IGiK–TVGMF: A MATLAB® GUIs package for computing, analysing and modelling temporal variations of gravity/mass functionals from

GRACE satellite based global geopotential modelsWalyeldeen Godah

Institute of Geodesy and Cartography, Centre of Geodesy and Geodynamics, 27 Jacka Kaczmarskiego St., 02-679 Warsaw, Poland

e-mail: walyeldeen.godah@igik.edu.pl

Abstract The determination, analysis and modelling of temporal variations of gravity/mass functionals (TVGMFs) are one of the most important tasks within the Earth-science disciplines. This contribution describes a MATLAB package, named IGiK–TVGMF (Instytut Geodezji i Kartografii–TVGMF), that allows computing, analysing and modelling TVGMF from Global Geopotential Models (GGMs) developed on the basis of a monthly interval GRACE (Gravity Recovery And Climate Experiment) satellite gravity mission data.The IGiK–TVGMF comprises of three graphical user interfaces (GUIs) developed with the use of a MATLAB R2017a App Designer. The first GUI, named the TVGMF–Computation is the main interface, in which thirteen TVGMFs can be determined at an individual point or on a grid of points, using GRACE-based GGMs from seven computation centres. Different parameters and filters can be specified in this GUI. The second and third GUI, named the TVGMF–Analysis(PCA/EOF) and the TVGMF–Analysis(SA), respectively, were developed to analyse and model the TVGMF using the Principle Component Analysis/ Empirical Orthogonal Function (PCA/EOF) and the Seasonal Adjustment (SA) methods. Numerical results and metadata obtained from the IGiK–TVGMF can be saved as text files by the user. The results obtained by the IGiK–TVGMF were validated using the corresponding ones obtained by the GRAVESOFT package and the International Centre for Global Earth Models (ICGEM) calculation service. The validation results were discussed.The IGiK–TVGMF, including Matlab functions and input datasets, is available as an open source at: https://github.com/Walyeldeen/Walyeldeen-Godah. The IGiK–TVGMF can easily be updated and modified by including new GGMs developed with the use of GRACE/GRACE Follow-On satellite missions.Keywords. GGM, GRACE, gravity/mass functionals, MATLAB, PCA/EOF, seasonal adjustment

GRACE-based GGMs The latest release, i.e. release five (RL05), of monthly GRACE-based GGMs from the GFZ (GeoForschungs Zentrum), the CSR (Centre for Space Research) and the JPL (Jet Propulsion Laboratory), the ITG (Institut für Geodäsie und Geoinformation of Bonn University), the AIUB (Astronomical Institute of Bern University), the HUST (Huazhong University of Science and Technology) and the Tongji (Tongji University) are used.These GGMs were obtained from the ICGEM ( ). http://icgem.gfz-potsdam.de/ICGEM/ICGEM.html

Second d/o spherical harmonic coefficients The monthly second d/o spherical harmonic coefficient C , that represents changes in the Earth's dynamic oblateness, is estimated from 20

GRACE, Ocean Bottom Pressure (GRACE-OBP) data with/without restoring the Glacial Isostatic Adjustment (GIA) signal ( ) and Sun et al., 2016satellite laser ranging (SLR) data ( ). Cheng et al., 2013

Reference GGM The EGM2008 (Earth Gravitational Model 2008), the EIGEN-6C4 (European Improved Gravity model of the Earth by New techniques), and the GECO (GOCE and EGM2008 COmbined model) were used as reference models.

Load Love numbers The load Love numbers are calculated using the Preliminary Reference Earth Model (PREM). The numerical values of load Love numbers were obtained from ( ).Wang et al. 2012

Computation of TVGMFs The TVGMFs were computed using the spherical harmonic analysis procedure (cf. ; ; Wahr et al., 1998 Torge and Müller, 2012 Barthelmes, 2013 Zhang et al., 2017; ).

Analysis and model TVGMFS Two methods, namely the seasonal adjustment (SA) method ( ), also called seasonal decomposition method (e.g. Findley et al., 1998Makridakis et al., 1998 Jolliffe, 2002), and the Principal Component Analysis/Empirical Orthogonal Function (PCA/EOF) method (e.g. ), were implemented in the IGiK–TVGMF to analyse and model the TVGMF.

The IGiK–TVGMF comprises of three graphical user interfaces (GUIs) developed with the use of a MATLAB R2017a App Designer (cf. ). https://www.mathworks.com/products/matlab/app-designer.html

AcknowledgementsThe research was conducted in the framework of the European Observation System-Poland (EPOS-PL), co-financed by the European Union from the European Regional Development Fund. The GRACE-based GGMs included in the IGiK–TVGMF are acquired from the ICGEM. The author is thankful to Prof. J. Krynski and Dr. M. Szelachowska from the IGiK for their fruitful discussions and suggestions to develop the IGiK–TVGMF.

ReferencesBarthelmes, F., 2013. Definition of functionals of the geopotential and their calculation from spherical harmonic models. . STR09/02.http://icgem.gfz-potsdam.de

Cheng, M.K., Tapley, B.D., Ries, J.C., 2013. Deceleration in the Earth's oblateness. Journal of Geophysical Research 118, 1–8.

Findley, D.F., Monsell, B.C., Bell, W.R., Otto, M.C., Chen, B.-C., 1998. New Capabilities and Methods of the X-12-ARIMA Seasonal-Adjustment Program. Journal of Business &

Economic Statistics 16(2), 127–152.

Godah et al., 2016. Analysis of RL05 GRACE-based and GOCE/GRACE-based GGMs using gravity measurements at Borowa Gora Geodetic-Geophysical Observatory, ESA special

Publication SP-740.

Godah, W., Szelachowska, M., Krynski, J., 2017a. .https://link.springer.com/article/10.1007/s11600-017-0064-3

Godah, W., Szelachowska, M., Krynski, J., 2017b. https://doi.org/10.1007/1345_2017_15.

Godah, W., Szelachowska, M., Krynski, J., 2018. .https://link.springer.com/article/10.1007/s40328-017-0206-8

Jolliffe, I., 2002. Principal component analysis. Springer-Verlag, New York.rdMakridakis, S., Wheelwright, S.C., Hyndman, R.J., 1998. Forecasting: methods and applications, 3 edn. Wiley, New York. ISBN 978-0-471-53233-0.

Sun, Y., Riva, R., Ditmar, P., 2016. .https://doi.org/10.1002/2016JB013073thTorge, W., Müller, J., 2012. Geodesy. 4 edition, Walter de Gruyter, Berlin-Boston.

Wahr, J., Molenaar, M., Bryan, F., 1998. .https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98JB02844

Wang et al., 2012. .https://doi.org/10.1016/j.cageo.2012.06.022

Zhang., X., Jin., S., Lu., X., 2017. .https://doi.org/10.3390/rs9101000

I G i K

The determination and the analysis of temporal mass variations within the Earth system are one of the main scientific objectives in the Earth science-related disciplines.

The Gravity Recovery and Climate Experiment (GRACE) satellite mission launched in March 2002 with its pioneering observation technology (satellite to satellite tracking in a low-low mode) provides valuable data for the determination of temporal variations of gravity/mass functionals (TVGMFs). The GRACE satellite mission was operated for three times longer than its initial planned duration. In October 2017, GRACE satellites ran out of fuel, and the mission was terminated. However, it emphasizes the need for the GRACE

ndFollow-On (GRACE-FO) satellite mission, which has been launched on 22 May 2018 (see ), for long-term https://gracefo.jpl.nasa.gov/sustainable information on TVGMFs.

In order to determine and analyse TVGMFs using GGMs from GRACE mission data, an appropriate computational tool is essentially needed. However, main limitations of currently available softwares and interactive online tools developed for the computation of static GMFs and TVGMFs can be ascribed as follows:

− The currently available softwares were mainly developed to determine static gravity functionals. Thus, in order to compute TVGMFs using these software, users may need to conduct a pre-processing or modify GRACE-based GGMs. Moreover, these softwares do not allow users to determine temporal variations of mass functionals from GRACE-based GGMs. Furthermore, for some of these softwares, users may need to determine TVGMFs from GRACE-based GGMs month by month, which would be very time and manpower consuming.

− The currently available softwares were mainly developed to determine static gravity functionals. Thus, in order to compute TVGMFs using these software, users may need to conduct a pre-processing or modify GRACE-based GGMs. Moreover, these softwares do not allow users to determine temporal variations of mass functionals from GRACE-based GGMs. Furthermore, for some of these softwares, users may need to determine TVGMFs from GRACE-based GGMs month by month, which would be very time and manpower consuming.

− These softwares and interactive online tools do not allow users analyse and model TVGMFs.

The GRACE/GRACE-FO Science Team Meeting 2018, The Helmholtz Centre Potsdam – GFZ German Research Centre for Geosciences, Potsdam, Germany, 9–11 October 2018

Examples of the validation of TVGMFs determined with use of the IGiK–TVGMF

The IGiK–TVGMF: A MATLAB GUIs package

Data used and methodsIntroduction

Ÿ The differences between temporal variations ofgeoid heights obtained from

(a) IGiK–TVGMF and GRAVSOFT,

(b) IGiK–TVGMF and ICGEM, (c) GRAVSOFT and ICGEM

Min Max Mean Std.

IGiK–TVGMF vs. GRAVSOFT –0.19 0.14 ‒0.02 0.06

IGiK–TVGMF vs. ICGEM –0.91 0.88 0.00 0.14

GRAVSOFT vs. ICGEM –0.84 0.92 0.02 0.15

TVGMF Computation

centres Degree-2

(C20) Ref.

Model Ref.

System Max. d/o

Filter

Geoid heights GFZ Release 05 SLR EIGEN-6C4 WGS84 60 DDK3

Equivalent water thickness

GFZ Release 05 SLR EIGEN-6C4 WGS84 60 DDK2

Gravity disturbance GFZ Release 05 GRACE EGM2008 GRS80 90 DDK1

Gravity anomalies GFZ Release 05 GRACE EGM2008 GRS80 90 DDK1

Deflections of the vertical (east–west component)

GFZ Release 05 GRACE GECO GRS80 90 DDK1

Gravity gradients (the 2nd vertical derivative)

GFZ Release 05 GRACE GECO GRS80 90 DDK1

Deflections of the vertical (north–south component)

GFZ Release 05 GRACE GECO GRS80 90 DDK1

1 Statistics of differences between temporal variations of geoid heights from the , the IGiK–TVGMFGRAFSOFT and the ICGEM [mm]

A comparison between TVGMFs from the IGiK‒TVGMF, GRAVSOFT and ICGEM CS (ICGEM Calculation Service) has been conducted.

Ÿ As an example, geoid heights for March 2005 and September 2005 are determined at a global grid of 3°×3° spatial resolution using GFZ GRACE-based GGMs truncated at d/o 60, the DDK3 filter, C from 20

GRACE data, GRS80 and these three softwares.

Ÿ The results obtained emphasize that temporal variations of geoid heihts from IGiK–TVGMF are conformable to the corresponding ones from the GRAVSOFT and the ICGEM.

Ÿ The main reason for these stripes might be ascribed to the fact that in the , a filter called is ICGEM gentlecutapplied to smooth long wavelength components, i.e. up to d/o 8, of the Earth gravity field.

Statistics of differences between TVGMFs from the IGiK‒TVGMF and the corresponding ones from the GRAVSOFT or ICGEM

Ÿ A very good agreement between the and IGiK–TVGMF

the . ICGEM

Seven time series of TVGMFs from the IGiK–TVGMF were validated with the corresponding ones from the GRAVSOFT package and the ICGEM .

The validation of TVGMFs – in the time domainThe validation of TVGMFs – in the space domain

Parameters applied in the TVGMF–Computation to determine and to validate TVGMFs

Ÿ The parameters utilized to validate temporal variations of gravity gradients and temporal variations of deflections of the vertical were arbitrary chosen. For the remaining TVGMFs, the selection of these parameters was based on previous investigations conducted in ( ; Godah et al. 2016 2017a; 2017b 2018; ).

Ÿ Standard deviations of the differences are at the level of sub-millimetre for the geoid height and sub-centimetre for the equivalent water thickness.

Ÿ Differences between the corresponding ∆g , ∆δg, ∆T sa rr

and ∆η from the and the are IGiK–TVGMF GRAVSOFT within or beneath numerical values that can be

–4provided by the GRAVSOFT, i.e. 1×10 or 4–decimals.

The TVGMF–Computation is a main GUI, in which, therteen TVGMFs can be computed from seven different computation centres.Different parameters, e.g. the Degree-2 (C ), the reference model, the reference system and the 20

maximum d/o, can be specified in this GUI.

1 The TVGMF–Analysis(SA) is a GUI developed for the analysis and model TVGMFs using the seasonal adjustment (SA) method.In this GUI, the seasonal, long-term and unmodelled components of the TVGMF analysed as well as are modelled TVGMF can be obtained.

2 The TVGMF–Analysis(PCA/EOF) is a GUI developed for the analysis and model TVGMFs using the seasonal adjustment (PCA/EOF) method.In this GUI, PCAs, EOFs and total variances of the TVGMF analysed as well as modelled TVGMF values can be obtained.

3

Ÿ The IGiK–TVGMF is a new MATLAB package developed for the determination and the analysis of TVGMFs using GRACE-based GGMs. It handles all available monthly GRACE-based GGMs from seven computational centres.

Ÿ The IGiK–TVGMF can easily be updated by including new GGMs developed from GRACE/GRACE-FO mission data.

Ÿ A very good agreement between TVGMFs determined from the IGiK-TVGMF and the corresponding ones from the GRAVSOFT and the ICGEM.

Ÿ Overall, the IGiK–TVGMF overcomes some of the limitations of currently available softwares and interactive online tools developed for the computation and the analysis of TVGMFs.

Ÿ The IGiK–TVGMF is available as an open source that can be redistributed and/or modified under the terms of the GNU Library General Public License 3. Currently, the IGiK–TVGMF, including MATLAB functions and input datasets, is available at: .https://github.com/Walyeldeen/Walyeldeen-Godah

Conclusions and future developments

-5

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IGiK–TVGMFICGEM G Browser

3–

∆N

[m

m]

time [year]2002 2004 2006 2008 2010 20162012 2014

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

∆E

WT

[m

]

time [year]2002 2004 2006 2008 2010 20162012 2014

IGiK–TVGMFICGEM G Browser 3–

-6

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∆δg

[]

μGal

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Δg

[]

saμGal

-4

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∆ [

Eö

tvö

s]T

rr

10-4

time [year]2008 2009 2010 2011 2012 20152013 2014 2016

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0

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∆η

[arc

sec]

10-4

-2.5

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0.0

0.5

1.0

1.5

2.010-3

∆ [ξa

rcse

c]

IGiK–TVGMF

GRAVSOFT

IGiK–TVGMF

GRAVSOFT

time [year]2008 2009 2010 2011 2012 20152013 2014 2016

time [year]2008 2009 2010 2011 2012 20152013 2014 2016

time [year]2008 2009 2010 2011 2012 20152013 2014 2016

time [year]2008 2009 2010 2011 2012 20152013 2014 2016

IGiK–TVGMF

GRAVSOFT

IGiK–TVGMF

GRAVSOFT

IGiK–TVGMF

GRAVSOFT

φ = 53.5° and λ = 16.5° φ = 52.47° and λ = 21.03°

φ = –2.163° and λ = 55.126°φ = 52.47° and λ = 21.03°

φ = –2.163° and λ = 55.126°

φ = 52.47° and λ = 21.03°

φ = 52.47° and λ = 21.03°

TVGMF Unit Min Max Mean Std.

IGiK–TVGMF vs. ICGEM ∆N m –0.0005 0.0007 –5×10–18 0.0002

∆EWT m –0.0172 0.0162 –8×10–17 0.0070

IGiK–TVGMF vs. GRAVSOFT

∆δg mGal –1×10–04 1×10–04 –7×10–17 6×10‒05

∆gsa mGal –7×10–05 7×10–05 –4×10–16 3×10‒05

∆η arcsec –9×10–05 1×10–04 –3×10–16 4×10‒05

∆ξ arcsec –9×10–04 7×10–04 –5×10–17 3×10‒04

∆Trr Eötvös –5×10–05 4×10–05 –1×10–16 3×10‒05

1

-80

-60

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latit

ud

e [

de

gre

e]

(a)

-80

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latit

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de

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(b)

-150 -100 -50 0 50 100 150

-80

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0.0 0.2 0.4 0.6 0.8[mm]

-0.2-0.4-0.6-0.8

longitude [degree]

latit

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e [

de

gre

e]

( )c

Taking into the consideration these limitations, the main objective and motivation of this contribution are to develop a novel MATLAB package, named the IGiK–TVGMF, for the determination, analysis and modelling of TVGMFs. This MATLAB package allows users to compute and analyse thirteen types of TVGMFs at a single point or a grid of points using GRACE-based GGMs.