Application of Wavelets for Along Track Multi-Resolution Analysis of GOCE SGG Data

Rossen Grebenitcharsky (rossen.grebenitcharsky@newcastle.ac.uk), Philip Moore Civil Engineering and Geosciences, Newcastle University, Newcastle, United Kingdom

Background MOTIVATION FOR ALONG TRACK GOCE SGG PRE-PROCESSING

• GLOBAL GEOPOTENTIAL MODELING: lost valuable gravity information from 8 km resolution along track to 80 km resolution in GPMs (see Table 1)!!!!!

• REGIONAL GEOID MODELING: need to use all available data with as fine as possible resolution • HOW? By localized base functions – Wavelets with very good localization properties both in space/time

and frequency/scale domain allow simultaneous analysis of SGG data in different levels (different bandwidths – marked by magenta lines in Figure 2 ) of wavelet decomposition & reconstruction

• WHY? Gravity gradients – poor accuracy in long-wavelengths; Wxx, Wyy, Wzz, Wxz – very accurate in short wavelength; Wxy & Wyz 10 times less accurate in short wavelengths; Possible leakage and amplifications of errors of non-accurate components due to rotations to LNOF; Changes in orbital altitude (up to 20 km range) lead to differential gravitational effects and changes in frequency content of the signal - need to bring all data on same mean orbital altitude

• WHAT TO DO? To filter out (time-wise approach per day) large spatially correlated long-medium wavelength errors (see Figure 2 – green spectrum) data along orbit; to reduce the errors from the rotation to LNOF (see Figure 2 – blue spectrum); to perform IOS for up/downward continuation to mean orbit

• Existing alternative: only filtering in spectral domain using cut-off frequencies determined for Effective Measurement Bandwidth (EMB – yellow area in Figure 2) of SGG data. ARMA modeling applied to filter out GOCE data

MAIN TASK: To extract as much as possible useful gravity information from along track GOCE SGG data necessary for regional geoid modeling (Figure 1) • To test a NEW globally applied procedure along daily GOCE satellite tracks, taking advantage of

relatively equally distributed SGG data

• By application of wavelet (WL) multiresolution-analysis (MRA) for up/downward continuation (to mean orbit – 260 km) combined with Input Output Systems (IOS) (Sideris, 1996)

• By application of wavelet multiresolution-analysis for filtering out large spatially correlated long-medium wavelength errors;

• By suppressing/filtering/ (substituting by Global Geo-potential Model) SGG frequencies with very large energy with respect to a reference GPM – for every level of WL decomposition and reconstruction

INPUT-OUTPUT SYSTEMS (IOS) FOR UP/DOWNWARD CONTINUATION AND FILTERING

Experiment description and first results

FOLLOWING SIDERIS (1996) AFTER SOME MODIFICATIONS FOR THIS STUDY where SA – satellite altitude; MO – mean satellite orbit at 260 km

Figure 2: Power Spectral Density for one day gravity gradient data: SGG (Wzz); Rotated to LNOF SGG (Tzz); DIR2I GPM data (Tzz) at mean orbit; Up/Downwarded & filtered out SGG data at 260 km Sat. altitude (Tzz) by Wavelets

Introduction

• Application area: Global processing and later to be utilized for regional gravity field studies

Applied on: daily along-track GOCE SGG data (Tzz – component presented) • Time interval covered: JUNE 01, 2010 – JUNE 30, 2010 (one month data) • Data type and source: EGG_NOM_2 & EGG_TRF_2, GOCE HLPF, ESA/ESRIN • Data management and processing: GOCE HLPF documents but not limited to: GO-MA-HPF-GS-0110, Issue 4, rev. 3, 09/12/2010 GO-TN -HPF-GS-0192, Issue 2, rev. 7.2, 07/09/2011

• The spectrums (Figure 2) of Tzz from GOCE DIR2 I (red) and Tzz from NEW APPROACH (cyan) show: 1)In EMB (yellow area) for 70 % frequencies the new approach almost replicates the model, but 30 % (higher freq. part) have greater energy; 2) For frequencies lower than EMB (up to 12 WL level) the spectrum from new approach is the same as the model spectrum; 3) For very low frequencies (included in approximated WL level) the signal energy from new approach is less than the model spectrum; For freq. high than EMB the new approach provides much greater energy than the model – more high frequency information is expected in Tzz from new approach

Conclusions • The first results indicate very promising performance of the NEW APPROACH in successful extracting higher frequency information for Tzz component of SGG w.r.t. Global Model (DIR2 I) and EGG_TRF_2 data. In addition, it is necessary to :

1) utilize other SGG components for complete GOCE data set; 2) validate with independent regional data sets; 3) improve the filtering for every WL level (utilizing the crossovers information); 4) reduce edge effects from application of WL and IOS

Acknowledgements: This study was funded by an NCEO award from the UK NERC. The authors thank Prof. Roland Pail from Technical University Munich , Germany and Dr. Johanes Bouman and Dr. Martin Fuchs from DGFI, Germany for supplying one day GOCE SGG data to validate the rotation procedure from GRF to LNOF also for their valuable advices regarding the rotation procedure intended to be used. The authors would like to thank Dr. Mehdi Eshagh from Royal Institute of Technology, Stockholm, Sweeden for providing public access to the software for generating gravity field gradients described in Mehdi and Abdollahzadeh (2012). References: Sideris, M.G. (1996), On the use of heterogeneous noisy data in spectral gravity field modelling methods, Journal of Geodesy 70: pp. 470-479, Springer-Verlag, 1996

Mehdi, E. and M. Abdollahzadeh (2012), Software for generating gravity gradients using a geopotential model on an irregular semivectorization algorithm, Computers & Geosciences 39: pp. 152-160, Elsevier, 2012

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FILTERING

IOS

WAVELETS

Table 1: Spherical harmonic degrees, frequencies and resolution limits for WL levels of decomposition

To suppress jth maximum frequency

𝐹𝐹𝑇𝑗 (𝑇𝑧𝑧 ,𝑜𝑏𝑠𝑆𝐴 = 𝐹𝐹𝑇𝑗 (𝑇𝑧𝑧 ,𝑚𝑜𝑑𝑒𝑙

𝑆𝐴

𝑃𝑆𝐷𝑗(𝑇𝑧𝑧𝑛𝑛 ) = 0

𝐹𝐹𝑇(𝑇𝑧𝑧𝑀𝑂) =

𝑃𝑆𝐷(𝑇𝑧𝑧 ,𝑚𝑜𝑑𝑒𝑙𝑀𝑂,𝑆𝐴 )

𝑃𝑆𝐷(𝑇𝑧𝑧 ,𝑚𝑜𝑑𝑒𝑙𝑆𝐴 ,𝑆𝐴 ) + 𝑃𝑆𝐷(𝑇𝑧𝑧

𝑛 ,𝑛) 𝐹𝐹𝑇(𝑇𝑧𝑧 ,𝑜𝑏𝑠

𝑆𝐴 )

𝑃𝑆𝐷(𝑇𝑧𝑧𝑛𝑛 ) = 𝐹𝐹𝑇 (𝑇𝑧𝑧 ,𝑜𝑏𝑠

𝑆𝐴 ) − (𝑇𝑧𝑧 ,𝑚𝑜𝑑𝑒𝑙𝑆𝐴 ) .∗ 𝐹𝐹𝑇 (𝑇𝑧𝑧 ,𝑜𝑏𝑠

𝑆𝐴 )− (𝑇𝑧𝑧 ,𝑚𝑜𝑑𝑒𝑙𝑆𝐴 )

𝐶𝑂𝑁𝐽

Figure 1: MAIN TASK DESCRIPTION

• Reference Geo-potential Model: GOCE DIR 2I, up to degree 240 • Model SGG data: generated according to Mehdi and Abdollahzadeh (2012) using the matlab codes provided by authors • Type of model SGG data: generated at satellite altitude and at mean orbital altitude of 260 km based on DIR 2I • Type of wavelets used: DB 10 (1D Daubechies wavelets) • Number of levels for signal decomposition and reconstruction: 12 • Frequency content limits for WL levels: Level 1 – highest frequency (resolution 8-16 km); Level 12 – lowest frequency (16000-32000km)

Figure 5. Tzz differences between EGG_TRF_2 & DIR2I model at sat. altitude Tzz differences between New Approach & DIR2I at 260 km altitude

DATE MIN MAX MEAN STD 01-06-2010 -0.310 0.234 0.001 0.106 02-06-2010

-0.623 0.441 0.003 0.167

Table 2. Tzz differences’ statistics at crossovers

•New approach can extract more high frequencies in SGG w.r.t. GPM, than EGG_TRF_2 (Figure 5) • Crossover differences are close to zero with standard deviation ~ 0.130 Eotvos (Figure 6, Table 2)

Figure 3. Tzz component of SGG at 260 km mean orbit – NEW APPROACH, units: [Eotvos]

Figure 4. Tzz component of SGG at 260 km mean orbit – GOCO DIR2 I, units: [Eotvos]

• Figure 3 and Figure 4 confirm that the NEW APPROACH extracts much more higher frequency information from SGG than the model especially visible in mountainous areas like Alpo-Himalaya mountain range, Alaska, Andes or areas with very large gravity variations like Indonesia and Japan Islands, Caribbean islands • Additionaly detected short wavelength information for SGG signal needs to be validated by independent data sets on land and at sea for different regions

Figure 6. Tzz differences at crossovers at 260 km