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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Climate studies require long data records extending the lifetime of a single remote sensing satellite mission. Precise satellite altimetry exploring global and regional evolution of the sea level has now completed a two decade data record. A consistent long-term data record has to be constructed from a sequence of different, partly overlapping altimeter systems which have to be carefully cross-calibrated. This cross-calibration is realized globally by adjusting an extremely large set of single- and dual-satellite crossover differences performed between all contemporaneous altimeter systems. The total set of crossover differences creates a highly redundant network and enables a robust estimate of radial errors with a dense and rather complete sampling for all altimeter systems analyzed. An iterative variance component estimation is applied to obtain an objective relative weighting between altimeter systems with different performance. The final time series of radial errors is taken to estimate (for each of the altimeter systems) an empirical auto-covariance function. Moreover, the radial errors capture relative range biases and indicate systematic variations in the geo-centering of altimeter satellite orbits. The procedure has the potential to estimate for all altimeter systems the geographically correlated mean errors which is not at all visible in single-satellite crossover differences but maps directly to estimates of the mean sea surface.

Electronic path delays, oscillator drifts, time tagging errors, antenna phase centre uncertainties, orbit errors, errors in geophysical correction models, improperly calibrated auxiliary sensors and even software conventions may cause systematic errors of the altimeter range (the distance from the satellite to the instantaneous sea level), the most important parameter to monitor the global and regional sea level evolution. On average these errors include a constant range bias, a drift term or even geographically correlated error pattern all of them mapping directly to the sea surface heights. This underpins the importance of a careful calibration of satellite based altimeter systems. A calibration is an indispensable prerequisite to construct a long-term data record to investigate sea level rise and to study regional sea level variability.

The range bias (and eventually, the drift term) can be estimated by a calibration scenario in which the altimeter derived sea surface heights are compared (over a sufficient long period of time) with geocentric sea level heights derived independently, e.g., by GPS-linked tide gauges or GPS equipped buoys. In the last two decades several permanent sites have been set up providing

For TOPEX (Ocean TOPography EXperiment), an absolute calibration procedure was first realized at the Harvest Platform. Up to now, the approach has been applied systematically for every 10-day repeat cycle for the full lifetime of the NASA/CNES (National Aeronautics and Space Administration/Centre National d’Etudes Spatiales) missions TOPEX/Poseidon and the follow-on missions Jason-1 and Jason-2 [

Not all missions (or sequence of missions with identical orbit configuration) have been calibrated with a comparable rigour. Initially, ERS-1 (1st Earth Remote Sensing satellite) was calibrated only by a few overflights of an oil platform near Venice [

ERS-2 had a few months of overlap operation with ERS-1, allowing the observation of identical sea surface height profiles with a delay of one day. Such formation flight configurations were also realized for the transition from ERS-2 to ENVISAT (30 min ahead of ERS-2), from TOPEX to Jason-1 and from Jason-1 to Jason-2 (with a delay of only 60 s) lasting each several months. With decreasing time difference, the measurement conditions of a formation flight become similar, such that sea level variability and sea state (wave heights) conditions cancel each other efficiently. These repeat pass analyses allow the most direct comparison of sea surface heights and provide rather small error estimates. On the other hand the relative calibration by repeat pass analyses is critical, as undetected drifts of the primary mission are projected to the follow-on mission.

However, also, the absolute

GPS equipped buoys provide another tool to directly measure the instantaneous sea surface height in the geocentric terrestrial reference system. This was e.g., applied for ERS-1 in the North Sea [

A comparison with global tide gauge networks have been proven to identify possible drifts and bias instabilities of altimeter systems over time [

The approach presented here is a

At this point, it should be already emphasized that all (residual) errors of correction models for sea state, radiometer, tides

The present paper is structured as follows: The comprehensive multi-mission altimeter data base used for the cross-calibration is described first, along with the efforts to gradually upgrade and harmonize the data. Section 3 explains in detail the cross-calibration methodology in terms of discrete modelling of radial errors, the treatment of multi-mission scenarios and the numerics applied to solve the least squares adjustment of a rather huge systems. Section 4 presents selected details of the extensive results of a multi-mission cross calibration for all altimeter systems operating the past two decades in order to indicate the specific potential of the approach presented here. The final conclusions summarizes the particular findings of the multi-mission cross-calibration.

Since the launch of TOPEX/Poseidon in 1992 at least two altimetry missions provide highly accurate contemporaneous measurements of the sea surface. In the last two decades data of 12 different altimeters were acquired, in some time periods, up to six missions were operated simultaneously.

The approach presented here not only rely on the so-called core missions (TOPEX and Jason-1/2) flying continuously on a fixed orbit with a repeat cycle of about 10 days. It also includes the ESA missions ERS1/2 and ENVISAT providing a denser ground track pattern with a 35 days repeat cycle and GFO (Geosat Follow-On) with 17 days repeat. In addition, this investigation also uses missions with very long repeat cycles, such as ERS-1 geodetic mission (GM) phases, Cryosat-2 LRM (Low Rate Mode) from ESA Baseline B and Jason-1 GM (completing a full geodetic mission phase just before its end-of-life). Even satellites with drifting orbits (ENVISAT extended mission (EM)) as well as the episodic laser altimeter measurements provided by ICESat (Ice, Cloud, and land Elevation Satellite) (Release 33) [

In order to reach the best possible consistency between the different altimetry missions re-processed orbits are used for most of the satellites. Whenever available they are based on the latest International Terrestrial Reference Frame (ITRF2008) and use the most recent gravity fields, correction models, and computation methodology.

In the following section we describe the discrete modelling of radial errors, an approach derived from the basic ideas of Cloutier [

The multi-mission crossover analysis aims to estimate by a least squares approach the radial errors for all altimeter systems operating simultaneously. Traditionally, radial errors in crossover adjustments are modelled by linear regressions, periodic functions, piecewise polynomials, or splines. In the present approach, no such analytical error model for the radial component is applied, because such models already imply more or less restrictive assumptions about the error characteristics. Fourier series, for example, pre-define frequencies that are supposed to explain a dominant periodicity of the error. Other possible frequencies or a-periodic errors, however, remain undetected. Instead of an analytic error model radial error components, _{i}_{j}_{i}_{j}_{i}_{j}_{lim}_{ij}

For the analysis, the observed crossover differences are simply modelled by the difference of two radial error components, _{i}_{j}_{ij}_{ij}_{x}_{ij}_{1}_{2}_{2}_{n}_{i}_{x}_{ij}_{ij}

By the first factor, the unit weight variance,
_{Δ} of the crossover differences. The second factor controls by the half weight width, _{x}_{ij}_{i}_{j}

The linear system, in _{m}_{m}_{i,i}_{+1})) and individual weights:
_{i,i}_{+1} = _{i}_{i}_{+1} of the times associated to two consecutive errors, _{i}_{i}_{+1}. The constant, _{m}_{m}_{m}

_{i}_{i}_{i}_{+1}, of a nearly simultaneous crossover event belong to

If Δ_{i}

Now, consecutive differences between radial error components of _{m}_{m}_{m}_{D}

Applying now the least squares minimization to both, the crossover differences, _{i}

The minimization

The Lagrange function:
^{−}^{1}. According to [

Finally, the least squares estimate of the radial errors are given by:
_{x}Δ

As will be discussed in the following section, the size of the system may become huge. Fortunately, the normals have a peculiar structure, suggesting to solve the system by an iterative solver.

The first term on the right hand side of _{x}X_{m}W_{m}D_{m}

Fixing a single error component maintains the sparsity of matrix

On the other hand, fixing a single error component is arbitrary and undesirable. Repeated absolute calibrations of altimeters (as performed over the absolute calibration sites) provide consolidated mean values for the range bias. In order to adapt the estimation

The discrete crossover analysis described so far is easily applied to any local or regional area of interest. In principle, only the crossover differences, Δ_{ij}_{i}_{j}

If the analysis is applied globally and the altimeter satellites to be included becomes two or more, then the number of single- and dual-satellite crossover events increases dramatically. In the last two decades, there were configurations with up to six altimeter systems operating simultaneously (_{lim}

In order to limit the numerical requirements, the global multi-mission crossover analysis is segmented and performed, for example, for successive 10-day periods with a two-day overlap to neighbouring periods. For example, for Jason-2, the number of crossovers per 10-day analysis period varied between some 20,000 (three missions) up to more than 150,000 (five missions) (see [

Comparing the radial error estimates within the overlapping period it was found that the radial errors were consistent to within 1–2 mm. This justifies to simply skipping the results in the overlapping periods and concatenate the radial errors of the central 10-day periods. By concatenating all 10-day analysis periods it is possible to construct a time series of radial errors covering the full life time of an altimeter mission.

Any multi-mission scenario implies in general that altimeter systems with different precision are to be cross-calibrated. It is therefore essential to obtain an objective relative weighting between all mission included in the common analysis. Therefore, the most appropriate tool is a variance component estimation (VCE) as described, e.g., in [

Partial redundancies can then be computed for every observation group by:
_{x}_{m}_{x}_{m}

The most expensive part in the VCE is the computation of the matrices in

Once, the radial error estimates,

After the least squares adjustment and the variance component estimate, a discrete, irregular spaced time series of radial errors are available for every altimeter mission included in the Cross-Calibration. Let _{m}_{1}_{2}_{n}_{i}_{i}_{i}_{i}/n_{rr}

A mean range bias, Δ

Alternatively, the radial errors may be approximated by a spherical harmonic series up to a degree of 2.

The harmonic coefficients up to a degree of 1 are equivalent to the mean radial error, _{00} ≡ Δ_{11} ≡ Δ_{11} ≡ Δ_{10} ≡ Δ_{20} indicates an error pattern affecting the flattening of the Earth figure. Finally, _{21} and _{21} are related to the orientation of the Earth rotation axis. This post-processing can be applied to the full time series of radial errors or just to those subsets of radial errors corresponding to the segmentation described in Section 3.6. The latter case allows us to monitor the temporal variation of parameters describing the mean range bias, the centre-of-origin shifts and _{2}_{x}

Based on a first order analytic solution of a satellite motion, Rosborough [

We generalize this property recognizing that any common component in the errors of ascending and descending passes will not cancel each other, but will directly map into the sea surface heights. It is therefore of general interest to quantify those common components, irrespectively of the cause of these error components. A simple bookkeeping about which radial error belongs to an ascending or descending ground track allows us to separate the radial errors into its mean and variable component. For every mission, the radial errors of ascending and descending tracks are averaged independent of each other, e.g., on a 2.5°

This section shows the results of a recent global multi-mission crossover analysis, version 14 (named MMXO14) using data of all missions available since the launch of TOPEX/Poseidon in 1992. Before this date, the processing is not possible, due to a lack of at least two simultaneous missions. Within this time period of more than 20 years, data of 12 different altimeter systems are available. Some of them were separated into different missions in order to account for orbit changes. The amount of contemporaneous missions varies between two (before 1995) and six (2004/2005),

The relative weighting between the data of the different missions is achieved by means of a VCE (see Section 3.7). The estimated variance components of the different missions are plotted in _{m}_{x}

The ratio of variance components (_{m}/σ_{x}

GFO gets the highest variances (lowest weights), especially in the last mission phase, with missing radiometer corrections and many data gaps; Jason-2, the most accurate mission, behaves more than two times better than GFO. In principle, missions with single-frequency altimeters (inevitably relying on ionospheric model corrections) show higher variance components than those with dual-frequency sensors. The level of SARAL is relatively high as this data set lacks, at the very beginning of the mission, a consolidated SSB corrections and an optimal calibration of the radiometer. In addition to the mean level, periods with higher variance components for all missions can be identified (e.g., years 2000 to 2002, due to the higher solar activity), as well as single peaks, indicating cycles with higher values. Most of them can be accounted for by special atmospheric conditions (solar storms), missing observations or measurement problems (including orbit maneuvers), and/or problems with applied corrections.

In contrast to the variance components, _{m}_{x}_{x}_{m}_{lim}_{max}_{Δ} _{0} = 0.01m (

In order to solve the rank defect, discussed in Section 3.5, one of the missions has to serve as reference mission by applying the constraint

In the following subsections, selected results of MMXO14 will be presented. The purpose is not to give a comprehensive or complete overview of all cross-calibration results for all missions. Instead, the focus will be on the most interesting results in order to demonstrate the potential and the application spectrum of the multi-mission cross-calibration approach.

The main result of the cross-calibration approach are time series of radial errors for all missions included in the analysis. These radial errors incorporate not only effects from the altimeter sensor (e.g., instrumental delays or drifts due to ageing effects), but also errors induced by the orbit determination process, by inaccurate geophysical corrections, or effects caused by sea state or variations of the sea level within the time period between the two crossover measurements.

The basic objective of the radial error analysis is to correct the altimeter ranges, allowing to consistently combine the data of all missions and deriving multi-mission products. The merging of altimeter data from different missions not only improves the spatio-temporal resolution, but also helps to construct time series, extending the lifetime of a single mission. In addition, the time series can be used to derive stochastic characterizations of the different missions by auto-covariance functions and power spectra (see Section 3.8). Moreover, the radial errors are the basis for further derivation of post-processing products, such as mean range biases or geographically correlated errors.

The first row in _{rr}

Range biases for each mission are computed for every 10-day period. For most of the missions, the time series of range biases over the whole mission lifetime shows little scatter and no systematic effects. Thus, the computation of one global mean range bias seems acceptable. These global mean range biases have been compiled in

Analyzing the time series of range bias clearly reveals systematic effects for some missions (we show only the most outstanding results of MMXO14). The ENVISAT mission, for example, exhibits a significant offset of about 3 cm between the range biases of Side A and Side B. This happens mid-2006, when the measurements were temporarily taken from the redundant system. For ICESat, operating only in short episodic laser periods, the estimated range biases of individual periods vary significantly. This explains the large standard deviation in

Typically, the range bias does not change in case of an orbit event,

The different noise levels of the Jason-1 range biases is due to the change of the reference mission (see Section 3.6). Between TOPEX cycle 366 and 603 (August 2002 to January 2009), Jason-1 serves as reference mission, causing a much smoother time series. After the end of the primary Jason-1 mission, Jason-2 is used as reference.

The last part of the Jason-1 EM range biases seems to include a moderate negative trend, which is not significant, due to the relative high scatter of the data within this time period. The source of both effects (higher scatter after cycle 655 and possible drift) is not yet understood. Unfortunately, the time series can not be extended to allow for a further inspection.

Investigating the radial errors separately for ascending and descending errors allows us to identify possible geographically correlated errors (GCEs) (Section 3.8). These errors are important for mapping and monitoring the sea surface, as the GCE cancel in single-satellite crossover differences, but map directly in the sea surface height. The GCE mainly reflects errors coming from the orbit determination process, but may also include other effects, as, for example, errors in the time tagging of the observations or effects of corrections applied to the data (SSB, wet troposphere,

In the past decade, the orbit quality has been gradually improved by new gravity field models from the dedicated mission, GRACE (Gravity Recovery and Climate Experiment) and GOCE (Gravity Field and Steady State Ocean Circulation Explorer), and consequenly, the GCE could be also reduced. Today, MMXO14 reveals RMS-values of GCE between three and 4 mm for most of the missions. An example how the GCE could be improved is shown in

The upper plot shows GCE based on the most recent SLCCI VER6 orbit [

Some dedicated pattern in GCE can already indicate systematic shifts of the centre-of-origin realization of the satellite orbits. For example, a clear north-south tilt in GCE illustrates a shift in the

For recent altimeter data sets and recent orbit solutions most of the centre-of-origin shift does not differ significantly from zero. However, some effects of the shift become visible if the gravity field models used for precise orbit determination do not properly account for temporal variations of gravity. This effect is mainly reflected in the

For most missions the degree 2 coefficients of the spherical harmonic series (_{21} and _{21}) do not differ significantly from the TOPEX realization, the _{20} term (corresponding to the flattening) shows a clear offset of 6.5 mm, yielding to a latitude dependent pattern in GCE (see _{20}-term is plotted in _{11} and _{11}.

The multi-mission cross-calibration of satellite altimeters is a powerful analysis tool to derive consistent multi-mission data, allowing one to merge data from different missions in order to improve the space-time sampling of sea surface heights and to construct a long-term data record (extending the lifetime of single missions) for global and regional sea level change studies. The modelling of radial errors is based on a rather simple concept, the discrete crossover analysis adjusting both crossover and consecutive differences. The system has to be regularized by a single constraint applied to a reference mission. In a multi-mission scenario, the normal equation becomes rather huge, but remains sparse and can be solved iteratively by a conjugate gradient algorithm. A variance component estimate ensures an objective relative weighting of missions with different precision. The redundancy of the adjustment grows with the number of missions, operating simultaneously. Already, with two contemporaneous missions, the strength of the analysis is sufficient to obtain a robust estimate of the radial errors. Once the time series of radial errors has been estimated, a post processing allows to characterize their stochastic properties by auto-covariance functions and to identify systematic error pattern in terms of centre-of-origin shifts or geographically correlated errors.

The present cross-calibration approach has been applied many times and is going to be updated whenever the data holdings have been extended, improved by new data releases, better correction models or reprocessed orbits. The selected results shown in the present paper demonstrate that robust relative range biases are obtained for the full lifetime of (nearly all) altimeter systems with completely different orbit configuration and sampling characteristics. The error analysis is able to identify systematic errors, alteration in the processing chain and inconsistent geocentring of the orbits. The present analysis also identified data problems of rather new missions (as HY-2A), but also weaknesses and unresolved issues of past missions (such as GFO or Jason-1).

A final remark to the GCE pattern and other systematic effects: In general, an improved orbit solution is taken to identfy the GCE pattern of the

This work was funded by the Deutsche Forschungsgemeinschaft (DFG), Bonn, Germany, under the grants, BO1228/5-3 and BO1228/6-3.

We would like to thank ESA, CNES and NASA, in particular the National Oceanic and Atmospheric Administration (NOAA) and the National Snow and Ice Data Center (NSIDC) for providing the altimeter mission and auxilary data, members of ESAs SLCCI team for providing new orbits for TOPEX and ERS-1/2, the Goddard Space Flight Centre for reprocessed GFO orbits and AVISO/CLS for the provision of the DAC correction. We would like to thank in particular the China National Space Administration (CNSA) and the National Satellite Ocean Application Service (NSOAS) for providing the HY-2A data.

Finally, we gratefully acknowledge the constructive comments of four anonymous reviewers.

The cross-calibration methodology has been developed and coded by Wolfgang Bosch. He also wrote the main part of the paper. Denise Dettmering performed the extensive computations of MMXO14 and described and discussed its results (chapter 4). Moreover, she gradually merged new orbits to the data holdings and wrote chapter 2. Christian Schwatke maintained and gradually updated the altimeter data base and the tools required to harmonize and upgrade the altimeter data.

The authors declare no conflict of interest.

Altimeter mission history of the last two decades.

A subset of MMXO14 results for three different missions: Jason-2 (in blue), SARAL (red), and ERS-2 (green). The top row illustrate two-day subsets of radial errors (m). The middle row shows the empirical auto-covariance functions, _{rr}

Estimated range bias for HY-2A.

Estimated range bias for Jason-1 (GDR-C data with GDR-D orbit).

Geographically correlated errors for ERS-2 (gridded to 2.5° cells) based on two different orbits: On top, the GCE for a recent SLCCI orbit is shown (with RMS =

Relative differences in the realization of the origin (

Geographically correlated errors for GFO. Map has no uniform scale.

Temporal evolution of coefficient _{20} for GFO.

Atmospheric corrections used in the version 14 processing standard of the multi-mission cross-calibration (MMXO14).

TOPEX | smoothed dual-freq. | TMR ^{a} |
model (ECMWF ^{b} |

Poseidon | NIC09 | TMR; enhanced product [ |
model (ECMWF) |

Jason-1 | smoothed dual-freq. | JMR ^{c} |
model (ECMWF) |

Jason-2 | smoothed dual-freq. | AMR ^{d} |
model (ECMWF) |

ERS-1 | NIC09 | MWR ^{e} |
model (ECMWF) |

ERS-2 | NIC09 (till cycle 35) | MWR; with CLS drift corr. [ |
model (ECMWF) |

GIM (from cycle 36) | |||

ENVISAT | smoothed dual-freq. (till cycle 64) | MWR | model (ECMWF) |

GIM + 8 mm (from cycle 65) | |||

GFO | GIM | MWR | model (NCEP) |

NCEP ^{f} |
|||

ICESat | none | NCEP | model (NCEP) |

Cryosat-2 | GIM (from GDR) | ECMWF | model (Meteo France) |

HY-2A | smoothed dual-freq. | MWR | model from IGDR |

SARAL | GIM (from GDR) | ECMWF | model (ECMWF) |

TOPEX Microwave Radiometer;

European Centre for Medium Weather Forecast;

Jason Microwave Radiometer;

Jason-2 Advanced Microwave Radiometer;

MicroWave Radiometer;

National Centers for Environmental Prediction.

Satellite orbits used in MMXO14.

TOPEX/Poseidon | SLCCI ^{a} |
ITRF2008 | DORIS ^{b}^{c} |
external, [ |

Jason-1 | GDR-D | ITRF2008 | DORIS/SLR/GPS | external, IDS [ |

Jason-1 GM | GDR-D | ITRF2008 | DORIS/SLR | internal L2 product |

Jason-2 | GDR-D | ITRF2008 | DORIS/SLR/GPS | internal L2 product |

ERS-1/2 | SLCCI VER6 | ITRF2008 | SLR/PRARE/XO ^{d} |
external, [ |

ENVISAT | GDR-D | ITRF2008 | DORIS/SLR | external, ESA [ |

GFO | GSFC std0905 ^{e} |
ITRF2005 | Doppler/SLR/XO | external, [ |

ICESat | GLA06, R33 ^{f} |
ITRF2005 | GPS | internal L1b ^{g} |

Cryosat-2 | GDR-D, Baseline B | ITRF2008 | DORIS/SLR | internal L2 product |

HY-2A | IGDR-D | ITRF2008 | DORIS/SLR/GPS | internal L2 product |

SARAL | GDR-D | ITRF2008 | DORIS/SLR | internal L2 product |

Sea Level Climate Change Initiative (European Space Agency (ESA));

Détermination d’Orbite et Radiopositionnement Intégrés par Satellite;

Satellite Laser Ranging;

Cross-over;

Goddard Space Flight Centre, version std0905;

Geoscience Laser data, Release 33;

Level 1b.

Global mean range bias in millimeters w.r.t. TOPEX for all missions included in MMXO14 and the corresponding absolute range bias estimates from

TOPEX-A (primary mission) | −0.2 ± 1.2 | 212 | 0 ± 8 | 8 ± 3 | 1.0 ± 1.8 |

TOPEX-B (primary mission) | −0.6 ± 1.5 | 122 | 0 ± 4 | 11 ± 4 | 10.1 ± 2.7 |

TOPEX (extended mission) | −0.0 ± 2.5 | 110 | – | – | – |

Poseidon | −1.1 ± 7.2 | 44 | −12 ± 10 | −2 ± 6 | – |

Jason-1 GDR-C (primary) | 97.3 ± 1.3 | 256 | 77 ± 3 | 96 ± 2 | 115.6 ± 2.1 |

Jason-1 GDR-C (extended) | 97.2 ± 2.6 | 112 | – | – | – |

Jason-1 GDR-C (geodetic) | 103.1 ± 1.7 | 33 | – | – | – |

Jason-2 GDR-D | −4.7 ± 1.0 | 174 | −4 ± 3 | 22 ± 2 | 2.0 ± 2.5 |

ERS-1 | 442.7 ± 8.3 | 133 | – | – | – |

ERS-2 | 71.2 ± 6.9 | 290 | −60 ± 18 | – | – |

ENVISAT (primary mission) | 450.8 ± 7.9 | 305 | 447 ± 7 | – | – |

ENVISAT (extended mission) | 441.2 ± 2.6 | 53 | – | – | – |

GFO | 21.0 ± 6.5 | 302 | – | – | – |

ICESat | −49.6 ± 23.7 | 68 | – | – | – |

HY-2A | 262.1 ± 170.1 | 52 | – | – | – |

Cryosat-2 (ESA Baseline B) | −244.0 ± 3.0 | 84 | – | – | – |

SARAL GDR-T | −67.5 ± 1.7 | 13 | −65 ± 5 | −58 ± 11 | – |

Estimated coefficients of a spherical harmonic series computed for radial errors of GFO in millimeters. Mean values for the whole mission lifetime.

_{00} |
24.3 ± 6.6 |

_{10} |
4.9 ± 5.9 |

_{11} |
−0.3 ± 2.4 |

_{11} |
−0.8 ± 3.6 |

_{20} |
−6.5 ± 2.9 |

_{21} |
0.1 ± 1.3 |

_{21} |
0.7 ± 1.5 |