Gravity Disturbances at Altitude and at the Surfaceˆ’0.0050 0 0.005 0.005 0.01 0.01 0.015 0.015...
1
-100 -50 0 50 100 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 Error due to typo in Hackney & Featherstone (2003) Free-Air Correction Latitude (degrees) Error (mGal) 1000 meters 3000 meters 5000 meters 7000 meters 9000 meters 11000 meters 25 30 35 40 45 0 500 1000 1500 2000 2500 3000 3500 GPS Benchmark Locations; those with gamma differences marked as circles Latitude (Degrees) Ellipsoidal Height (meters) Zero Difference +1 uGal Difference -1 uGal Difference -0.005 0 0 0.005 0.005 0.01 0.01 0.015 0.015 0.015 0.02 0.02 0.02 0.02 0.025 0.025 0.025 0.03 0.03 0.03 0.035 0.035 0.04 0.04 0.045 Difference of NGS 3rd Order Correction from Confocal Correction (mGal) Latitude (decimal degrees) Altitude (m) 0 10 20 30 40 50 60 70 80 90 0 2000 4000 6000 8000 10000 0 0 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.05 0.05 0.06 Difference of NGS 2nd Order Correction from Confocal Correction (mGal) Latitude (decimal degrees) Altitude (m) 0 10 20 30 40 50 60 70 80 90 0 2000 4000 6000 8000 10000 -11 -10 -10 -9 -9 -8 -8 -7 -7 -7 -6 -6 -6 -6 -5 -5 -5 -5 -4 -4 -4 -4 -3 -3 -3 -3 -2 -2 -2 -2 -1 -1 -1 -1 0 0 0 0 0 Difference of Linear Correction from Confocal Correction (mGal) Latitude (decimal degrees) Altitude (m) 0 10 20 30 40 50 60 70 80 90 0 2000 4000 6000 8000 10000 0 0 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.04 0.05 0.05 0.05 0.06 0.06 0.06 0.06 0.07 0.07 0.07 0.08 0.08 0.09 0.09 0.1 Difference of H&M 2nd Order Correction from Confocal Correction (mGal) Latitude (decimal degrees) Altitude (m) 0 10 20 30 40 50 60 70 80 90 0 2000 4000 6000 8000 10000 -100 -80 -60 -40 -20 0 20 40 60 80 100 -308.7 -308.6 -308.5 -308.4 -308.3 "Free-Air Corrections" at 1000 meters (3280.84 feet) above the Ellipsoid Latitude (degrees) Correction (mGal) Linear 2nd order H&M Explicit 2nd order NGS 3rd order NGS Confocal -100 -80 -60 -40 -20 0 20 40 60 80 100 -1543.5 -1543 -1542.5 -1542 -1541.5 -1541 -1540.5 -1540 -1539.5 "Free-Air Corrections" at 5000 meters (16404.2 feet) above the Ellipsoid Latitude (degrees) Correction (mGal) Linear 2nd order H&M Explicit 2nd order NGS 3rd order NGS Confocal -100 -80 -60 -40 -20 0 20 40 60 80 100 -3396 -3394 -3392 -3390 -3388 -3386 -3384 -3382 "Free-Air Corrections" at 11000 meters (36089.24 feet) above the Ellipsoid Latitude (degrees) Correction (mGal) Linear 2nd order H&M Explicit 2nd order NGS 3rd order NGS Confocal Normal gravity is gravity due to an ellipsoidal representation of the Earth, at or above the surface of the ellipsoid. A common ellipsoid is WGS-84. Gravity is negative here, by the convention that positive is directed upward. A scalar gravity disturbance (δg) is the difference between scalar measured full-field gravity (g) and scalar normal gravity (γ) at the exact measurement point (P) : δg P = g P - γ P (1) g P is provided online as the official GRAV-D airborne gravity product. Normal gravity has often been calculated on the surface of the ellipsoid (γ 0 ) and then a height (or “free-air”) correction applied. The correction can be represented as a series (Heiskanen and Moritz, 1967). Thus, the first-order free-air correction approximates the vertical gradient in normal gravity over the distance (h) between the measurement and surface of the ellipsoid, such that: δg FA ≈ γ ' h P (2), where γ ' = δγ/δh and h P is ellipsoidal height of point (P) Plugging into Equation (1): δg P ≈ g P - γ 0 + δg FA (3) The second order and third order forms of the free-air correction (in brackets) are such that: δg P ≈ g P - γ 0 + [ (γ ' h P ) + (1/2 γ '' h P 2 ) ] (4) δg P ≈ g P - γ 0 + [ (γ ' h P ) + (1/2 γ '' h P 2 ) + (1/3 γ ''' h P 3 ) ] (5) The accuracy of the gravity disturbance is dependent on our ability to either accurately: 1. calculate normal gravity at the measurement point; or 2. calculate normal gravity at the surface and calculate the derivates of noraml gravity with respect to height G11A-0908 Gravity Disturbances at Altitude and at the Surface Theresa M. Damiani* NOAA-National Geodetic Survey, 1315 East-West Hwy, SSMC3, Silver Spring, MD 20910; *Contact: [email protected] 1. Background 2. Normal Gravity and the Free-Air Correction 6. References and Acknowledgements 4. Results 3. Available Methods The GRAV-D (Gravity for the Redefinition of the American Vertical Datum) Project of the U.S. National Geodetic Survey plans to collect airborne gravity data across the entire U.S. and its holdings over the next decade. The goal of the project is to create a gravimetric geoid model to use as the national vertical datum by 2022. The project plan and more details are available: http://www.ngs.noaa.gov/GRAV-D GRAV-D (as of August 2013) has publicly released full-field gravity products from these high-altitude flights for >15% of the country. The full-field gravity (FFG) at altitude product is versatile because it allows the user to calculate any disturbance or anomaly that is appropriate for their application- based on any datum and height above the datum desired. A. Linear Free-Air Correction B. Heiskanen and Moritz (H&M) 1967 Explicit 2nd order Free-Air Correction C. NGS-Improved H&M Free-Air Correction [1] Featherstone, W.E. (1995) “On the Use of Australian Geodetic Datums in Gravity Field Determination.” Geomatics Research Australasia, 62, pp. 17-36. [2] Featherstone, W.E. and M.C. Dentith (1997) “A Geodetic Approach to Gravity Data Reduction for Geophysics.” Computers & Geosciences, 23(10), pp. 1063±1070 [3] Hackney, R. I. and W. E. Featherstone (2003). “Geodetic vs. geophysical perspectives of the ‘gravity anomaly’.” Geophysical Journal International, 154, pp. 35-43. [4] Heiskanen, W.A. and H. Mortiz (1967). Physical Geodesy. W.H. Freeman and Company: San Francisco. 364 pp. [5] Pavlis, N.K., S.A. Holmes, S.C. Kenyone, and J.K. Factor (2012). “The development and evaluation of the Earth Gravitational Model (EGM2008).” Journal of Geophysical Research, 117, B04406, doi:10.1029/2011JB008916. [6] NGA (2010). “Earth Gravitational Model 2008 (EGM2008).” and hsynth_WGS84.f available online at: http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008/ The least precise correction, but one derived from experimental data, is a linear (first-order) approximation of the vertical gravity gradient near the surface of the Earth: γ ' = 0.3086 mGal/meter δg FA ≈ 0.3086 h P (6) IMPORTANCE OF THIS STUDY: Accurate combination (or comparisons) of airborne gravity data with marine or terrestrial data require accounting for normal gravity at the measurement points and differences in the data spectral content. This poster compares available methods of calculating gravity disturbances and assesses them for their errors across all latitudes and from zero to 11 km altitude. In the conclusions, we recommend the most accurate method for calculating disturbances. Also, before comparing two sets of gravity disturbances collected at different heights, an additional step must be taken to filter or continue the data sets until they have matching signal content. Given: The value of γ 0 , normal gravity at the surface of the ellipsoid, is given to a precision of 1 microGal by the Somigliana-Pizetti formula. In the literature, it is written in two ways, which are equivalent: Heiskanen and Moritz (1967) Eqn 2-78; Hackney and Featherstone (2003) Eqn 5. An ellipsoid must be chosen as the reference. WGS-84 is used here. They give a 2nd-order free-air correction, for “small” heights above the ellipsoid (H&M Equation 2-124): δg FA ≈ -2γ a /a [ 1 + f +m + (-3f + 5/2 m) sin 2 Φ ] h + 3γ a h 2 /a 2 (7) where γ a is normal gravity at the equator at the surface of the ellipsoid, a, f, and m are properties of the ellipsoid Φ is latitude h is ellipsoidal height of the measurement Cons: This equation approximates γ 0 and its derivatives in terms of γ a and parameters of the ellipsoid. Also, the second derivative of normal gravity is derived for a spheroid, not an ellipsoid. Subtracting γ 0 from H&M Eqn 2-132 for γ P , we can solve for second order and third order free-air corrections without approximating γ 0 : δg FA ≈ -2γ 0 /a [ 1 + f +m - 2f sin 2 Φ ] h + 3γ 0 h 2 /a 2 (8) δg FA ≈ -2γ 0 /a [ 1 + f +m - 2f sin 2 Φ ] h + 3γ 0 h 2 /a 2 + 4γ 0 h 3 /a 3 (9) These equations still approximate the normal gravity derivatives, but can be solved using the precise Somigliana-Pizetti γ 0 value. The second and third derivatives of normal gravity are derived for a spheroid, not an ellipsoid. Adding a 4th term changes the correction by < 0.06 microGals (for maximim altitude of 11km). Important Note: My Equation 8 is the same as presented in Featherstone and Dentith (1997) and Featherstone (1995). However, Hackney and Featherstone’s (2003) Equation 6, which should be the same, has a typo where γ a should be γ 0 . The result of this typo is that there is an error in the free-air correction that grows with increasing latitude when using the 2003 version as printed (see right). D. Confocal Ellipsoid for γ P For any point (P) in space, an ellipsoid can be drawn through that point that has the same foci as the reference ellipsoid. This “confocal” ellipsoid has an associated Somigliana-Pizetti equation that will precisely calculate normal gravity on confocal ellipsoid’s surface. Inside the National Geospatial Intelligence Agency’s (NGA’s) Fortran routines of the hsynth_WGS84.f program (NGA, 2010; Pavlis, et al., 2012), is a subroutine called “radgrav” that calculates: 1. a confocal ellipsoid that passes through any point (P) and 2. γ P , normal gravity at P. P Confocal Ellipsoid Reference Ellipsoid Foci h An advantage of this method is that the gravity disturbance can be calculated without applying a free-air correction, using equation (1). But, for comparisons, the free-air correction from this method would be: δg FA = γ 0 - γ P (10) E. Harmonic Synthesis of Reference Ellipsoid’s Zonal Coefficients Spherical harmonics divide a sphere into a series of compartments, as in the figure below (Heiskanen and Moritz, 1967). The ellipsoid, being a simple surface that only varies with respect to latitude, can be represented well with the first 10 even zonal harmonics (2, 4, 6, ..., 20). To be zonal, these harmonics all have orders equal to zero. In NGA’s hsynth_WGS84.f program, a subroutine called “grs” can calculate the harmonic coefficients for an ellipsoid. Once the coefficients are obtained (shown in table above), the hsynth_WGS84.f program can be run with those coefficients as the input to obtain a very accurate value of normal gravity at any point on or above the ellipsoid, γ P . This method also offers the advantages of Method D, using no free-air correction to calculate the disturbance. Again, for comparions only, the free-air correction for this method would be Equation 10, as well. I would like to thank Simon Holmes and Ajit Singh for their assistance with this work. degree (n) order (m) normalized C nm normalized N nm 2 0 -0.48416678176D-03 0 4 0 0.79030375916D-06 0 6 0 -0.16872497152D-08 0 8 0 0.34605251221D-11 0 10 0 -0.26500241424D-14 0 12 0 -0.41079006120D-16 0 14 0 0.44717732450D-18 0 16 0 -0.34636255309D-20 0 18 0 0.24114560041D-22 0 20 0 -0.16024329348D-24 0 WGS-84 Spherical Harmonic Coefficients Comparison of Two Most Accurate Methods (D & E) Since Methods D & E both provide a disturbance without needing a free-air correction, these are the two most accurate methods available. For comparison, a test data set with the locations of 2789 of NGS’ most reliable GPS Benchmarks in the continental United States was used to calculate normal gravity γ P with both Methods D and E, and the results compared. F. H&M 1967 Equation for γ P They give a explicit 2nd order equation for γ P (H&M Equation 2-123): γ P ≈ γ 0 [ 1 - 2/a ( 1+ f +m + -2f sin 2 Φ ) h + 3 h 2 /a 2 (11) This equation is most accurate when solved with the γ 0 from the Somigliana-Pizetti formula. It has the same advantages of Methods D&E, however contains approximations that make it less accurate. The results of the comparison showed that Methods D and E returned identical results in mGal (to three decimal places) for all but 10 of the GPS Benchmarks. The remaining 10 have a +1 or -1 microGal difference due to rounding error. Thus, the two methods both yeild results precise to 1 microGal. Comparison of Methods A through C with Method D - The 2nd order NGS formula (also, Featherstone & Dentith, 1997) is more accurate at nearly all latitudes and altitudes compared to the Heiskanen and Moritz 2nd order formula. The exception is that the two 2nd order corrections are nearly equivalent at very low latitudes (< +/- 10 degrees). - The addition of the 3rd term to the NGS formula further benefits higher latitudes (> +/- 50 deg.). - Recommended correction: Confocal or Zonal Coefficient Methods If not available, then use the NGS 3rd order formula (error <0.05 mGal) *Disturbances at different heights must still be continued to equal height before comparison.* 5. Conclusions
Transcript of Gravity Disturbances at Altitude and at the Surfaceˆ’0.0050 0 0.005 0.005 0.01 0.01 0.015 0.015...
−100 −50 0 50 100−18
−16
−14
−12
−10
−8
−6
−4
−2
0
2Error due to typo in Hackney & Featherstone (2003) Free−Air Correction
Latitude (degrees)
Erro
r (m
Gal
)
1000 meters3000 meters5000 meters7000 meters9000 meters11000 meters
25 30 35 40 450
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GPS Benchmark Locations; those with gamma differences marked as circles
Latitude (Degrees)
Ellip
soid
al H
eigh
t (m
eter
s)
Zero Difference+1 uGal Difference−1 uGal Difference
−0.005 00 0.005
0.0050.01
0.010.015
0.0150.015
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5
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Difference of NGS 3rd Order Correction from Confocal Correction (mGal)
Latitude (decimal degrees)
Alti
tude
(m)
0 10 20 30 40 50 60 70 80 900
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0.010.01
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0.020.02
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Difference of NGS 2nd Order Correction from Confocal Correction (mGal)
Latitude (decimal degrees)
Alti
tude
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−11−10−10−9
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−4−4
−4−4
−3−3
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−2−2
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Difference of Linear Correction from Confocal Correction (mGal)
Latitude (decimal degrees)
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tude
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0.09 0.1Difference of H&M 2nd Order Correction from Confocal Correction (mGal)
Latitude (decimal degrees)
Alti
tude
(m)
0 10 20 30 40 50 60 70 80 900
2000
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−100 −80 −60 −40 −20 0 20 40 60 80 100
−308.7
−308.6
−308.5
−308.4
−308.3
"Free−Air Corrections" at 1000 meters (3280.84 feet) above the Ellipsoid
Latitude (degrees)
Corr
ectio
n (m
Gal
)
Linear2nd order H&M Explicit2nd order NGS3rd order NGSConfocal
−100 −80 −60 −40 −20 0 20 40 60 80 100−1543.5
−1543
−1542.5
−1542
−1541.5
−1541
−1540.5
−1540
−1539.5"Free−Air Corrections" at 5000 meters (16404.2 feet) above the Ellipsoid
Latitude (degrees)
Corr
ectio
n (m
Gal
)
Linear2nd order H&M Explicit2nd order NGS3rd order NGSConfocal
−100 −80 −60 −40 −20 0 20 40 60 80 100−3396
−3394
−3392
−3390
−3388
−3386
−3384
−3382"Free−Air Corrections" at 11000 meters (36089.24 feet) above the Ellipsoid
Latitude (degrees)
Corr
ectio
n (m
Gal
)
Linear2nd order H&M Explicit2nd order NGS3rd order NGSConfocal
Normal gravity is gravity due to an ellipsoidal representation of the Earth, at or above the surface of the ellipsoid. A common ellipsoid is WGS-84. Gravity is negative here, by the convention that positive is directed upward.
A scalar gravity disturbance (δg) is the difference between scalar measured full-field gravity (g) and scalar normal gravity (γ) at the exact measurement point (P) :
δgP = gP - γP (1)
gP is provided online as the official GRAV-D airborne gravity product.
Normal gravity has often been calculated on the surface of the ellipsoid (γ0) and then a height (or “free-air”) correction applied. The correction can be represented as a series (Heiskanen and Moritz, 1967). Thus, the first-order free-air correction approximates the vertical gradient in normal gravity over the distance (h) between the measurement and surface of the ellipsoid, such that:
δgFA ≈ γ' hP (2), where γ' = δγ/δh and hP is ellipsoidal height of point (P)
Plugging into Equation (1):δgP ≈ gP - γ0 + δgFA (3)
The second order and third order forms of the free-air correction (in brackets) are such that:
δgP ≈ gP - γ0 + [ (γ' hP) + (1/2 γ'' hP2) ] (4)
δgP ≈ gP - γ0 + [ (γ' hP) + (1/2 γ'' hP2) + (1/3 γ''' hP
3) ] (5)
The accuracy of the gravity disturbance is dependent on our ability to either accurately:1. calculate normal gravity at the measurement point; or2. calculate normal gravity at the surface and calculate the derivates of noraml gravity with respect to height
G11A-0908
Gravity Disturbances at Altitude and at the SurfaceTheresa M. Damiani*
NOAA-National Geodetic Survey, 1315 East-West Hwy, SSMC3, Silver Spring, MD 20910; *Contact: [email protected]
1. Background
2. Normal Gravity and the Free-Air Correction
6. References and Acknowledgements
4. Results3. Available MethodsThe GRAV-D (Gravity for the Redefinition of the American Vertical Datum) Project of the U.S. National Geodetic Survey plans to collect airborne gravity data across the entire U.S. and its holdings over the next decade. The goal of the project is to create a gravimetric geoid model to use as the national vertical datum by 2022. The project plan and more details are available: http://www.ngs.noaa.gov/GRAV-D
GRAV-D (as of August 2013) has publicly released full-field gravity products from these high-altitude flights for >15% of the country. The full-field gravity (FFG) at altitude product is versatile because it allows the user to calculate any disturbance or anomaly that is appropriate for their application- based on any datum and height above the datum desired.
A. Linear Free-Air Correction
B. Heiskanen and Moritz (H&M) 1967 Explicit 2nd order Free-Air Correction
C. NGS-Improved H&M Free-Air Correction
[1] Featherstone, W.E. (1995) “On the Use of Australian Geodetic Datums in Gravity Field Determination.” Geomatics Research Australasia, 62, pp. 17-36.[2] Featherstone, W.E. and M.C. Dentith (1997) “A Geodetic Approach to Gravity Data Reduction for Geophysics.” Computers & Geosciences, 23(10), pp. 1063±1070[3] Hackney, R. I. and W. E. Featherstone (2003). “Geodetic vs. geophysical perspectives of the ‘gravity anomaly’.” Geophysical Journal International, 154, pp. 35-43.[4] Heiskanen, W.A. and H. Mortiz (1967). Physical Geodesy. W.H. Freeman and Company: San Francisco. 364 pp.[5] Pavlis, N.K., S.A. Holmes, S.C. Kenyone, and J.K. Factor (2012). “The development and evaluation of the Earth Gravitational Model (EGM2008).” Journal of Geophysical Research, 117, B04406, doi:10.1029/2011JB008916.[6] NGA (2010). “Earth Gravitational Model 2008 (EGM2008).” and hsynth_WGS84.f available online at: http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008/
The least precise correction, but one derived from experimental data, is a linear (first-order) approximation of the vertical gravity gradient near the surface of the Earth:γ' = 0.3086 mGal/meter
δgFA ≈ 0.3086 hP (6)
IMPORTANCE OF THIS STUDY:Accurate combination (or comparisons) of airborne gravity data with marine or terrestrial data require accounting for normal gravity at the measurement points and differences in the data spectral content.
This poster compares available methods of calculating gravity disturbances and assesses them for their errors across all latitudes and from zero to 11 km altitude. In the conclusions, we recommend the most accurate method for calculating disturbances.
Also, before comparing two sets of gravity disturbances collected at different heights, an additional step must be taken to filter or continue the data sets until they have matching signal content.
Given: The value of γ0, normal gravity at the surface of the ellipsoid, is given to a precision of 1 microGal by the Somigliana-Pizetti formula. In the literature, it is written in two ways, which are equivalent: Heiskanen and Moritz (1967) Eqn 2-78; Hackney and Featherstone (2003) Eqn 5.
An ellipsoid must be chosen as the reference. WGS-84 is used here.
They give a 2nd-order free-air correction, for “small” heights above the ellipsoid (H&M Equation 2-124):
δgFA ≈ -2γa/a [ 1 + f +m + (-3f + 5/2 m) sin2Φ ] h + 3γah2/a2 (7)whereγa is normal gravity at the equator at the surface of the ellipsoid,a, f, and m are properties of the ellipsoidΦ is latitudeh is ellipsoidal height of the measurement
Cons: This equation approximates γ0 and its derivatives in terms of γa and parameters of the ellipsoid. Also, the second derivative of normal gravity is derived for a spheroid, not an ellipsoid.
Subtracting γ0 from H&M Eqn 2-132 for γP, we can solve for second order and third order free-air corrections without approximating γ0:
δgFA ≈ -2γ0/a [ 1 + f +m - 2f sin2Φ ] h + 3γ0h2/a2 (8) δgFA ≈ -2γ0/a [ 1 + f +m - 2f sin2Φ ] h + 3γ0h2/a2 + 4γ0h3/a3 (9)
These equations still approximate the normal gravity derivatives, but can be solved using the precise Somigliana-Pizetti γ0 value. The second and third derivatives of normal gravity are derived for a spheroid, not an ellipsoid. Adding a 4th term changes the correction by < 0.06 microGals (for maximim altitude of 11km).
Important Note: My Equation 8 is the same as presented in Featherstone and Dentith (1997) and Featherstone (1995). However, Hackney and Featherstone’s (2003) Equation 6, which should be the same, has a typo where γa should be γ0.
The result of this typo is that there is an error in the free-air correction that grows with increasing latitude when using the 2003 version as printed (see right).
D. Confocal Ellipsoid for γP For any point (P) in space, an ellipsoid can be drawn through that point that has the same foci as the reference ellipsoid. This “confocal” ellipsoid has an associated Somigliana-Pizetti equation that will precisely calculate normal gravity on confocal ellipsoid’s surface.
Inside the National Geospatial Intelligence Agency’s (NGA’s) Fortran routines of the hsynth_WGS84.f program (NGA, 2010; Pavlis, et al., 2012), is a subroutine called “radgrav” that calculates: 1. a confocal ellipsoid that passes through any point (P) and 2. γP, normal gravity at P.
P ConfocalEllipsoid
ReferenceEllipsoid
Focih
An advantage of this method is that the gravity disturbance can be calculated without applying a free-air correction, using equation (1).But, for comparisons, the free-air correction from this method would be:δgFA = γ0 - γP (10)
E. Harmonic Synthesis of Reference Ellipsoid’s Zonal Coe�cientsSpherical harmonics divide a sphere into a series of compartments, as in the figure below (Heiskanen and Moritz, 1967). The ellipsoid, being a simple surface that only varies with respect to latitude, can be represented well with the first 10 even zonal harmonics (2, 4, 6, ..., 20). To be zonal, these harmonics all have orders equal to zero.
In NGA’s hsynth_WGS84.f program, a subroutine called “grs” can calculate the harmonic coefficients for an ellipsoid. Once the coefficients are obtained (shown in table above), the hsynth_WGS84.f program can be run with those coefficients as the input to obtain a very accurate value of normal gravity at any point on or above the ellipsoid, γP.
This method also offers the advantages of Method D, using no free-air correction to calculate the disturbance. Again, for comparions only, the free-air correction for this method would be Equation 10, as well.
I would like to thank Simon Holmes and Ajit Singh for their assistance with this work.
degree (n) order (m) normalized Cnm normalized Nnm
2 0 -0.48416678176D-03 04 0 0.79030375916D-06 06 0 -0.16872497152D-08 08 0 0.34605251221D-11 0
10 0 -0.26500241424D-14 012 0 -0.41079006120D-16 014 0 0.44717732450D-18 016 0 -0.34636255309D-20 018 0 0.24114560041D-22 020 0 -0.16024329348D-24 0
WGS-84 Spherical Harmonic Coefficients
Comparison of Two Most Accurate Methods (D & E) Since Methods D & E both provide a disturbance without needing a free-air correction, these are the two most accurate methods available. For comparison, a test data set with the locations of 2789 of NGS’ most reliable GPS Benchmarks in the continental United States was used to calculate normal gravity γP with both Methods D and E, and the results compared.
F. H&M 1967 Equation for γP They give a explicit 2nd order equation for γP (H&M Equation 2-123):
γP ≈ γ0 [ 1 - 2/a ( 1+ f +m + -2f sin2Φ ) h + 3 h2/a2 (11)
This equation is most accurate when solved with the γ0 from the Somigliana-Pizetti formula. It has the same advantages of Methods D&E, however contains approximations that make it less accurate.
The results of the comparison showed that Methods D and E returned identical results in mGal (to three decimal places) for all but 10 of the GPS Benchmarks. The remaining 10 have a +1 or -1 microGal difference due to rounding error. Thus, the two methods both yeild results precise to 1 microGal.
Comparison of Methods A through C with Method D
- The 2nd order NGS formula (also, Featherstone & Dentith, 1997) is more accurate at nearly all latitudes and altitudes compared to the Heiskanen and Moritz 2nd order formula. The exception is that the two 2nd order corrections are nearly equivalent at very low latitudes (< +/- 10 degrees).- The addition of the 3rd term to the NGS formula further benefits higher latitudes (> +/- 50 deg.).- Recommended correction: Confocal or Zonal Coefficient Methods If not available, then use the NGS 3rd order formula (error <0.05 mGal)*Disturbances at different heights must still be continued to equal height before comparison.*
5. Conclusions
![Notes on Maxwell & Delaney PSYCH 710 Multiple Comparisons · [1] "C 4: F=9.823, t=3.134, p=0.005, psi=11.944, CI=(5.620,18.269), adj.CI= (1.485,22.403)" Notice that I divided the](https://static.fdocument.org/doc/165x107/5f36dd51e181c47b74575aa9/notes-on-maxwell-delaney-psych-710-multiple-comparisons-1-c-4-f9823.jpg)
