Resolution and significant contributions of tidalforcing in flexible harmonic grouping computed

using Singular Value Decomposition

Adam Ciesielski, Thomas Forbriger

Karlsruhe Institute of Technology, GPI (KIT), BFO

EGU 2020 Display

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Contents

I Introduction: SVD in tidal analysis

I Methods: RATA-software

I 1. Investigations of atmospheric cross-talk

I 2. Separation of tides with degree 2 and 3

I Methods: Synthetic harmonic study

I 3. Resolution loss in central frequencies

I 4. Intrinsic wave grouping

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Contents

Introduction

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Tidal AnalysisInterest

I Our main interest of tidal analysis is the accurate and precisedetermination of tidal parameters, the quantities describingthe Earth response to the tidal forcing.

I Widely used software, like ETERNA or Baytap-G, use ana-priori grouping of harmonics present in a tidal catalogues(e.g. Venedikov 1961, Hartmann & Wenzel 1995, Tamura1987).

I Wave grouping of a harmonic development is a modelparameterisation used to make the analysis problemoverdetermined.

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MotivationAnalysis defficiency

The common analysis methods minimize the data misfit only (e.g.Eterna 3.4).

I If model assumptions (e.g. credo of smoothness, knownfree-core resonance parameters, known ratio between responseto degree 2 and degree 3 forcing) are incorrect, the analysiscan lead to unnoticed artefacts

I Moving window tidal analyses of gravity recordings showtemporal variations of tidal parameters for different stations(Meuers, 2004; Meuers et al. 2016; Schroth, 2018).

I We search for the causes of this phenomenon.

We abandoned the a-priori wave grouping and investigated theintrinsic nature of tidal harmonics using Singular ValueDecomposition.

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Tidal AnalysisSingular Value Decomposition

SVD is a factorisation of a linear regression matrix. The regressionmatrix consists of tidal harmonics in-phase and quadrature signalfor rigid Earth (tidal forcing to Earth surface, also described as“equilibrium tide”).That concept from signal processing allows us to linearize theproblem such that gravity signal is linearly scaled to the forcingsignal

A cos(ωt + φ) = A · cos(ωt) · cos(φ)− A · sin(ωt) · sin(φ).

A cos(ωt + φ) = AC (t) · cos(φ) + AS(t) · sin(φ).

And further to determine gravimetric parameters.SVD allows us to study the significance of tidal harmonics,“natural” wave grouping, possible cross-talk between harmonics orgroups and matrix null space.

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Tidal analysisSingular Value Decomposition

The tidal prediction problem:

~s = G ~m

~s: synthetic gravity signal vectorG : rigid earth (equilibrium) tide signals matrix (forcing)1

~m: gravimetric factors (tidal parameters) vector (response)

SVD factorization:

G = UΛV T

Diagonal Λ: singular values (SV). Large SV indicate essentialcomponents of ~s.Orthonormal V : a kind of natural wave grouping.Orthonormal U : the corresponding gravity variation.

1Forward operator G may also include air pressure as a part of the forcingmodel.

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Singular Value DecompositionMatrix V

It turns out in the investigation, that Matrices V indicatecross-talk.On all the diagrams:

I Harmonic vectors are sorted with decreasing singular value.

I Absolute values of coefficients are plotted.

I Black points on displayed matrix V mean that absolute valuesof coefficients exceed 1/

√2.

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Singular Value DecompositionThe cut-off

In the analysis, we would like to specify the level which is a limit ofthe significance of the model parameters.

k = log10

(λ1λp

)Singular values λ are sorted, so λ1 is the largest. The ratiocorresponds to the signal/noise range. We distinguish two differentlevels of cut-off k :

I k = 3, that corresponds to the usual range in superconductinggravimeter tidal observations

I k = 7, the level above the range present in tidal catalogues

Often we refer to them as the “Singular value threshold level”. Inmost diagrams presented, we apply the cut-off level k = 3.

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Singular Value DecompositionModel resolution matrix

Rm = VpVTp

Model resolution matrix properties:

I Describes how the (generalised) inverse solution “smears” theoriginal model into a recovered model.

I The true model “leaks” into adjacent model parameters and isreduced in its maximum amplitude in the recovered model.

I If we consider a model parameter for a harmonic, it may bepartially parameter obtained for another harmonic.

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Contents

Methods - RATA softwareRobust Approach to Tidal Analysis

The tidal forward operator G is a set of tidal harmonics that would berecorded on rigid Earth at a fixed place.

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Methods - RATA softwareAlternative tidal analysis with SVD

We compute time series for each harmonic present in the Tamuratidal catalogue by using a modified version of “Predict” (ETERNApackage).

I Resulting values can be, but do not need to be, groupedprior to SVD analysis.

I Other than with conventional programs, wave groups can bedefined not only as frequency intervals.

I One possibility is separation harmonics of degree 2 and 3

I We may investigate which singular vectors do notsignificantly contribute to the predicted tidal data or arenoise-sensitive

I With proper time-domain signal filters, we may investigate thecross-talk between harmonics and air pressure variations.

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Methods - RATA softwareAlternative tidal analysis with SVD

PlaceTimePredict(ETERNA) Rigid Earth Tide Each harmonicin-phase and quadrature IGETS(database) Pressure recordingsGravity recordings Grouping function

SingularValueDecomposition

Data selectionandtime-domain fIlteringWaveGroups

Forward operator G

Catalogue

UV ΛGravityvector gInversion

Gravimetric parameters m

Grouping

VThe scope of the display13 / 40

1. Atmospheric cross-talkMotivation

Frequency of S1 is exactly 1 CpD. Earth response is also affectedby “radiation tides” which are induced by air pressure diurnal cycle.

That variation is included in the forcing model together with wavegroups. We study the properties of S1 group and air pressuregravity signal.

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1. Atmospheric cross-talkDescription

The next few slides present investigation of atmosphericcross-talk to tidal groups.

I We use 1 year length tidal forward operator withcorresponding air pressure records at BFO.

I Standard grouping has been applied. All the data have beenfiltered.

I We expect to observe cross-talk between air pressure inducedgravity and tidal harmonics from expected group K1 (S1)

I We verify if there is a cross-talk from the other harmonics,previously noticed for unfiltered data (Phi1, O1, MF).

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1. Atmospheric cross-talkMatrix V - absolute values

K1-SK1-CM2-CM2-SO1-SO1-CS2-SS2-CP1-CP1-SN2-SN2-CQ1-CQ1-SK2-CK2-SM1-S

M1-CJ1-SJ1-CL2-CL2-S

MI2-S

MI2-C

OO1-C

OO1-S

FI1-S

FI1-C

M3-CM3-S

PSI1-C

PSI1-S

AIR-X

S1-CS1-S

M0S0-C

MF-C

M0S0-S

MF-S

0 10 20 30 400.11101001000104

Vector index

nm/s2

SingularValues RMS; Observatory: BFO; Date: 2011-11-01; Timespan: 365 days.

C: in-phase, S: quadrature signal 16 / 40

1. Atmospheric cross-talkResolution matrix: 35/39 harmonics

C: in-phase, S: quadrature signal 17 / 40

1. Atmospheric cross-talkConclusion

I Singular value decomposition indicates cross-talk to otherharmonics in our dataset.

I While correlation with S1 group was expected, the reason foreffect on other harmonics (e.g. Psi1, Phi1) remains unclear.

I The long-period trends have been correctly filtered, so they donot contribute to the signal.

I SVD indicates that all model parametres should be resolvable.

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2. Separation of degree 2 and 3 tidesDescription

The next few slides present an investigation of the possibility todetermine the ratio of gravimetric factors of degree 2 anddegree 3 tides. In this attempt, however, the groups combine ofeven or odd tides, respectively.

I We investigated properties of tidal forward operator for 1 yearat BFO.

I Standard grouping has been applied, but harmonics of degree2 and 3 have been separated.

I We suppose, that harmonics should be well resolvable.

We expect ratios to be close to the a-priori values, but this analysisgoes beyond our current study.

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2. Separation of degree 2 and 3 tidesMatrix V - absolute values

K1o-S

K1o-C

M2o-C

M2o-S

O1o-S

O1o-C

S2o-C

S2o-S

N2o-S

N2o-C

Q1o-C

Q1o-S

M1o-S

M1o-C

J1o-S

J1o-C

MIo-S

MIo-C

L2o-C

L2o-S

OO1o-S

OO1o-C

M3o-S

M3o-C

M1+-S

M1+-C

L2+-S

L2+-C

N2+-C

N2+-S

MI+-C

MI+-S

S2+-S

S2+-C

O1+-S

O1+-C

Q1+-S

Q1+-C

J1+-S

J1+-C

K1+-S

K1+-C

M2+-C

M2+-S

M4+-S

M4+-C

OO1+-S

OO1+-C

M0+-C

M0o-C

M0o-S

M0+-S

0 10 20 30 40 500.11101001000104

Vector index

nm/s2

SingularValues RMS; Observatory: BFO; Date: 2011-11-01; Timespan: 365 days.

C: in-phase, S: quadrature signal; o: even (2), +: odd (3) harmonics20 / 40

2. Separation of degree 2 and 3 tidesResolution matrix: 38/52 harmonics

C: in-phase, S: quadrature signal; o: even (2), +: odd (3) harmonics 21 / 40

2. Separation of degree 2 and 3 tidesConclusion

I As expected, SVD indicates the ability to determine the rationbetween harmonics of degree 2 and 3.

I However, parameters of some counterparts, such as odd K1 orM2, may not be determined.

I The results are reasonable since harmonics of the largestamplitudes are quite often of even degree.

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Contents

Methods: synthetic harmonics

The tidal forward operator G is a set of harmonic signals with arbitraryfrequencies.

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Methods - synthetic harmonic studyIntrinsic wave grouping

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Methods - synthetic harmonic studyIntrinsic wave grouping

I We also study synthetic sinusoidal harmonics and theirintrinsic grouping behaviour.

I For that purpose, the Mathematica script is developed.

I It allows us to study the difference from uniformly distributed,equal harmonics, to the wave group that amplitudes are fromthe normal distribution.

Gaussian distribution is the first approach of the real harmonicdistribution.

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3. Resolution loss in central frequenciesDescription

The next few slides present SVD analysis with fixed numbers ofharmonics, fixed upper frequency and fixed length of time series.The sets differ by the lower frequency in each set, whatmodifies the separation between harmonics.

I Harmonics are distributed equidistantly (uniformly) within thespecified frequency range.

I All harmonics have the same amplitude.

I We present results for two different cut-off conditions.

We expect the smaller the frequency interval between harmonics,the larger the off-diagonal values of the resolution matrix. Thecentral frequencies are primarily affected.

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3. Resolution loss in central frequenciesDifferent frequency distribution

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3. Resolution loss in central frequenciesSmall threshold (SV ratio: 107)

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3. Resolution loss in central frequenciesLarge threshold (SV ratio: 103)

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3. Resolution loss in central frequenciesConclusion

Considering constant time series length, the modified Rayleighcriterion gives fixed separation condition (Munk & Hasselmann,1964, Godin, 1970).

I The smaller the frequency separation, the fewer harmonicsmeet the criterion.

I The corner frequencies become indistinguishable from onlyone close frequency

I The central frequencies cannot be discriminated between twoneighbouring harmonics.

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4. Intrinsic wave groupingDescription

The last few slides present two different kinds of (intrinsic) wavegrouping based on Singular Value Decomposition. Sets differ bythe amplitude distribution.

I Harmonics are equally distributed in the frequency domain.

I The left plots display harmonics with equal amplitudes, theright plots represent harmonics have amplitudes fromGaussian distribution.

I The first slide displays two tidal operators and harmonicfrequency distribution.

I Further slides present resolution matrices with correspondingsingular value spectra.

We expect that the questionable wave grouping apparent in theanalysis of harmonics from uniform distribution clarifies for morerealistic, normally distributed harmonics.

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4. Intrinsic wave groupingExample groups

Constant amplitude

1.25 1.30 1.35 1.40 1.45 1.500.0

0.5

1.0

1.5

2.0

1.25 1.30 1.35 1.40 1.45 1.50

0.0

0.5

1.0

1.5

2.0

Frequency (CpD)

Amplitude(a.u.)

Frequency distribution

Gaussian amplitude

1.25 1.30 1.35 1.40 1.45 1.500.0

0.2

0.4

0.6

0.8

1.0

1.25 1.30 1.35 1.40 1.45 1.50

0.0

0.2

0.4

0.6

0.8

1.0

Frequency (CpD)

Amplitude(a.u.)

Frequency distribution

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4. Intrinsic wave groupingOperator G (harmonic signal)

Constant amplitude Gaussian amplitude33 / 40

4. Intrinsic wave groupingStandard grouping

Constant amplitude Gaussian amplitude

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4. Intrinsic wave groupingStandard grouping: Conclusions

In Tidal Analysis we group harmonics with the most resolvable(largest) harmonics.

I Blue boxes indicate such groups on both resolution matrices.

I On the right-hand side, the neighbouring frequencies aroundcentral frequency are combined together in one group.

I On the left-hand side, they form two such groups.

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4. Intrinsic wave groupingResolution grouping

Constant amplitude Gaussian amplitude

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4. Intrinsic wave groupingResolution grouping: Conclusions

The new blue boxes indicate apparent groups from resolutionmatrices.

I The off-diagonal coefficients between harmonic 5 and 6 issmaller than between 4 and 5.

I Therefore model parameter obtained for harmonic 5 would bepartially harmonic 5 (0.29), harmonic 4 (0.28) and harmonic 6in (0.21).

I Smearing between harmonic 4 and 6 is negligible. Thus, thismight indicate another kind of wave grouping.

Similar indications we may notice on the second plot. However,

I They do not differ that much from the “standard” approachsince the other harmonics are resolvable too.

I Probably the results for more realistic values of harmonicswould be consistent with the common wave grouping.

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Summary

I 1. Atmospheric pressure may be properly distinguished fromother forcings, but some cross-talk is unclear

I 2. Model parameters of degree even and odd tides should beresolvable (in general)

I 3. There is apparent consistency with Rayleigh criterion

I 4. Resolution matrix may be a tool to investigate (or confirm)the wave grouping

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Future plans

I Detailed investigation of demonstrated problems

I Add penalty terms to the inversion problem

I Introduce model constraints (e.g. “credo of smoothness”)

I Allow for data residuals in inversion (e.g. mentionedair-pressure “radiation tides”)

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References

Ducarme, B.; 2009: Limitations of High Precision Tidal Prediction

Godin, G.; 1970: The resolution of tidal constituents

Meurers, B.; 2004:Investigation of temporal gravity variations in SG-records

Meurers, B., M. Van Camp, O. Francis, and V. Palinkas; 2016:Temporal variation of tidal parameters in superconducting gravimeter time-series

Munk & Hasselmann; 1964: Super-resolution of tides

Schroth, E., Forbriger, T. and Westerhaus, M.; 2018:A catalogue of gravimetric factor and phase variations for twelve wave groups

Tamura, Y.; 1987: A harmonic development of the tide-generatingpotential, BIM, 99, pp. 6813-6855.

Venedikov; 1961: Application a l’analyse harmonique desobservations des marees terrestres de la methode des moindrescarres, Comptes Rendus Acad. Bulgare des Sciences, 14

Wenzel, H.G.; 1996: The nanogal software: Earth tide dataprocessing package ETERNA 3.30, BIM, 124, pp. 9425-9439.

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