1. POTENTIAL METHODS 2016-2017 Spherical Harmonics Part 3b
Carla Braitenberg
Trieste University, DMG
Home page:

2. Contents Utility of Spherical Harmonic Expansion of the gravity field
Properties of Spherical Harmonics
Usage and Applications of existing global fields
9 nov 2016

3. Introduction to Spherical Harmonic Expansion
Goal: to map the Earth gravity field.
Contour Map
G(i,j), i=1,Nx; J=1,Ny
Single observations point
Gi, i=1,N
Averaging process:
Calculate gravity values on a regular grid with Nx by Ny values.
Grid values through interpolation method: weighted sum of observation points.

4. Introduction to Spherical Harmonic Expansion
Create global map with grid g(i,j):
G(i,j): grid of gravity values at zero height level (on geoid). Gravity known only at exactly these points and at this height.
Modern observations include:
gravimeter on land
Shipborne observations
Airborne observatons
Satellite observations
-> observations at different heights

5. Introduction to Spherical Harmonic Expansion
Standard averaging: interpolation methods require all data at the same height level.
But different types of observations are at different levels.
Mathematical procedure required to deal with different heights for obtaining the averaged field.

6. Solution: Spherical Harmonic Expansion
What it is: Particular mathematical description of the field.
Possible because the gravitational potential obeys the Laplace equation:
Sum of second derivatives of potential equals to zero.
Consists of:
a table of numbers (coefficients of expansion), that are unique to the field
A finite set of well defined mathematical functions which are numbered (f1,f2,f3,f4...)

7. Solution: Spherical Harmonic Expansion
Advantage: allows to calculate the gravity field in any point.
-> Not tied to fixed grid points.
-> any height above earth surface.
Field is reproduced up to a certain amount of detail: smallest detail depends on total number of coefficients contained in the table of coefficients.

8. Properties: Spherical Harmonic Expansion
Functions: well defined functions which are numbered.
The higher the number, the more detail do the functions contain.
Orthogonal: product between two functions is zero
Normalized: product of the function with itself is equal to 1
Spanning: Any function U(r) can be written as the linear combination of the functions:
U(r)=a1F1 (r) + a2F2 (r) +a3F3 (r) aNFN (r)

9. Analogy to rocks
Rocks are described in terms of percentage of elementary costituents.
Example: Magma defined by percentage of a set of base-rocks.
Good set of base-rocks should have independent constituents. Mathematically: orthogonality.
The percentage of the rock-type is the analogue to the coefficients.

10. Base functions in spherical coordinates
Spherical coordinates: latitude, longitude, vertical height.
Local cartesian coordinates: Easting, Northing, vertical height.
(Wikipedia Commons)

11. Base functions in spherical coordinates
Base functions in spherical coordinates satisfying the Laplace equation:
Pnl(sin ): polynomials in sin() of order n. Given n, there are 2n+1 base functions:
Analyze base function: you find that there is separation of the three variables r, , .

12. Resolution of base functions
Resolution of base-functions (smallest detail described) increases with n. Observe variation in longitude :

13. Resolution of base functions
Test different values of lN: N
Half-wavelength in minutes
Half-wavelength in km
Max degree N 70
120
200
250
360
720
2160
154.3 90 54
43.2
30. 15 5
285.7
166.7
100 80
55.6
27.8
9.26

14. Resolution of base functions
The resolution is given by the amount of nodal lines of the functions. In longitude: Nodal lines are great circles passing through the poles: L nodal lines present. In latitude: Nodal lines are small circles parallel to the equator. N-L nodal lines present.

15. Resolution of base functions
order l=0: no dependence on longitude (zonal harmonics)
When n = l (bottom-right in the figure) no zero crossings in latitude (sectoral harmonics)
For other cases, the functions have an alternating positive-negative pattern on the sphere (tesseral harmonics)
n=3
l=0
n-l=3
n=3
l=1
n-l=2
n=3
l=2
n-l=1
n=3
l=3
n-l=0
n=5
l=2
n-l=3

16. N=4,L=0
N=5,L=0
N=20,L=0

17. N=20,L=3
N=20,L=20
N=20,L=10

18. Illustrazione armoniche sferiche
Scopo: utilizzare la rappresentazione grafica messa a disposizione del GFZ-Potsdam per conoscere in maggior dettaglio le proprietà delle armoniche sferiche.
Collegarsi a:
Aprire l finestra Visualisation of spherical harmonics
Una descrizione dello sviluppo del potenziale gravitazionale e delle variabili usate nel sito si trova in:
Si vedano anche gli appunti del corso.
Fare variare solo il grado “l”, lasciando l’ordine “m” pari a zero (armoniche zonali)
Descrivere le proprietà delle armoniche
Fissare un grado “l” ed aumentare l’ordine “m” da 1 ad “l-1” (armoniche tesserali)
Creare le armoniche settoriali (“l”=”m”).
Eduroam: usare credenziale ateneo. e password
Accesso wifi tramite posta elettronica: e password

19. Expansion of the gravitational potential
M = Earth mass G=Gravitational constant
a= equatorial radius
Cnl Snl: Harmonic coefficients (Stokes coefficients)
Pnl(sin): Associated Legendre Polynmials
N=maximum degree of expansion

20. Expansion of the gravitational potential
Coefficients of expansion obtained from best fit of model to gravity observations.
Multiple satellite missions and terrestrial measurements can be used, with different heights.
Even different kind of measurements, as gravity gradient, gravity, orbital parameters of satellites.
End user: will use final grid or point values synthesized by expansion.
Lowering N equivals to low pass filtering of field, as less detail is included.

21. Derivatives of the gravitational potential
Derivatives of potential field gives gravity and Marussi tensor
-> derivatives applied to base functions. Coefficients are not touched.
-> the table of coefficients is useful to calculate gravity anomaly and full gravity tensor at any geographical coordinate and height above surface.

22. Table of coefficients
The Earth Gravity Model is published as table of numbers, which are the coefficients of the expansion.
Some of the coefficients of the GOCE model TIM 3 (Pail et al., 2011)
n l Cnl Snl dCnl dSnl
E E E E-11
E E E E-11
E E E E-11
E E E E-11
E E E E-11
E E E E-11
E E E E-11
E E E E-11
Fine 9 nov 2016

23. Geoid undulation - EIGEN-GL04C
Start

24. Gravity anomaly - EIGEN-GL04C

25. Degree variances and error degree variances
The amplitude of the field at varying degree is described by the degree variances cn . In same way also the degree errors are defined.
Consider degree variances cl and error degree variances
Empirical rule of Kaula (1966) for variaion of degree variances:

26. Cumulative degree variances and error degree variances
Cumulative degree variances cn and error degree variances give the cumulative amplitude and error up to a a certain degree.

27. Estimated degree variances
Error degree variance spectra for gravity mission concepts SST-hl, SST-ll and satellite
Gradiometry (SGG) compared to pre-Champ Gravity model (GM) and with the signal
degree variances of the gravity field (Kaula). The high precision of SST-ll at long and
medium length scales and the highspatial-resolution of gradiometry are apparent here.
(Balmino et al., 2002)

28. Complementarity of missions
CHAMP:
Proof of validity of of SST-concept. First time combination of SST-hl tracking and observed 3D-accelerometry.
Increase resolution of previous models at low degree.
GRACE:
First SST-ll mission.
Improvement of coefficients at medium.long wavelengths (Max L= ) up to 3 orders of magnitude.
Temporal variations
GOCE (launch 17 march 2009):
Higher spatial resolution (Max expected L=280; corresponds to 65 km)

29. Current global gravity fields
EIGEN-6C4 (Förste et al., 2012): N=2190
EGM2008 (Pavlis et al., 2008): N=2159
GOCE TIM R5 (Pail et al., 2011): N=280
GOCO03S (Mayer-Gürr T. et al. (2012)): N=250
Many more are published here:
(International Centre for Global Earth Models)

30. Degree Error curves for Goce derived models TIM and GOCO
Comparison of GOCE-only models and the newest combined model GOCO (GOCE plus different satellite missions)
After GFZ_ICGM_Homepage

31. Error curves for gravity anomaly
The degree variances can be used to estimate the precision of gravity of the global models.
Precision of gravity at zero height of GOCE and EGM08
Braitenberg et al., 2010

32. Modern global gravity fields
GRACE satellite derived fields
Pure satellite derived field: N=120
Coefficients of higher order: integration of satellite and terrestrial data.
Max degree and order N: refers to spherical harmonic development

33. (Shin et al., 2008)

34. How to use EGMs in practice
In exploration: calculate gravity anomaly or gradients through global model at desired locations, or as gridded values. Use them as you do normally with terrestrial observations.
Understanding of Expansion is important in maximizing efficiency of calculation:
Given the maximum degree N, choose grid-sampling interval accordingly.
N=250: no short wavelengths contained. A smooth field with 80 km resolution. You may oversample, but short wavelengths are not present in field.
Important utility: integration with smaller scale terrestrial observations.
Define regional field from global model-> subtract from your local terrestrial observations.

35. How to use EGMs in practice
Local terrestrial observations: the regional field cannot be determined correctly.
Normal procedure: detrend data.
But: local signal can be distorted this way.
Far better: use regional field from global models to clean the terrestrial observations from deeper sources.
Here: regional field does not need to have high resolution, a lower resolution is sufficient. All the details are contained in your local data.

36. How to use EGMs in practice
Mass corrections done as with terrestrial observations:
All mass corrections must be applied as standard procedure:
Effect of topography -> from gravity anomaly to Bouguer anomaly
Effect of isostatic crustal variations -> from Bouguer anomaly to isostatic residual
Superficial-deep masses separation: frequency filtering.
Fine 15 nov 2016

37. Details on the EGM2008 Spherical Harmonic Expansion

38. Contents Details on the model EGM2008
Database of terrestrial observations
Method of Fill in values
Coastal areas.
Inizio 16 nov 2016

40. EGM2008
Details on the construction of EGM2008 are published in Pavlis et al., 2012, The development and evaluation of the Earth Gravitational Model 2008 (EGM2008) , JGR, Vol. 117.
Important: know where the field is reliable. What is the realistic resolution and precision.
Danger of overinterpretating the field due to varying quality of terrestrial gravity data entering the model.

41. Least Squares construction
Spherical harmonic coefficients fom least square adjustment of satellite and terrestrial observations in iterative procedure.
Satellite: GRACE SST (Satellite-Satellite tracing)
Altimetric satellites: sea surface in ocean and some inland lakes (e.g. Kaspian Sea)
At each iteration the ocean dynamic topography (ODT) is determined more closely: Sea surface is observed from altimeter, but must be corrected for ocean currents in order to give correct information on gravity field. Ocean currents can be determined from the deviation of the ocean surface from the geoid. Use a starting geoid from satellite only, determine ODT, then the gravity over ocean, and make a correction to the geoid at higher frequencies. Then repeat procedure.

42. Digital Terrain Model DTM used in several steps of adjustment:
Upward continuation of observed gravity data to fixed level
Estimate of gravity effect of topography for fill in areas and for remove-restore method in interpolating the observations onto a regular grid.
Topo: SRTM. Shuttle Radar Terrain Model. Coverage: Lat 60°N to 56°S. 90% absolute error between 6 m and 10 m, according to terrain.
Over Ice: used DTM Contains Ice elevation.
Topo decomposed into spherical harmonic expansion.

43. Satellite Altimetry
Satellite altimetry derived gravity anomaly (1arcmin resolution) covers 70% of the globe.
Sources:
- DNSC (Danish National Space Center) -> DNSC08GRA
Scripps Inst. Of Oceanography and National Oceanic and Atmosphere Administration (SIO/NOAA) -> SSv.18.1
DNSC: uses height of sea surface
SIO/NOAA: uses slopes of sea surface
Near Coast: DNSC superior, as slopes need two datapoints and coastal datapoint is missing.
Oceanwide difference is 2 mGal. Greater near coast.

44. Oceanic and terrestrial data
Pavlis et al., 2012

45. Terrestrial data Terrestrial data on land in three classes:
5’ unrestricted: 17.6% of surface
15’ proprietary data: 42.9% of land
Data unavailable: 12%
The class b) corresponds to expansion up to N=720. For 720 N2016 gravity field of topography is calculated for add-on of smaller wavelengths. In these areas only 15arcmin resolution should be used.
Class c): Fill-in with GRACE only (N<60), EGM96 (60 N360), Gravity of Topography for (360 N2190).

46. Terrestrial data 5 ‘ and 15’ grid
Pavlis et al., 2012

47. Database over land and ocean
Land Coast Ocean
65km
195 km
280 km
PGM2007B DNSC07 DNSC DNSC SSv. 18.7
PGM2007B SSv. 18.7
PGM2007B: previous gravity model from Pavlis - group
Altimetric data available at 1 arcmin, they were reduced to 5 arcmin, the resolution of the final grid of terrestrial data (ocean and land)

48. Final remarks
Nominal resolution is 5arcmin, corresponding to N=2190. The resolution is valid only where terrestrial data were available at this resolution. Over most continental areas resolution of 15 arcmin (N=720) is more realistic, as the higher frequencies only reflect topography.
Ovear oceanic areas the resolution of 5’ is realistic, due to the input of the altimetric satellites. Ocean only model has even higher resolution of 1 arcmin.
This makes the GOCE (N=250) the best field today available in regions where terrestrial data are scarce.

49. Thank you for your attention!