The radar altimetric observations 1:

1
Part 2.

• The radar altimetric observations (1)
• Altimetry data
• Contributors to sea level
• Crossover adjustment
• From altimetric heights to Gravity (2)
• Geodetic theory
• FFT for global gravity fields
• Least Squares Collocation
• The Global Altimetric Gravity Field (3)
• Accuracy assesment
• Applications

2
The Anomalous Potential.

• The anomalous potential T is the difference
  between the actual gravity potential W and the
  normal potential U
• T is a harmonic function outside the masses of
  the Earth satisfying
• (?²T 0) Laplace (outside the masses)
• (?²T -4???) Poisson (inside the masses (? is
  density))
• Expanding T in spherical harmonic functions
• Pij are associated Legendre's functions of degree
  i and order j
• Geoid heights, multiplying the coefficients by
  1/?.
• Gravity anomalies, multiplying the coefficients
  by (i-1)/R

3
Bruns formula links N with T N can be
expressed in terms of a linear functional applied
on T (? is the normal gravity) Gravity and T
Deflection of the vertical
(n,e)Deflection of the vertical is related
to geoid slope Geoid slopes (east, west) can be
obtained from altimetry by tranformingthe
along-track slopes to east-west slopes.
4
Three ways to get gravity from altimetry.
1) Integral formulas (Stokes Vening Meinesz
Inverse) Requires extensive computations over
the whole earth. Replace analytical integrals
with grids and is combined with FFT 2) Fast
Fourier Techniques. Requires gridded data (will
return to that). Very fast computation.
Presently the most widely used method. 3)
Collocation. Requires big computers. Rene
Forsberg on 1. Main focus on 2 3
5
2D FFT Flat Earth approximation

• Flat Earth approx is valid (2-300 km from
  computation point, Sideris, 1997).
• The geodetic relations with T are then
• Where F is the 2D planar FFT transform

6
From height to gravity using 2D FFT - KMS
approach

• An Inverse Stokes problem
• High Pass filter operation enhance high
  frequency.
• Optimal filter was designed to handle white noise
  power spectral decay obtained using
• Frequecy domain LSC with a Wiener Filter
  (Forsberg and Solheim, 1990)
• Power spectral decay follows Kaulas rule (k-4)
• Resolution is where wavenumber k yields ?(k)
  0.5
• For KMS 12.5 km is used.

7
From Deflections of Vertical to gravity.

• Laplaces Equation (with approximation) gives.
• In Frequency domain this is
• Sandwell and Smith uses this formulation to go
  from deflections of
• Vertical to gravity. (the deflections of
  vertical are easily computed
• From along track slopes using a transformation).

8
Least Squares Collocation for altimetry.

• Advantage
• The ability in-corporate randomly spaced data of
  various types
• Predicting related geodetic quantities taking
  into account the different statistics of the
  input data.
• Predicting both signal and error components.
• No interpolation is required.

9
Least Squares Collocation

• The optimum estimate using geoid height or geoid
  slopes is
• With aposteori error variance
• Chh , C?gh , C?g?g , C?? , C?g? Covariance
  matrices between height-height,
• gravity height, gravity-gravity, slope-slope and
  slope-gravity.
• Covariance matrices Dhh and D?? contain the noise
  variance

10
Covariance estimation.

• Empirically (I.e., Knudsen 1987)
• GRAVSOFT Empcov. Compute cov by comparing all
  points with all points.
• Through Covariance propagation CNNLN(LN(K(P,Q)))
  
• The covariance between the anomalous potential
  is
• siTT are degree variances

11
Covariance propagation

• The degree variance of Tscherning Rapp (1984)
• Example using degree variances from OSU91A,
• A 1571496 m4/s4, and
• RB R - 7.4 km is the depth to the Bjerhammer
  sphere
• Height 1500 km

12
NN gg ee
Ng ge eN
13
Pros and Cons of LSC

• Ability to handle ir-regular sampled data
  without the degradation though interpolation
• Very suited for local fields.
• Coastal regions (Advantage over FFT method -
  Gibbs phenomenon.)
•  
• Its not computational feasible for global high
  res marine gravity fields yet.
• LSC requires data within a cell of at least 1?
  containing easily 2000 data points in order to
  compute accurate covariance functions for each
  prediction point.
• Compromise
• LSC are more conveniently used in combination
  with FFT methods.
• LSC can very efficiently interpolation of the
  corrected altimetric observations
• (heights or deflections of the vertical onto
  regular grid.
• LSC interpolation requires only few points
  compared with full LSC (10-100)
•  
•  

14
2D FFT Interpolation using collocation.

• GEOGRID (GRAVSOFT) can perform interpolation
  using second order Gauss Markov Covariance
  function.
• r is the distance, C0 is the signal variance, a
  is the correlation length

• Modify covariance function to model residual
  surface height variability or along track
• Errors (causing cross-track gradients between
  parallel tracks-orange skin effect)
• Adding or modifying the covariance function to
  account for this error in the interpolation.
• Additive error covariance only applied to
  observations on the same track
• ? fixed to 100km, but other parameters determined
  empirically

15
Signal variance C0
16
Correlation length a
17
Additional error variance computed from aposteori
xover variance (D0)
18
Residual altimetric ?N (from lecture 1. )
19
After Interpolation Using GEOGRIDxx
20
After conversion to gravity Using GEOFOURxx
21
Restoring longwavelength Geoid field (EGM)
22
GRAVSOFT and Altimetry.

• Least Squares collocation
• GEOCOL - least-squares collocation and comp.of
  reference fields.
• EMPCOV, COVFIT empirical covariance function
  estimation and fitting.
•  
• Interpolation
• GEOGRID fast and reliable program for handling
  the interpolation of the observations onto a
  regular grid is the collocation based algorithm
• FFT
• The GRAVSOFT software library has facilities to
  do two-dimensional planar FFT via the routine
  geofour
• Multiband spherical 2D FFT can be handled using
  the routine spfour.
• Finally software for carrying out the
  one-dimensional FFT is available via the routine
  sp1d.