1 Basic statistical concepts and least-squares. Sat05_61.ppt, Statistical concepts 2.Distributions Normal-distribution 2. Linearizing 3. Least-squares The overdetermined problem The underdetermined problem

2 Histogram. Describes distribution of repeated observations At different times or places ! Distribution of global 1 0 mean gravity anomalies

3 Statistic: Distributions describes not only random events ! We use statistiscal description for ”deterministic” quantities as well as on random quantities. Deterministic quantities may look as if they have a normal distribution !

4 ”Event” Basic concept: Measured distance, temperature, gravity, … Mapping: Stocastisc variabel. In mathematics: functional, H – function-space – maybe Hilbertspace. Gravity acceleration in point P: mapping of the space of all possible gravity-potentials to the real axis.

5 Probability-density, f(x): What is the probability P that the value is in a specific interval:

6 Mean and variance, Estimation-operator E:

7 Variance-covariance in space of several dimensions: Mean value and variances:

8 Correlation and covariance-propagation: Correlation between two quantities: = 0: independent. Due to linearity

9 Mean-value and variance of vector If X and A 0 are vectors of dimension n and A is an n x m matrix, then The inverse P = generally denoted the weight- matrix

10 Distribution of the sum of 2 numbers: Exampel Here n = 2 and m = 1. We regard the sum of 2 observations: What is the variance, if we regard the difference between two observations ?

11 Normal-distribution 1-dimensional quantity has a normal distribution if Vektor of simultaneously normal-distributed quantities if n-dimensional normal distribution.

12 C ovarians-propagation in several dimensions: X: n-dimensional, normally distributed, D nxm matrix, then Z=DZ also normal distributed, E(Z)=D E(X)

13 Estimate of mean, variance etc

14 Covarians function If the covariance COV(x,y) is a function of x,y then we have a Covarians-function May be a function of Time-difference (stationary) Spherical Distance, ψ on the unit-sphere (isotrope)

15 Normally distributed data and resultat. If data are normaly dsitributed, then the resultats are also normaly distributed If they are linearily related ! We must linearize – TAYLOR-Development with only 0 and 1. order terms. Advantage: we may interprete error-distributions.

16 Distributions in infinite-dimensional spaces V( P) element in separable Hilbert-space: Normal distributed with sum of variances finite !

17 Stochastic process. What is the probability P for the event is located in a specific interval Exampel: What is the probability that gravity in Buddinge lies in between -20 and 20 mgal and that gravity in Rockefeller lies in the same interval

18 Stokastisc process in Hilbertspace What is the mean value and variance of ”the Evaluation- functional”,

19 Covariance function of stationary time-series. Covariance-function depends only on |x-y| Variances called ”Power-spectrum”.

20 Covariance function – gravity-potential. Suppose X ij normal-distributed with the same variance for constant ”i”.

21 Isotropic Covariance-function for Gravity potential

22 Linearizering: why ? We want to find best estimate (X) for m quantities from n observations (L). Data normal-distributed, implies result normaly distributed, if there is a linear relationship. If m > n there exist an optimal metode for estimating X: Metode of Least-Squares

23 Linearizing – Taylor-development. If non-linear: Start-værdi (skøn) for X kaldes X 1 Taylor-development with 0 og 1. order terms after changing the order

24 Covariance-matrix for linearizered quantities If measurements independently normal distributed with varians-covariance Then the resultatet y normal-dsitributed with variance- covarians:

25 Linearizing the distance-equation. Linearized based on coordinates

26 On Matrix form: If 3 equations with 3 un-knowns !

27 Numerical-example If (X 11, X 12,X 13 ) = ( m, m, m). Satellite: ( , , ) Computed distance: m Measured distance: m (( )dX 1 + ( ) dX 2 +( ) dX 3 )/ = ( ) or: dX dX dX 3 = -2.7

28 Linearizing in Physical Geodesy based on T=W-U In function-spaces the Normal-potential may be regarded as a 0-order term in a Taylor- development.We may differentiate in Metric space (Frechet-derivative).

29 Method of Least-Square. Over-determined problem. More observations than parameters or quantities which must be estimated: Examples: GPS-observations, where we stay at the same place (static) We want coordinates of one or more points. Now we suppose that the unknowns are m linearily independent quantities !

30 Least-squares = Adjustment. Observation-equations: We want a solution so that Differentiation:

31 Metod of Last-Squares. Variance-covariance.

32 Metod of Least-Squares. Linear problem. Gravity observations: H, g= /-0.02 mgal G I / / /-0.03

33 Observations-equations..

34 Method of Least-Squares. Over-determined problem. Compute the varianc-covariance-matrix

35 Method of Least-Squares. Optimal if observations are normaly distributed + Linear relationship ! Works anyway if they are not normally distributed ! And the linear relationship may be improved using iteration. Last resultat used as a new Taylor-point. Exampel: A GPS receiver at start.

36 Metode of Least-Squares. Under-determined problem. We have fewer observations than parameters: gravity- field, magnetic field, global temperature or pressure distribution. We chose a finite dimensional sub-space, dimension equal to or smaller than number of observations. Two possibilities (may be combined): We want ”smoothest solution” = minimum norm We want solution, which agree as best as possible with data, considering the noise in the data

37 Method of Least-Squares. Under-determined problem. Initially we look for finite-dimensional space so the solution in a variable point P i becomes a linear- combination of the observations y j : If stocastisk process, we want the ”interpolation-error” minimalized

38 Method of Least-Squares. Under-determined problem. Covariances: Using differentiation: Error-variance:

39 Method of Least-Squares. Gravity-prediction. Example: Covarianses: COV(0 km)= 100 mgal 2 COV(10 km)= 60 mgal 2 COV(8 km)= 80 mgal 2 COV(4 km)= 90 mgal 2 P 6 mgal Q 10 mgal R 10 km 8 km 4 km

40 Method of Least-Squares. Gravity prediction. Continued: Compute the error-estimate for the anomaly in R.

41 Least-Squares Collocation. Also called: optimal linear estimation For gravity field: name has origin from solution of differential-equations, where initial values are maintained. Functional-analytic version by Krarup (1969) Kriging, where variogram is used closely connected to collocation.

42 Least-Squares collocation. We need covariances – but we only have one Earth. Rotate Earth around gravity centre and we get (conceptually) a new Earth. Covariance-function supposed only to be dependent on spherical distance and distance from centre. For each distance-interval one finds pair of points, of which the product of the associated observations is formed and accumulated. The covariance is the mean value of the product-sum.

43 Covarians-function for gravity anomalies, r=R.

44 Covarians function for gravity-anomalies: Different models for degree-variances (Power-spectrum): Kaula, 1959, (but gravity get infinite variance) Tscherning & Rapp, 1974 (variance finite).