int MAX_EVALUATIONS = 100;
/**
* the default date to use when evaluating the {@link #harmonics}. Used when
* no other dates are available. Should be removed in a future release.
*/
private final AbsoluteDate defaultDate;
/**
* the reference ellipsoid.
*/
private final ReferenceEllipsoid referenceEllipsoid;
/**
* the geo-potential combined with an algorithm for evaluating the spherical
* harmonics. The Holmes and Featherstone method is very robust.
*/
private final transient HolmesFeatherstoneAttractionModel harmonics;
/**
* Creates a geoid from the given geopotential, reference ellipsoid and the
* assumptions in the comment for {@link Geoid}.
*
* @param geopotential the gravity potential. Only the anomalous
* potential will be used. It is assumed that the
* {@code geopotential} and the {@code
* referenceEllipsoid} are defined in the same
* frame. Usually a {@link GravityFields#getConstantNormalizedProvider(int,
* int, AbsoluteDate) constant geopotential} is used to define a
* time-invariant Geoid.
* @param referenceEllipsoid the normal gravity potential.
* @throws NullPointerException if {@code geopotential == null ||
* referenceEllipsoid == null}
*/
public Geoid(final NormalizedSphericalHarmonicsProvider geopotential,
final ReferenceEllipsoid referenceEllipsoid) {
// parameter check
if (geopotential == null || referenceEllipsoid == null) {
throw new NullPointerException();
}
// subtract the ellipsoid from the geopotential
final SubtractEllipsoid potential = new SubtractEllipsoid(geopotential,
referenceEllipsoid);
// set instance parameters
this.referenceEllipsoid = referenceEllipsoid;
this.harmonics = new HolmesFeatherstoneAttractionModel(
referenceEllipsoid.getBodyFrame(), potential);
this.defaultDate = AbsoluteDate.ARBITRARY_EPOCH;
}
@Override
public Frame getBodyFrame() {
// same as for reference ellipsoid.
return this.getEllipsoid().getBodyFrame();
}
/**
* Gets the Undulation of the Geoid, N at the given position. N is the
* distance between the {@link #getEllipsoid() reference ellipsoid} and the
* geoid. The latitude and longitude parameters are both defined with
* respect to the reference ellipsoid. For EGM96 and the WGS84 ellipsoid the
* undulation is between -107m and +86m.
*
* <p> NOTE: Restrictions are not put on the range of the arguments {@code
* geodeticLatitude} and {@code longitude}.
*
* @param geodeticLatitude geodetic latitude (angle between the local normal
* and the equatorial plane on the reference
* ellipsoid), in radians.
* @param longitude on the reference ellipsoid, in radians.
* @param date of evaluation. Used for time varying geopotential
* fields.
* @return the undulation in m, positive means the geoid is higher than the
* ellipsoid.
* @see Geoid
* @see <a href="http://en.wikipedia.org/wiki/Geoid">Geoid on Wikipedia</a>
*/
public double getUndulation(final double geodeticLatitude,
final double longitude,
final AbsoluteDate date) {
/*
* equations references are to the algorithm printed in the geoid
* cookbook[2]. See comment for Geoid.
*/
// reference ellipsoid
final ReferenceEllipsoid ellipsoid = this.getEllipsoid();
// position in geodetic coordinates
final GeodeticPoint gp = new GeodeticPoint(geodeticLatitude, longitude, 0);
// position in Cartesian coordinates, is converted to geocentric lat and
// lon in the Holmes and Featherstone class
final Vector3D position = ellipsoid.transform(gp);
// get normal gravity from ellipsoid, eq 15
final double normalGravity = ellipsoid
.getNormalGravity(geodeticLatitude);
// calculate disturbing potential, T, eq 30.
final double mu = this.harmonics.getMu(date);
final double T = this.harmonics.nonCentralPart(date, position, mu);
// calculate undulation, eq 30
return T / normalGravity;
}
@Override
public ReferenceEllipsoid getEllipsoid() {
return this.referenceEllipsoid;
}
/**
* This class implements equations 20-24 in the geoid cook book.(Losch and
* Seufer) It modifies C<sub>2n,0</sub> where n = 1,2,...,5.
*
* @see "DMA TR 8350.2. 1984."
*/
private static final class SubtractEllipsoid implements
NormalizedSphericalHarmonicsProvider {
/**
* provider of the fully normalized coefficients, includes the reference
* ellipsoid.
*/
private final NormalizedSphericalHarmonicsProvider provider;
/**
* the reference ellipsoid to subtract from {@link #provider}.
*/
private final ReferenceEllipsoid ellipsoid;
/**
* @param provider potential used for GM<sub>g</sub> and a<sub>g</sub>,
* and of course the coefficients Cnm, and Snm.
* @param ellipsoid Used to calculate the fully normalized
* J<sub>2n</sub>
*/
private SubtractEllipsoid(
final NormalizedSphericalHarmonicsProvider provider,
final ReferenceEllipsoid ellipsoid) {
super();
this.provider = provider;
this.ellipsoid = ellipsoid;
}
@Override
public int getMaxDegree() {
return this.provider.getMaxDegree();
}
@Override
public int getMaxOrder() {
return this.provider.getMaxOrder();
}
@Override
public double getMu() {
return this.provider.getMu();
}
@Override
public double getAe() {
return this.provider.getAe();
}
@Override
public AbsoluteDate getReferenceDate() {
return this.provider.getReferenceDate();
}
@Override
public NormalizedSphericalHarmonics onDate(final AbsoluteDate date) {
return new NormalizedSphericalHarmonics() {
/** the original harmonics */
private final NormalizedSphericalHarmonics delegate = provider.onDate(date);
@Override
public double getNormalizedCnm(final int n, final int m) {
return getCorrectedCnm(n, m, this.delegate.getNormalizedCnm(n, m));
}
@Override
public double getNormalizedSnm(final int n, final int m) {
return this.delegate.getNormalizedSnm(n, m);
}
@Override
public AbsoluteDate getDate() {
return date;
}
};
}
/**
* Get the corrected Cnm for different GM or a values.
*
* @param n degree
* @param m order
* @param uncorrectedCnm uncorrected Cnm coefficient
* @return the corrected Cnm coefficient.
*/
private double getCorrectedCnm(final int n,
final int m,
final double uncorrectedCnm) {
double Cnm = uncorrectedCnm;
// n = 2,4,6,8, or 10 and m = 0
if (m == 0 && n <= 10 && n % 2 == 0 && n > 0) {
// correction factor for different GM and a, 1 if no difference
final double gmRatio = this.ellipsoid.getGM() / this.getMu();
final double aRatio = this.ellipsoid.getEquatorialRadius() /
this.getAe();
/*
* eq 20 in the geoid cook book[2], with eq 3-61 in chapter 3 of
* DMA TR 8350.2
*/
// halfN = 1,2,3,4,5 for n = 2,4,6,8,10, respectively
final int halfN = n / 2;
Cnm = Cnm - gmRatio * FastMath.pow(aRatio, halfN) *
this.ellipsoid.getC2n0(halfN);
}
// return is a modified Cnm
return Cnm;
}
@Override
public TideSystem getTideSystem() {
return this.provider.getTideSystem();
}
}
/**
* {@inheritDoc}
*
* <p> The intersection point is computed using a line search along the
* specified line. This is accurate when the geoid is slowly varying.
*/
@Override
public GeodeticPoint getIntersectionPoint(final Line lineInFrame,
final Vector3D closeInFrame,
final Frame frame,
final AbsoluteDate date) {
/*
* It is assumed that the geoid is slowly varying over it's entire
* surface. Therefore there will one local intersection.
*/
// transform to body frame
final Frame bodyFrame = this.getBodyFrame();
final StaticTransform frameToBody =
frame.getStaticTransformTo(bodyFrame, date);
final Vector3D close = frameToBody.transformPosition(closeInFrame);
final Line lineInBodyFrame = frameToBody.transformLine(lineInFrame);
// set the line's direction so the solved for value is always positive
final Line line;
if (lineInBodyFrame.getAbscissa(close) < 0) {
line = lineInBodyFrame.revert();
} else {
line = lineInBodyFrame;
}
final ReferenceEllipsoid ellipsoid = this.getEllipsoid();
// calculate end points
// distance from line to center of earth, squared
final double d2 = line.pointAt(0.0).getNormSq();
// the minimum abscissa, squared
final double n = ellipsoid.getPolarRadius() + MIN_UNDULATION;
final double minAbscissa2 = n * n - d2;
// smaller end point of the interval = 0.0 or intersection with
// min_undulation sphere
final double lowPoint = FastMath.sqrt(FastMath.max(minAbscissa2, 0.0));
// the maximum abscissa, squared
final double x = ellipsoid.getEquatorialRadius() + MAX_UNDULATION;
final double maxAbscissa2 = x * x - d2;
// larger end point of the interval
final double highPoint = FastMath.sqrt(maxAbscissa2);
// line search function
final UnivariateFunction heightFunction = new UnivariateFunction() {
@Override
public double value(final double x) {
try {
final GeodeticPoint geodetic =
transform(line.pointAt(x), bodyFrame, date);
return geodetic.getAltitude();
} catch (OrekitException e) {
// due to frame transform -> re-throw
throw new RuntimeException(e);
}
}
};
// compute answer
if (maxAbscissa2 < 0) {
// ray does not pierce bounding sphere -> no possible intersection
return null;
}
// solve line search problem to find the intersection
final UnivariateSolver solver = new BracketingNthOrderBrentSolver();
try {
final double abscissa = solver.solve(MAX_EVALUATIONS, heightFunction, lowPoint, highPoint);
// return intersection point
return this.transform(line.pointAt(abscissa), bodyFrame, date);
} catch (MathRuntimeException e) {
// no intersection
return null;
}
}
@Override
public Vector3D projectToGround(final Vector3D point,
final AbsoluteDate date,
final Frame frame) {
final GeodeticPoint gp = this.transform(point, frame, date);
final GeodeticPoint gpZero =
new GeodeticPoint(gp.getLatitude(), gp.getLongitude(), 0);
final StaticTransform bodyToFrame =
this.getBodyFrame().getStaticTransformTo(frame, date);
return bodyToFrame.transformPosition(this.transform(gpZero));
}
/**
* {@inheritDoc}
*
* <p> The intersection point is computed using a line search along the
* specified line. This is accurate when the geoid is slowly varying.
*/
@Override
public <T extends CalculusFieldElement<T>> FieldGeodeticPoint<T> getIntersectionPoint(final FieldLine<T> lineInFrame,
final FieldVector3D<T> closeInFrame,
final Frame frame,
final FieldAbsoluteDate<T> date) {
final Field<T> field = date.getField();
/*
* It is assumed that the geoid is slowly varying over it's entire
* surface. Therefore there will one local intersection.
*/
// transform to body frame
final Frame bodyFrame = this.getBodyFrame();
final FieldStaticTransform<T> frameToBody = frame.getStaticTransformTo(bodyFrame, date);
final FieldVector3D<T> close = frameToBody.transformPosition(closeInFrame);
final FieldLine<T> lineInBodyFrame = frameToBody.transformLine(lineInFrame);
// set the line's direction so the solved for value is always positive
final FieldLine<T> line;
if (lineInBodyFrame.getAbscissa(close).getReal() < 0) {
line = lineInBodyFrame.revert();
} else {
line = lineInBodyFrame;
}
final ReferenceEllipsoid ellipsoid = this.getEllipsoid();
// calculate end points
// distance from line to center of earth, squared
final T d2 = line.pointAt(0.0).getNormSq();
// the minimum abscissa, squared
final double n = ellipsoid.getPolarRadius() + MIN_UNDULATION;
final T minAbscissa2 = d2.negate().add(n * n);
// smaller end point of the interval = 0.0 or intersection with
// min_undulation sphere
final T lowPoint = minAbscissa2.getReal() < 0 ? field.getZero() : minAbscissa2.sqrt();
// the maximum abscissa, squared
final double x = ellipsoid.getEquatorialRadius() + MAX_UNDULATION;
final T maxAbscissa2 = d2.negate().add(x * x);
// larger end point of the interval
final T highPoint = maxAbscissa2.sqrt();
// line search function
final CalculusFieldUnivariateFunction<T> heightFunction = z -> {
try {
final FieldGeodeticPoint<T> geodetic =
transform(line.pointAt(z), bodyFrame, date);
return geodetic.getAltitude();
} catch (OrekitException e) {
// due to frame transform -> re-throw
throw new RuntimeException(e);
}
};
// compute answer
if (maxAbscissa2.getReal() < 0) {
// ray does not pierce bounding sphere -> no possible intersection
return null;
}
// solve line search problem to find the intersection
final FieldBracketingNthOrderBrentSolver<T> solver =
new FieldBracketingNthOrderBrentSolver<>(field.getZero().newInstance(1.0e-14),
field.getZero().newInstance(1.0e-6),
field.getZero().newInstance(1.0e-15),
5);
try {
final T abscissa = solver.solve(MAX_EVALUATIONS, heightFunction, lowPoint, highPoint,
AllowedSolution.ANY_SIDE);
// return intersection point
return this.transform(line.pointAt(abscissa), bodyFrame, date);
} catch (MathRuntimeException e) {
// no intersection
return null;
}
}
@Override
public TimeStampedPVCoordinates projectToGround(
final TimeStampedPVCoordinates pv,
final Frame frame) {
throw new UnsupportedOperationException();
}
/**
* {@inheritDoc}
*
* @param date date of the conversion. Used for computing frame
* transformations and for time dependent geopotential.
* @return The surface relative point at the same location. Altitude is
* orthometric height, that is height above the {@link Geoid}. Latitude and
* longitude are both geodetic and defined with respect to the {@link
* #getEllipsoid() reference ellipsoid}.
* @see #transform(GeodeticPoint)
* @see <a href="http://en.wikipedia.org/wiki/Orthometric_height">Orthometric_height</a>
*/
@Override
public GeodeticPoint transform(final Vector3D point, final Frame frame,
final AbsoluteDate date) {
// convert using reference ellipsoid, altitude referenced to ellipsoid
final GeodeticPoint ellipsoidal = this.getEllipsoid().transform(
point, frame, date);
// convert altitude to orthometric using the undulation.
final double undulation = this.getUndulation(ellipsoidal.getLatitude(),
ellipsoidal.getLongitude(), date);
// add undulation to the altitude
return new GeodeticPoint(
ellipsoidal.getLatitude(),
ellipsoidal.getLongitude(),
ellipsoidal.getAltitude() - undulation
);
}
/**
* {@inheritDoc}
*
* @param date date of the conversion. Used for computing frame
* transformations and for time dependent geopotential.
* @return The surface relative point at the same location. Altitude is
* orthometric height, that is height above the {@link Geoid}. Latitude and
* longitude are both geodetic and defined with respect to the {@link
* #getEllipsoid() reference ellipsoid}.
* @see #transform(GeodeticPoint)
* @see <a href="http://en.wikipedia.org/wiki/Orthometric_height">Orthometric_height</a>
*/
@Override
public <T extends CalculusFieldElement<T>> FieldGeodeticPoint<T> transform(final FieldVector3D<T> point, final Frame frame,
final FieldAbsoluteDate<T> date) {
// convert using reference ellipsoid, altitude referenced to ellipsoid
final FieldGeodeticPoint<T> ellipsoidal = this.getEllipsoid().transform(
point, frame, date);
// convert altitude to orthometric using the undulation.
final double undulation = this.getUndulation(ellipsoidal.getLatitude().getReal(),
ellipsoidal.getLongitude().getReal(),
date.toAbsoluteDate());
// add undulation to the altitude
return new FieldGeodeticPoint<>(
ellipsoidal.getLatitude(),
ellipsoidal.getLongitude(),
ellipsoidal.getAltitude().subtract(undulation)
);
}
/**
* {@inheritDoc}
*
* @param point The surface relative point to transform. Altitude is
* orthometric height, that is height above the {@link Geoid}.
* Latitude and longitude are both geodetic and defined with
* respect to the {@link #getEllipsoid() reference ellipsoid}.
* @return point at the same location but as a Cartesian point in the {@link
* #getBodyFrame() body frame}.
* @see #transform(Vector3D, Frame, AbsoluteDate)
*/
@Override
public Vector3D transform(final GeodeticPoint point) {
try {
// convert orthometric height to height above ellipsoid using undulation
// TODO pass in date to allow user to specify
final double undulation = this.getUndulation(
point.getLatitude(),
point.getLongitude(),
this.defaultDate
);
final GeodeticPoint ellipsoidal = new GeodeticPoint(
point.getLatitude(),
point.getLongitude(),
point.getAltitude() + undulation
);
// transform using reference ellipsoid
return this.getEllipsoid().transform(ellipsoidal);
} catch (OrekitException e) {
//this method, as defined in BodyShape, is not permitted to throw
//an OrekitException, so wrap in an exception we can throw.
throw new RuntimeException(e);
}
}
/**
* {@inheritDoc}
*
* @param point The surface relative point to transform. Altitude is
* orthometric height, that is height above the {@link Geoid}.
* Latitude and longitude are both geodetic and defined with
* respect to the {@link #getEllipsoid() reference ellipsoid}.
* @param <T> type of the field elements
* @return point at the same location but as a Cartesian point in the {@link
* #getBodyFrame() body frame}.
* @see #transform(Vector3D, Frame, AbsoluteDate)
* @since 9.0
*/
@Override
public <T extends CalculusFieldElement<T>> FieldVector3D<T> transform(final FieldGeodeticPoint<T> point) {
try {
// convert orthometric height to height above ellipsoid using undulation
// TODO pass in date to allow user to specify
final double undulation = this.getUndulation(
point.getLatitude().getReal(),
point.getLongitude().getReal(),
this.defaultDate
);
final FieldGeodeticPoint<T> ellipsoidal = new FieldGeodeticPoint<>(
point.getLatitude(),
point.getLongitude(),
point.getAltitude().add(undulation)
);
// transform using reference ellipsoid
return this.getEllipsoid().transform(ellipsoidal);
} catch (OrekitException e) {
//this method, as defined in BodyShape, is not permitted to throw
//an OrekitException, so wrap in an exception we can throw.
throw new RuntimeException(e);
}
}
}