licentiate presentation

IntroductionMethodology

PapersConclusion

On Optimisation and Design ofGeodetic Networks

Licentiate Thesis in Geodesy

Mohammad Amin Alizadeh Khameneh

Royal Institute of Technology (KTH)

12 June 2015

Mohammad Amin Alizadeh Khameneh On Optimisation and Design of Geodetic Networks 1 / 31


PapersConclusion

Outline1 Introduction

Network Quality CriteriaNetwork Optimal Design

2 MethodologyMethodsModels

3 PapersPaper IPaper IIPaper IIIPaper IV

4 ConclusionRound-upFuture Works



PapersConclusion









PapersConclusion


Introduction

Need for geodetic networks in urban management,engineering projects, hydrography, geo-hazardassessment , aerial photogrammetry, ...

Deformation monitoringCrustal movements, ground subsidenceDeformation of man-made structures: water powerdams, underground tunnels, bridges, high buildings, ...



PapersConclusion


Establishment of a Geodetic Network:

Network DesignWhere the network points should be located?How the network should be measured?

Network qualityExecution: designed network to realityNetwork Analysis



PapersConclusion


PrecisionReliabilityEconomy

Sensitivity(in case of deformation monitoring)

Optimal design leads to:Avoiding unnecessary observationsSaving a considerable amount of time and effortIdentifying and eliminating gross errors



PapersConclusion


Zero-Order Design (ZOD)optimum reference datum

First-Order Design (FOD)optimum locations for the stations

Second-Order Design (SOD)which observations with what precision and reliability

THird-Order Design (THOD)how to improve the existing network



PapersConclusion

MethodsModels








PapersConclusion

MethodsModels

Methodology

Since the last few decades, the network design approacheshave evolved from the intuition/empirical to analyticalmethods.

Trial and Error MethodAnalytical Method

minimise/maximise some Object Function (OF) thatdescribes precision, reliability and cost by a scalarvalue.Criterion matrix as an ideal variance covariancematrix, which should be best approximated by theactual covariance matrix of estimated parameters.



PapersConclusion

MethodsModels

Single-Objective Optimisation Model (SOOM)minimising or maximising any of the objective functions for:

Precision‖Cx −Cs‖ = minimum

Reliability‖r‖ = maximum

Cost‖P‖ = minimum

Bi-Objective Optimisation Model (BOOM)A pair combination of any of these criteriaMulti-Objective Optimisation Model (MOOM)All the quality requirements are considered in the OF

Sensitivity in a Network:The capability of a network to detect displacements ordeformation parameters of a certain magnitude

λi = dTi C−1

didi ∼ χ2

1−α (df )



PapersConclusion

Paper IPaper IIPaper IIIPaper IV








PapersConclusion


Paper I Acta Geodaetica et Geophysica, 2014, Published

titleThe Effect of Constraints on Bi-Objective Optimisation ofGeodetic Networks

objectives of the studyContradiction of the controlling constraints in a SOOM,which may lead to an infeasibility in the optimisation pro-cess causes a problem in these models.A BOOM of precision and reliability can solve the problem,but how important is using the controlling constraints?



PapersConclusion


Objective Function (OF) for BOOM of Precision andReliability[

‖Hw− u‖‖vec (Cs)‖

+ ‖R11w− (rm − r00)‖‖rm‖

]→ min

subject to { [DT 0

]w = 0

A00w ≤ b00

By applying L2-norm to the above OF, the BOOM is convertedto a quadratic programming model and by solving that, one canget the optimal values.



PapersConclusion


The BOOM of precision andreliability can optimise thenetwork properly even with-out controlling constraints.

Unconstrained BOOM ismore economical in practi-cal considerations as moreobservables are removedfrom the plan, while theaccuracy and reliability ofthe network almost meetthe network requirements.

Precision of the net points: 3mm, reliability of the observations: ≥ 0.4

(a) Unconstrained (b) Constrained to Precision

(c) Constrained to Reliability (d) Constrained to Precision andReliability



PapersConclusion


Although the standard er-rors of the net points afterunconstrained BOOM area bit larger than the con-strained results, the uncon-strained model can success-fully fulfil the precision andreliability requirements.



PapersConclusion


Paper II Boletim de Ciencias Geodesicas, 2015, Accepted

titleTwo-Epoch Optimal Design of Displacement MonitoringNetworks

objective of the studyTo design an optimal displacement monitoring network intwo epochs by estimating the variances of the observationsrather than their weights.



PapersConclusion


One-Epoch Optimisation

Cx = σ20

[(ATPA + EET

)−1− E

(ETEETE

)−1ET]

The main idea of the one-epoch optimisation is to change the configuration and determine theweight of observations by fitting the following mathematical model:

Hw = u

, where

H =[vec

(∂C∆x∂x1− ∂C

2∂x1

)vec

(∂C∆x∂y1− ∂C

2∂y1

)· · ·

vec(∂C∆x∂xm− ∂C

2∂xm

)vec

(∂C∆x∂ym− ∂C

2∂ym

)vec

(∂C∆x∂p1

)· · · vec

(∂C∆x∂pn

)],with C

2 as a criterion matrix, and

w = (∆x1 ∆y1 · · · ∆xm ∆ym ∆p1 ∆pn)T

as an improvement vector for net points position and observation weights, and

u = vec(C

2

)− vec (C∆x)

In order to derive w, the following optimization model should be solved:

‖Hw− u‖2 → min



PapersConclusion


Two-Epoch Optimisation

Using Gauss-Helmert model instead of Gauss-Markov, gives this ability to consider allobservations of two epochs:

A∆x + B[ε1ε2

]= w = B

[L1L2

]with B =

[−In In

]

C∆x =(ATK−1A + DDT

)− E

(ETDDTE

)−1ET where K = BQBT and

Q = diag(

Q1 Q2)

The expansion of C∆x can be presented in a matrix form as:H′w′ − u′

where

H′ =[vec

(∂C∆x∂x1− ∂C

∂x1

)vec

(∂C∆x∂y1− ∂C

∂y1

)· · · vec

(∂C∆x∂xm− ∂C

∂xm

)vec

(∂C∆x∂ym− ∂C

∂ym

)vec

(∂C∆x∂q1

1

)· · · vec

(∂C∆x∂q1

n

)vec

(∂C∆x∂q2

1

)· · · vec

(∂C∆x∂q2

n

)]w′ = (∆x1 ∆y1 · · · ∆xm ∆ym ∆q1

1 · · · ∆q1n ∆q2

1 · · · ∆q2n)T

u′ = vec (C)− vec (C∆x)

An optimisation model can be formulated as ‖H′w′ − u′‖2 → min subject to physicalconstraints.



PapersConclusion


Comparing the two methods, wehave similar accuracies of dis-placementsThe same configuration and posi-tion changes of the points in bothapproaches

Based on the two-epoch optimisa-tion procedure, less observationsare needed in each epoch. It canbe concluded as a more economi-cal solution

(e) One-Epoch Optimisation

(f) Two-Epoch Optimisation (Epoch 1)

(g) Two-Epoch Optimisation (Epoch 2)

(h) Observation weights



PapersConclusion


Paper III Journal of Geodetic Science, 2015, Accepted

titleOptimisation of Lilla Edet Landslide GPS MonitoringNetwork

objective of the studyTo implement different optimisation models in a real casestudy and design an optimal network. This network is sup-posed to be sensitive in detecting possible displacements.



PapersConclusion


Study Area

Lilla Edet village

Landslide andsubduction alongGota river

35 GPS stations



PapersConclusion


Basic equation for measured GPS baselines:

∆xpq∆ypq∆zpq

− ε =

xq − xpyq − ypzq − zp

− ε =

1 −1 0 0 0 00 0 1 −1 0 00 0 0 0 1 −1

xqxpyqypzqzp

and Pi = σ2

0

σ2∆xi 0 00 σ2

∆yi 00 0 σ2

∆zi

−1

The statistical test to figure out if the displacements can be detected or not: λk = dTk C−1

xkdk ∼ 2χ2

0.95 (3) ,λk = dT

k C−1xk

dk ∼ 2χ20.95 (3) → H0 is rejected if λk > 2χ2

0.95 (3)

Criterion matrix based on the sensitivity of the network in detecting the displacements:

Cs =

σ2

s1I3 0 0 00 σ2

s2I3 0 00 0 . . . 00 0 0 σ2

skI3

, k = 1, 2, ...,m with σ2sk

= dTk dk

2χ20.95(3) , k = 1, 2, · · · ,m

SOOM of Precision: ‖Cs −Cx‖2 = min

we try to minimise the linearised form of the variance matrix as: ‖Hw− u‖2 = min , where

H =[

vec(∂Cx∂P1

)vec

(∂Cx∂P2

)· · · vec

(∂Cx∂Pn

) ]u = vec (Cs)− vec (Cx)

w =[

∆P1 ∆P2 · · · ∆Pn]T

SOOM of Reliability: min (‖diag (R)‖∞) = min(∥∥∥∥diag

(R0 +

n∑i=1

∂R∂Pi

∆Pi

)∥∥∥∥∞

)= max

SOOM of Cost: ‖P‖∞ =∥∥∥∥P0 +

n∑i=1

∂P∂Pi

∆Pi

∥∥∥∥∞

= min

BOOM/MOOM:[‖Hw−u‖2‖vec(Cs)‖2

+ ‖R2w−(ro−R1)‖2‖ro‖2

+ ‖C2w−(co−C1)‖2‖co‖2

]→ min



PapersConclusion


(i) SOOM of reliability with precision constraint andsensitive to detect 5 mm displacement

(j) BOOM of precision and reliability, sensitive todetect 5 mm displacement

(k) Redundancy number (reliability) of theobservations after optimisation procedure

(l) Precision of the net points afteroptimisation based on different models

(m) Results of different optimisation modelsperformed on the GPS network



PapersConclusion


RemarksUnconstrained SOOM of precision had no control onreliability; precise but not reliable.The SOOM of reliability, constrained to precision,yielded better results (in sense of precision andreliability).BOOM or MOOM provided the network qualityrequirements with less number of baselines (costlyefficient).Insignificant effect of cost criterion on MOOM in GPSnetworks with short baselines. BOOM can be efficientenough.



PapersConclusion


Paper IV Acta Geodaetica et Geophysica, 2015, Submitted

titleThe Effect of Instrumental Precision on Optimisation ofDisplacement Monitoring Networks

objective of the studyTo investigate the effect of observation precision in optimisa-tion of the Lilla Edet GPS displacement monitoring network.It has been assumed that the precision of GPS observationscan be increased in the subsequent epochs.



PapersConclusion


If we increase the weight of observations in the second epoch P2, we will acquirea more precise network by the second observation plan, so we can write:

P2 = 1k P1 → Cx2 = k Cx1 with k < 1

The displacement vector: d = x2 − x1 → Cd = Cx1 + Cx2

Cd − k Cx1 = Cx1 = C0x1 +

n∑i=1

∂Cx1

∂Pi∆Pi

We try to minimise the differences between the VC matrix of the first epoch anda defined ideal criterion matrix:∥∥∥Cs −C0

x1

∥∥∥2

= min

subjected to precision, reliability and physical constraints.



PapersConclusion


(n) Optimised observation plans for the first and second epoch,considering k=0.5

(o) Number of removed baselines due to precision improvements.The number of initial baselines is 245.



PapersConclusion

Round-upFuture Works








PapersConclusion


Conclusion

1 In Paper I: Unconstrained BOOM was also efficient.2 In Paper II: The two-epoch method removed more ob-

servations.3 In Paper III: The Lilla Edet GPS monitoring network

was optimised by different optimisation models.4 In Paper IV: Significant changes in designing an obser-

vation plan of epoch-wise measurements by assuminghigher observation precision for the latter epoch.



PapersConclusion


Investigating the effect of possible correlations on GPSbaseline processing.Developing the optimisation technique to design defor-mation monitoring networks, using Finite Element Method.Considering a direction constraint in optimisation pro-cedure.Applying intelligent optimisation techniques beside theclassical methods.



PapersConclusion


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