12.540 Principles of the globa Positioning System Lecture 10 Prof. Thomas Herring 03/1203 12540Lec10 Estimation Introduction · Homework review Overview Basic concepts in estimation Models: Mathematical and Statistical Statistical concepts
03/12/03 12.540 Lec 10 1 12.540 Principles of the Global Positioning System Lecture 10 Prof. Thomas Herring 03/12/03 12.540 Lec 10 2 Estimation: Introduction – Basic concepts in estimation – Models: Mathematical and Statistical – Statistical concepts • Homework review • Overview 1
Basic concepts Basic problem: We measure range and phase data that are related to the positions of the ground receiver, satellites and other quantities How do we determine the " best position for the receiver and other quantities What do we mean by best " estimate? Inferring parameters from measurements is estimation 03/1203 12540Lec10 Basic estimation Two styles of estimation(appropriate for geodetic type measurements) Parametric estimation where the quantities to be that express the observablesarables in equations Condition estimation where conditions can be formulated among the observations. Rarely used, most common application is leveling where the sum of the height differences around closed circuits must be zero
03/12/03 12.540 Lec 10 3 Basic concepts estimation 03/12/03 12.540 Lec 10 4 Basic estimation – Parametric estimation where the quantities to be that express the observables – formulated among the observations. Rarely used, most common application is leveling where the sum of the height differences around closed circuits must be zero • Basic problem: We measure range and phase data that are related to the positions of the ground receiver, satellites and other quantities. How do we determine the “best” position for the receiver and other quantities. • What do we mean by “best” estimate? • Inferring parameters from measurements is • Two styles of estimation (appropriate for geodetic type measurements) estimated are the unknown variables in equations Condition estimation where conditions can be 2
Basics of parametric estimation All parametric estimation methods can be broken into a few main steps Observation equations: equations that relate the arameters to be estimated to the observed quantities(observables). Mathematical model position sat elite position (implicit in po clocks receiver Stochastic model: Statistical description that describes the random fluctuations in the measurements and maybe the parameters Inversion that determines the parameters values from the mathematical model consistent with the statistical mode 03/1203 12540Lec10 Observation model Observation model are equations relating observables to parameters of model Observable= function(parameters) Observables should not appear on right-hand-side of equation Often function is non-linear and most common method is linearization of function using Taylor senes expansion Sometimes log linearization for f=a bc ie Products fo parameters
03/12/03 12.540 Lec 10 5 Basics of parametric estimation – Observation equations: equations that relate the parameters to be estimated to the observed position, satellite position (implicit in r), clocks, atmospheric and ionosphere delays – Stochastic model: Statistical description that describes the random fluctuations in the measurements and maybe the parameters – Inversion that determines the parameters values from the mathematical model consistent with the statistical model. 03/12/03 12.540 Lec 10 6 Observation model – – of equation • All parametric estimation methods can be broken into a few main steps: quantities (observables). Mathematical model. • Example: Relationship between pseudorange, receiver • Observation model are equations relating observables to parameters of model: Observable = function (parameters) Observables should not appear on right-hand-side • Often function is non-linear and most common method is linearization of function using Taylor series expansion. • Sometimes log linearization for f=a.b.c ie. Products fo parameters 3
Taylor series expansion In most common Taylor series approach y=f(x1,x2,x3,x4) yo+ ay=f(x)+(xAx x=( x2,x3, 4) The estimation is made using the difference between the observations and the expected values based on apriori values for the parameters The estimation returns adjustments to apriori parameter values 03/1203 12540Lec10 Linearization Since the linearization is only an approximation, the estimation should be iterated until the adjustments to the parameter values are zero For GPs estimation Convergence rate is 100 1000: 1 typically(ie, a 1 meter error in apriori coordinates could results in 1-10 mm of non linearity error)
03/12/03 12.540 Lec 10 7 Taylor series expansion • In most common Taylor series approach: • The estimation is made using the difference between • The estimation returns adjustments to apriori y = f (x1, x2, x3, x4 ) y0 y = f (x) x 0 + ∂f (x) ∂x Dx x = (x1, x2, x3, x4 ) the observations and the expected values based on apriori values for the parameters. parameter values + D 03/12/03 12.540 Lec 10 8 Linearization • Since the linearization is only an approximation, the estimation should be iterated until the adjustments to the parameter values are zero. • For GPS estimation: Convergence rate is 100- 1000:1 typically (ie., a 1 meter error in apriori coordinates could results in 1-10 mm of nonlinearity error). 4
Estimation (Will return to statistical model shortly) Most common estimation method is"least-squares"in which the parameter estimates are the values that minimize the sum of the squares of the differences between the observations and modeled values based on parameter estimates For linear estimation problems, direct matrix formulation for solution For non-linear problems: Linearization or search technique where parameter space is searched for minimum value Care with search methods that local minimum is not found (will not treat in this course) 12540Lec10 Least squares estimation Originally formulated by Gauss Basic equations: Ay is vector of observations A is linear matrix relating parameters to observables; Ax is vector of parameters; v is esidual △y=AAx+v minimize(v v); superscript T means transpose △x=(AA)A△y
03/12/03 12.540 Lec 10 9 • (Will return to statistical model shortly) • minimize the sum of the squares of the differences on parameter estimates. • For linear estimation problems, direct matrix formulation for solution • minimum value • found (will not treat in this course) Estimation Most common estimation method is “least-squares” in which the parameter estimates are the values that between the observations and modeled values based For non-linear problems: Linearization or search technique where parameter space is searched for Care with search methods that local minimum is not 5 03/12/03 12.540 Lec 10 10 Least squares estimation D observables; D residual Dy = ADx + v minimize vT ( ) v ; Dx = (AT A) -1 AT Dy • Originally formulated by Gauss. • Basic equations: y is vector of observations; A is linear matrix relating parameters to x is vector of parameters; v is superscript T means transpose