Original paper: Aydin C and Demirel H, (2007), Effect of estimated variance components for different gravity meters on analysis of gravity changes, Festschrift zum 65. Geburtstag von Prof. Dr.-Ing. Carl-Erhard Gerstenecker, Technische Universitaet Darmstadt, December 2007, pp.1-11, Darmstadt.
Effect of estimated variance components for different gravity meters on analysis of gravity changes Cüneyt Aydin, Hüseyin Demirel Yildiz Technical University, Department of Geodesy and Photogrammetry Engineering, 34349 Besiktas-Istanbul, Turkey, e-mails:
[email protected] ,
[email protected] Tel.: +90-212-2597070; Fax: +90-212-2596762
Abstract. In this study, a Gauss-Markoff model with unknown variance components belonging to each gravity meter used for measuring of a gravity network is considered. The Marmara Gravity Network is used for the numerical examples. The periods of the network are adjusted separately for the two cases; (1) the gravity meters’ observations are weighted equally, (2) weighted according to the estimated components with BIQUE method. Then, the annual gravity changes referred to the statistically verified points using geometrical deformation analysis are compared for each case. As a result of the comparison, it is shown that the changes vary each other significantly.
Key words: Gravity changes−Variance component estimation−BIQUE−Marmara Region
1. Introduction The observed−temporal but non-tidal−gravity changes are important constraints for modeling deformations occured due to an earthquake activity, movements in a volcanic region, post-glacial rebounds (Berrino 2000; Demirel and Gerstenecker 1990; Ergintav et al. 2007; Kuo et al. 1999; Mäkinen 2000) etc. The most practical way to monitor the changes in time and space domain is setting a gravity network and measure the points of the network with different relative spring gravity meters simultaneously in definite periods. The approximately calibrated and reduced reading of a meter, say c(t), can be written as a function of gravity (g) and some instrumental functions c(t)=g−N−f(c)+D(t)
(1)
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Original paper: Aydin C and Demirel H, (2007), Effect of estimated variance components for different gravity meters on analysis of gravity changes, Festschrift zum 65. Geburtstag von Prof. Dr.-Ing. Carl-Erhard Gerstenecker, Technische Universitaet Darmstadt, December 2007, pp.1-11, Darmstadt.
where N is the gravity level, f(c) is the calibration correction function, and D(t) is the drift function of the meter (see, e.g. Torge 1989). Because of the g, the observations of the meters at the same point are related with each other and, hence, this leads to consider those heterogeneous data under the same functional model. For weighting of the observations, the previous experiences on the errors are commonly used. However, the data sets of the meters may follow different distributions and have statistical discrepancies which have to be balanced in the stochastic part of the Gauss-Markoff model. It is well known that improper weighting of the observations in least-squares adjustment may cause incorrect estimation of the unknown parameters and error assessment. If we use the results of such an adjustment in testing of some hypotheses, especially in deformation analysis, the statistical decisions can be misleading (Crocetto 2000; Fotopoulos 2005). Thus, it is advisable to determine the weights using the quality of both observations and design of the experiment (Hsu 1999). The best way for doing this is the estimation variance/covariance components, which is indispensable in attaining the accuracy of “reallife” measurements (Xu et al. 2007). There are numerous studies on the estimation of the components both in geodesy and statistics, starting with Helmert (1924) and Rao (1971). The applied methods are classified in Crocetto (2000). Some of the estimation approaches are minimum norm of quadratic unbiased estimation (MINQUE), almost unbiased estimation (AUE) and best invariant quadratic unbiased estimation (BIQUE) (Crocetto 2000; Hsu 1999; Rao 1971). Considering all, we take the different gravity meters’ observations in a gravity network as independent groups whose variance components are unknowns to be estimated through BIQUE method. Then, the weighting of the observations in a period is performed using the estimated components. According to the results of the applications to Marmara Gravity Network (MAGRANET) which is measured as 5 periods between 2003-2005, it is shown that such a weighting affects the statistical decisions in deformation analysis.
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Original paper: Aydin C and Demirel H, (2007), Effect of estimated variance components for different gravity meters on analysis of gravity changes, Festschrift zum 65. Geburtstag von Prof. Dr.-Ing. Carl-Erhard Gerstenecker, Technische Universitaet Darmstadt, December 2007, pp.1-11, Darmstadt.
2. Adjustment of gravity observations With g=go+δ(g), N=No+δ(N), f(c)=δ(λ)c(t) and l=c(t)−(go−No), the observation equation for a reading of a meter is written in a simple form using Eq. (1) as follows E(l)=l+e= δ(g)−δ(N)−c(t) δ(λ)+D(t)
(2)
where ( )o and δ(...) denote approximate values and unknowns, respectively (δ(λ) is the calibration correction), l is the dimilished observation, e is the error of the observation and E( ) represents the expectation. For all observations in the network, the Gauss-Markoff model holds E(l)= l+e=Ax=[ Ag Am ] [ xg xm ]T with E(e)=0 , Σ= σ 02 P−1
(3)
where l is the n×1 vector of observations, e is the n×1 vector of errors, A is the n×u matrix of known coefficients, x is the u×1 vector of unknowns, Σ is the n×n variance-covariance (VC) matrix, P is the n×n weight matrix of the observations, σ 02 is the variance of unit weight and the terms with (…)g and (…)m correspond to the gravity and intrumental ~ 2, unknowns, respectively. For the uncorrelated observations with the variances σ 1
~ 2 , the weight matrix is set ~ 2 ,…, σ σ n 2 ~ 2 , σ2 / σ ~ 2 ,..., σ 2 / σ ~ 2) . P=diag( σ 02 / σ 1 0 2 0 n
(4)
Let us assume that one meter’s calibration coefficient is known (for relative calibration, see, e.g. Atzbacher and Gerstenecker 1993) and all points’ gravity values are unknowns. Using the orthogonal projection R = A m ( A Tm PA m ) −1 A m P , the instrumental unknowns can be eliminated from the linear model (3). To do this, the observation vector l and the matrix ~ ~ ~ A are transformed with (In−R) into l = (In−R) l and A = (In−R) [ A g A m ] =[ A g 0 ] , where In is the n×n unit matrix. Then, the estimation of ug gravity unknowns becomes similar to the estimation procedure in a leveling network. For a trace-minimum solution, the constraint equation hTxg=0 is incorporated into the adjustment where h is a vector fulfilling ~ A g h=0, hence hT=( 1 / u g )[ 1 1 1,...,1 ] (see, e.g. Koch 1999, p.190). 3
Original paper: Aydin C and Demirel H, (2007), Effect of estimated variance components for different gravity meters on analysis of gravity changes, Festschrift zum 65. Geburtstag von Prof. Dr.-Ing. Carl-Erhard Gerstenecker, Technische Universitaet Darmstadt, December 2007, pp.1-11, Darmstadt.
Accordingly, we get the trace-minimum solution for the gravity unknowns as ~ ~ xˆ g = Q g A Tg P l with its cofactor matrix Q g , ~ ~ Q g = ( A Tg P A g + hh T ) −1 − hh T .
(5)
The estimator σˆ 02 of the variance σ 02 of unit weight then follows with σˆ 02 =(1/f) ~ ~ ~ ~ ( l − A g xˆ g ) T P ( l − A g xˆ g ) where f is the degrees of freedom. 3. Estimation of variance components A Gauss-Markoff model with k unknown variance components σ12 , σ 22 ,…, σ 2k is given as follows (Koch 1999, p.226; Rao 1971) E(l)=Ax
with Σ= σ12 V1+ σ 22 V2+…+ σ 2k Vk
(6)
where Vj (j=1, 2, ..., k) is the given non-zero symmetrical informative matrix of the jth group of observations. 2
If we define the jth unknown variance component σ j as a function of a known variance
(σ 2j ) 0 and an unknown factor βj, 2
σ j = β j (σ 2j ) 0 ,
(7)
the VC matrix in the model (6) is rewritten as Σ=β1 T1+β2T2+…+βk Tk
with Tj= (σ 2j ) 0 Vj (j=1, 2, ..., k) .
(8)
Hence, the model (6) is being transformed to the one with k unknown variance factors βj (j=1, 2,..., k) (Lucas and Dillinger 1998). According to the best quadratic invariant unbiased estimation (BIQUE) method (Koch 1999, p. 225-233; Crocetto 2000), the vector of variance factors β is estimated as βˆ = K −1q
(9) 4
Original paper: Aydin C and Demirel H, (2007), Effect of estimated variance components for different gravity meters on analysis of gravity changes, Festschrift zum 65. Geburtstag von Prof. Dr.-Ing. Carl-Erhard Gerstenecker, Technische Universitaet Darmstadt, December 2007, pp.1-11, Darmstadt.
where K = (tr( WTi WT j )) , q = (qi ) = ( l T WTi Wl ) , i, j ∈ {1, 2,..., k}, and W is the symmetrical matrix computed with the approximate VC matrix Σ 0 = T1 + T2 + ... + Tk as W = Σ o−1 − Σ o−1 A ( A T Σ o−1 A ) − A T Σ o−1 .
(10)
It is expected that the estimated variance factors are equal to “1”. However, because of the approximate variances (σ 2j ) 0 , they are not. Therefore, the estimated variances (σˆ 2j )1 = (βˆ j )1 (σ 2j ) 0 from Eq. 7 are taken as the new approximate variance components and the estimation procedure given in Eq (8)−(10) is repeated. Such an iterative estimation is stopped as soon as all factors converge to “1” in iteration w and we get the estimated variance components as σˆ 2j = (σˆ 2j ) w = (βˆ j ) w (σˆ 2j ) w−1 .
(11)
With Eq (11) and (6), the estimated VC matrix Σˆ is obtained by Σˆ = σˆ 12 V1+ σˆ 22 V2+…+ σˆ 2k Vk .
(12)
The iterated estimation procedure is independent from the selection of the starting values
(σ 2j ) 0 (Koch 1999, s. 233). Furthermore, if we use the weight matrix P= σ 02 Σˆ −1 according to the estimated VC matrix in the least-squares adjustment, the variance of unit weight does not change after the adjustment, i.e. σˆ 02 = σ 02 holds. (see, e.g. Aydin (2007)). An another expression for this equality is given in Hsu (1999) for iterated-AUE. 4. Geometrical deformation analysis Let gˆ I and gˆ II be the estimated gravity values of the initial and present periods in a gravity network with ug points, respectively. With their cofactor matrices Q g, I and Q g, II from Eq. (5), the difference vector dˆ and its weight matrix Pd are obtained by
dˆ = gˆ II − gˆ I , Pd =( Q g, I + Q g, II )+
(13)
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Original paper: Aydin C and Demirel H, (2007), Effect of estimated variance components for different gravity meters on analysis of gravity changes, Festschrift zum 65. Geburtstag von Prof. Dr.-Ing. Carl-Erhard Gerstenecker, Technische Universitaet Darmstadt, December 2007, pp.1-11, Darmstadt.
where ( )+ denotes the pseudo-inverse. In order to learn if there is any significant change globally, the null hypothesis H0: E( dˆ )=0 is tested against its alternative HA: E( dˆ )≠0. If the null hypothesis is true, the test statistic T follows (central) F-distribution, T=( dˆ T Pd dˆ ) /( s 2 h) ∼F(h, fd)
(14)
where s2 is the pooled variance (=(fI σˆ 02,I + fII σˆ 02,II )/ fd), h is the rank of Pd and fd=fI+ fII. If the T is bigger than the upper percentage point of the F-distribution Fh , f d , 1−α , then it is concluded that the difference vector has some significant gravity changes with the confidence level of (1−α). In order to find the responsible points for the global change, the next step is the localization of the differences (Cooper 1989; 334-338). For one dimensional network, the simplified localization method given in Aydin (2001) can be used: First, the difference vector dˆ is transformed with its weight matrix to an another form, δ= Pd dˆ . Then the individual contribution of the point i to the quadratic form
dˆ T Pd dˆ is computed as follows ν i = δ i2 / Pd,i
,
(i=1, 2,…, ug)
(15)
where Pd,i is the ith diagonal element of the Pd . The point for which the maximum ν is computed is accepted as the deformed point and then extracted from the datum definition with the S-transformation. For the other points which define the new datum, the global test and the localization procedure are repeated until the test gives no any significance change in the network. 5. Numerical example Around the Marmara Sea in Turkey, a network with 25 gravity points was established in 2003 (Fig.1). The network named as MAGRANET (Marmara Gravity Network) was measured with two relative gravity meters (LaCoste&Romberg model G-85 and G-858) in June and October between 2003-2005 as 5 periods (Table 1).
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Original paper: Aydin C and Demirel H, (2007), Effect of estimated variance components for different gravity meters on analysis of gravity changes, Festschrift zum 65. Geburtstag von Prof. Dr.-Ing. Carl-Erhard Gerstenecker, Technische Universitaet Darmstadt, December 2007, pp.1-11, Darmstadt.
Table 1 Numbers of lines and observations in the periods of MAGRANET Period No.
Number of lines 6 7 7 7 7
Period Operators
1 2 3 4 5
October 2003 June 2004 October 2004 June 2005 October 2005
Gerstenecker-Tiede Gers.-Aydin-Tiede Gers.-Steineck Gers.-Steineck-Aydin Steineck-Aydin
YACT
KANT
YILD
41.0
KRDM
Latitude (° )
IBBT
OVCT KAMT
AVCT
MER1
HART
SILE
SELP
BGNT
Number of observations 391 499 444 436 491
YIGI TUBI
Marmara Sea
DOKU
CINA
KDER
OLUK
KVAK DUMT
40.5
KUTE
FIST
SEVK
ERCT
26.5
27
27.5
28
28.5
29
29.5
30
30.5
31
Longitude (° )
Fig. 1. The distribution of the points of MAGRANET 5.1 Adjustment of MAGRANET The periods were processed with a Matlab program GrAd (Aydin 2007). In the preprocessing stage, the readings were calibrated according to the meters’ calibration tables and the necessary reductions (earth tide, atmospheric pressure and height) were made. Then, the median method was applied to the readings in a session in order to search for any possible gross-errors. About 2% of the observations of a period were cleaned with the method in the pre-processing stage. Before network adjustment, first, the network’s lines consisting the points measured in each day were adjusted separately for the aims of i) the detection of any possible jumps and ii) the definition of the degree of the most proper drift polynomial model. The both meters’ observations in a corresponding line were adjusted
~ 2 =1/100. The jump detection was implemented together using the equal weight of P= σ 02 / σ using the robust method with Denmark weighting and about 7% of the observations of a period were detected as they had jumps. 7
Original paper: Aydin C and Demirel H, (2007), Effect of estimated variance components for different gravity meters on analysis of gravity changes, Festschrift zum 65. Geburtstag von Prof. Dr.-Ing. Carl-Erhard Gerstenecker, Technische Universitaet Darmstadt, December 2007, pp.1-11, Darmstadt.
The cleaned observations of the lines were combined successfully in the network
~ 2 ) I =(1/100)I and the adjustment using the model (3) with the weight matrix P=( σ 02 / σ n n calibration of G-858 relative to G-85. The square roots of the estimated variances of unit weight ( σˆ 0 ) and the mean standard deviations of the gravity unknowns (msd) of traceminimum solutions are given in Table 2.
Table 2. Error estimates of the periods October 2003
June 2004
October 2004
June 2005
October 2005
σˆ 0 (µGal)
1.06
0.95
1.16
1.04
0.94
msd (µGal)
5.49
3.83
4.22
3.85
3.33
Period
5.2 Estimation of variance components For estimating the variance components of the meter’s observations, the VC matrix in the model (6) was taken as Σ= σ12 V1+ σ 22 V2 ( σ12 and σ 22 correspond to G-85 and G-858, respectively) with ⎡I V1 = ⎢ n 1 ⎣ 0
⎡0 0⎤ V = , ⎢0 2 0⎥⎦ ⎣
0 ⎤ I n 2 ⎥⎦
where n1 and n2 are the numbers of the observations of the meters in a period. Using the BIQUE method in Sect.3, the variance components of the meters were iteratively estimated for each period (Table 3). As can be seen from the table, the variance components, i.e. the variances of the observations here in this example, differ from 100 µGal2 assumed in the previous adjustment procedures.
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Original paper: Aydin C and Demirel H, (2007), Effect of estimated variance components for different gravity meters on analysis of gravity changes, Festschrift zum 65. Geburtstag von Prof. Dr.-Ing. Carl-Erhard Gerstenecker, Technische Universitaet Darmstadt, December 2007, pp.1-11, Darmstadt.
Table 3. Estimated variance components of the meters in the periods October 2003
June 2004
October 2004
June 2005
October 2005
σˆ 12 (µGal2)
91.36
60.26
153.15
70.03
52.15
σˆ 22 (µGal2)
136.42
121.00
119.33
148.08
133.32
Period
5.3 Deformation analysis The question we mainly pose is: Do the statistical decisions in a deformation analysis change if we use the results of the adjustments with the weight matrices in the form of
~ 2 ) I ? For this purpose, first, the periods were re-adjusted P= σ 02 Σˆ −1 instead of P=( σ 02 / σ n using the new weight matrices produced according to the estimated variances in Table 3. Comparing the previous adjustment results, the followings have been observed; i) the gravity values differ irregulary between the magnitudes of −7 and 11 µGal, ii) the standard deviations of the gravity values are almost same (just decrease of 10% ), iii) the cofactor matrices of the gravity unknowns are different, iv) as might be expected, σˆ 02 = σ 02 (=1 µGal). Then, for the analysis of the changes between October 2005-October 2004 and June 2005June 2004 (October 2003 period was not considered for the analysis in this study), the deformation analysis procedure in Sect.4 was applied using the results of the adjustment with equal weights (Case I) and the new adjustment (Case II) separately. For both cases, the global test showed that some significant changes occured between the periods with the confidence level of 95%. However, the localization procedures gave different results, especially in comparison of June 2005-June 2004. Table 4 summarizes the obtained gravity changes for the cases after S-transformation in which the reference points define the current datum of the network (bold values belong to the detected object points). From the comparison of Case I and II, it is seen obviously that the statistical decisions in deformation analysis are influenced by using the results of the adjustment with stochastic model established according to the estimated variance components. 9
Original paper: Aydin C and Demirel H, (2007), Effect of estimated variance components for different gravity meters on analysis of gravity changes, Festschrift zum 65. Geburtstag von Prof. Dr.-Ing. Carl-Erhard Gerstenecker, Technische Universitaet Darmstadt, December 2007, pp.1-11, Darmstadt.
6. Conclusion Estimation of variance/covariance components in linear models are essential if we deal with the combination of heterogeneous data or use heteroscedastic observations. This study considers the different gravity meters’ observations as independent groups whose variance components are unknowns. For the estimation of the components, the BIQUE method is used. According to the numerical examples in Marmara Gravity Network (MAGRANET), it is shown that two gravity meters’ stochastic structures should be represented by different variance components and the adjustment results can change significantly if we use incomplete weighting of the observations. Accordingly, the importance of proper stochastic modelling for the gravity observations in deformation analysis is highlighted.
Table 4 The gravity changes after localization procedures Oct. 2005-Oct. 2004
Point TUBI OLUK CINA FIST ERCT KUTE DUMT OVCT KRDM SILE YILD AVCT SELP MER1 BGNT DOKU KVAK SEVK KAMT IBBT HART YACT KANT YIGI KDER
June 2005-June 2004
Case I
Case II
Case I
Case II
9.84 16.53 −14.36 −4.30 15.29 29.33 8.88 −0.42 −15.09 −2.97 −5.18 −4.16 −6.38 −17.10 −1.26 −34.23 −10.38 −10.48 10.77 32.82 −2.49 25.78 −15.63 31.63 23.85
13.84 12.42 −10.90 −5.58 17.65 31.75 9.49 1.99 −9.32 1.49 −6.95 −5.76 −11.72 −19.78 4.31 −34.65 −11.75 −6.51 3.83 31.48 −4.19 25.34 −17.96 32.19 24.44
1.47 18.60 −1.72 −17.03 −7.59 9.78 37.34 −15.31 −16.75 −38.20 9.78 −33.36 23.61 −25.29 −5.04 −4.45 −2.76 −7.70 −8.84 14.67 −1.20 −16.45 18.95 30.51 3.34
12.39 30.01 9.13 −5.83 3.11 22.79 50.35 −3.90 −5.65 −21.21 23.36 −20.38 33.74 −12.02 9.83 7.95 7.48 6.03 −3.13 18.92 9.27 −5.63 35.02 38.61 10.95
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Original paper: Aydin C and Demirel H, (2007), Effect of estimated variance components for different gravity meters on analysis of gravity changes, Festschrift zum 65. Geburtstag von Prof. Dr.-Ing. Carl-Erhard Gerstenecker, Technische Universitaet Darmstadt, December 2007, pp.1-11, Darmstadt.
Acknowledgements. The authors are thankful to Prof. Dr.-Ing. Carl Gerstenecker for his remarkable collaboration and helps on improvement of gravimetric works in YTU. References Atzbacher K, Gerstenecker C (1993) Secular gravity variations: Recent crustal movements or scale factor changes? Journal of Geodynamics, 18: 107-121. Aydin C (2001) Investigation on reliability of static models in analysis of vertical crustal deformations, MSc Thesis (in Turkish), Yildiz Technical University, Science Institute, Istanbul. Aydin C (2007) Model design for monitoring gravity changes in Marmara Region, PhD Thesis (in Turkish), Yildiz Technical University, Science Institute, Istanbul. Berrino G (2000) Combined gravimetry in the observation of volcanic processes in Southern Italy, Journal of Geodynamics, Vol. 30: 371-388. Cooper MAR (1987) Control surveys in civil engineering, Collins, London. Crocetto N, Gatti M, Russo P (2000) Simplified formulae for the BIQUE estimation of variance components in disjunctive observation groups, Journal of Geodesy, 74: 447-457. Demirel H, Gerstenecker C (1990), Secular gravity variations along the North Anatolian Fault, In: R. Rummel & R.G. Hipkin (Eds.), Gravity, Gradiometry and Gravimetry, Symposium No. 103, Edinburgh, Scotland, Aug. 8-10, 163-171, New York. Ergintav S, Dogan U, Gerstenecker C, Cakmak R, Belgen A, Demirel H, Aydin C, Reilinger R (2007) A snapshot (2003–2005) of the 3D postseismic deformation for the 1999, Mw=7.4 İzmit earthquake in the Marmara Region, Turkey, by first results of joint gravity and GPS monitoring, Journal of Geodynamics, 44/1-2: 1-18. Fotopoulos G (2005) Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data, Journal of Geodesy, 79: 111-123. Helmert FR (1924) Die Ausgleichungsrechnung nach der Methode der kleinsten Quadrate, 3. AUFL., Leipzig/Berlin. Hsu R (1999) An alternative expression for the variance factors using iterated almost unbiased estimation, Journal of Geodesy, 73:173-179. Koch KR (1999) Parameter estimation and hypothesis testing in linear models, Springer-Verlag, Berlin. Kuo JT, Zheng JH, Song SH, Liu KR (1999) Determination of earthquake epicentroids by inversion of gravity variation data in the BTTZ Region, China, Tectonophysics, 312: 267-281.
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Original paper: Aydin C and Demirel H, (2007), Effect of estimated variance components for different gravity meters on analysis of gravity changes, Festschrift zum 65. Geburtstag von Prof. Dr.-Ing. Carl-Erhard Gerstenecker, Technische Universitaet Darmstadt, December 2007, pp.1-11, Darmstadt.
Lucas R, Dillinger WH (1998) MINQUE for block diagonal bordered systems such as those encountered in VLBI data analysis, Journal of Geodesy, 72: 343-349. Mäkinen J (2000) Geodynamical studies using gravimetry and levelling, Academic Dissertation in Geophysics, the Faculty of Science of the University Helsinki, Helsinki. Rao CR (1971) Minimum variance quadratic unbiased estimation of variance components, Journal of Multivariate Analysis, 1: 445-456. Torge W (1989) Gravimetry, Walter de Gruyter, Berlin. Xu P, Liu Y, Shen Y, Fukuda Y (2007) Estimability analysis of variance and covariance components, Journal of Geodesy, 81: 593-602.
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