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Since its launch in the year 2002, the space satellite mission GRACE provides spherical harmonic coefficients, which can be used to observe the time-variable part of the Earth’s gravity field. It was initially assumed that the derived gravitational quantities from these coefficients are of high accuracy and would thus deliver reliable large scale m...
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Terrestrial water storage (TWS) in high mountain areas contributes large runoff volumes to nearby lowlands during the low-flow season when streamflow is critical to downstream water supplies. The potential for TWS from GRACE (Gravity Recovery and Climate Experiment) satellites to provide long-lead streamflow forecasting in adjacent lowlands during...
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... The resulting differences in modeled variables thus reveal uncertainty ranges of present-day reanalysis models (see Table 2.1 for further details of these datasets). 1979-present 1979-present 1979-2009 6h, daily, monthly 1h, 6h, daily, monthly 1h, 6h, monthly Assimilation scheme 4D-Var 3D-Var 3D-Var ...
In the discussion about the global climate change, one central topic are the projected changes in the water cycle. It is predicted that there will be an increase of extreme hydrometeorological events like heavy precipitation or droughts. It is obvious that this intensification of the hydrological cycle will have a significant impact on the society. Such predictions require reliable and consistent datasets for the major hydrological variables precipitation, evapotranspiration, runoff, and water storage changes. Today, there are various data sources for each of these variables. While some of these datasets are still based on in situ measurements, there are alternative data sources available, which are often derived from satellite-based measurements. The advantages of these observations are obviously the homogeneous spatial and temporal resolutions on the global scale. Besides such satellite-based products, state-of-the-art hydrological and hydrometeorological models and reanalyses also provide consistent long-term estimates for the major hydrological variables. In order to evaluate the past, present, and future state of the climate system, it is mandatory that there is both temporal and spatial consistency between these data sources. Otherwise, the mismatch between the different water cycle variables cause imbalances in the empirical evaluation of the hydrological cycle, which, in the end, hinder the analysis of extreme events or variations on climatic time scales. It is thus of major importance to investigate the strengths and weaknesses of the data sources for precipitation, evapotranspiration, runoff, and water storage changes, but also the level of consistency between different water cycle variables. This doctoral thesis, which comprises of four articles, shall therefore serve as a comprehensive overview over the current status of our knowledge about and our data basis for the large-scale water cycle. Therefore, various data sources for the four major water cycle variables are compared and evaluated on different temporal and spatial scales. These sources comprise gridded observations (GPCC, GPCP, CRU, DEL, CPC), atmospheric reanalysis models (ERA-Interim, MERRA, CFSR), partially model-based datasets (GLEAM, MOD16, FLUXNET MTE), land surface models (GLDAS, MERRA Land), satellite-derived water storage changes from GRACE, and in situ runoff observations from GRDC. The study reveals serious shortcomings in the empirical evaluation of the large-scale water cycle. On the global scale, significant differences can be identified when comparing the model estimates from the three reanalyses against gridded observations of precipitation and temperature. However, differences with similar magnitudes can also be observed between the applied observation-based datasets. A catchment-scale analysis over 96 catchments of different sizes and climatic conditions worldwide confirms that these differences occur on both the global- and the catchment-scale. In the context of the gridded precipitation observations, this can be (at least partly) explained with a significant decrease of rain gauges worldwide. Looking at the spatial distribution of the gauges reveals large data gaps e.g. in the Tropics or the African continent, which leads to a high level of uncertainty in these regions. The shortcomings in the data sources for each of the four water cycle variables are confirmed by an analysis of the global- and basin-scale water budgets. Due to their ability to simulate the whole climate system, the three reanalysis models allow in principle a consistent evaluation of the global water cycle. However, it is shown that there are significant imbalances and numerical artifacts in their oceanic and continental water budgets, which obviously hinder the use of such model estimates for e.g. extreme value or climate trend studies. On the basin-scale, the evaluation of the water budgets from different combinations of widely used data sources for precipitation, evapotranspiration, runoff, and water storage changes reveal imbalances of more than 25% of the mean annual runoff over most of the 96 study regions. Even if some data combinations allow a reasonable closure of the water budgets over certain catchments, it is not possible to identify a single best dataset which performs consistently on the global scale. That being said, the significant decrease in the number of stream gauges worldwide further aggravates a continuous analysis of the basin-scale water cycle. The study therefore presents an approach, with which basin-scale time series of precipitation, evapotranspiration, runoff, and water storage changes can be predicted or corrected. The method is based on an Ensemble Kalman Filter framework, where all required input parameter are derived from an ensemble of hydrological and hydrometeorological datasets. In order to evaluate the performance of the proposed framework, the filter is used for predicting runoff over 16 catchments. A comparison with observed runoff shows correlations larger than 0.5, relative errors lower than 20%, and NSE-values larger than 0.5 for most of the study regions. Overall, the study shows that our current datasets for the major water cycle variables have to be used with care. The large imbalances and inconsistencies in the water budgets on both the global- and basin-scale deny the direct use of such estimates for e.g. climate trend studies or the analysis of extreme events. Thus, in order to use our current datasets for studying the projected changes in the global water cycle, a careful analysis and data correction has to be performed. It is further stressed that, despite the promising performance of certain alternative methods like e.g. the presented EnKF-approach, there is still an urgent need for in situ observations for precipitation and runoff. Otherwise, it will become more difficult in the near future to perform water budget studies or climate analyses, but also to validate hydrological or hydrometeorological models.
... two cases arise depending on whether the spectrum B lm lm is a real number (e.g., Han et al. 2005) or a complex number (e.g., Lorenz 2009). If the spectral weights are complex, then the amplitude and phase are given bȳ ...
Representing the spherical harmonic spectrum of a field on the sphere in terms of its amplitude and phase is termed as its polar form. In this study, we look at how the amplitude and phase are affected by linear low-pass filtering. The impact of filtering on amplitude is well understood, but that on phase has not been studied previously. Here, we demonstrate that a certain class of filters only affect the amplitude of the spherical harmonic spectrum and not the phase, but the others affect both the amplitude and phase. Further, we also demonstrate that the filtered phase helps in ascertaining the efficacy of decorrelation filters used in the grace community.
... This idea is illustrated in terms of a multi-resolution concept by Kusche [2007]. The optimum value of the regularization parameter is estimated via variance component estimation [Lorenz, 2009], but can also be tweaked to the desired smoothness as demonstrated by Kusche [2007]. Figure 3.5 shows the spatial structure of the regularization filter for nine different calculation points spread over three different latitudes and three different longitudes. ...
... This is clear in the smoothing operator at the pole ( Figure 3.5), where we see that the smoothing operator is isotropic, which is the case for a latitude dependent anisotropic kernel. Despite this fact, Lorenz [2009] found benefit in using the full covariance matrix for filtering the datasets. In this work we will follow the method of Sasgen et al. [2006] to model the signal covariance, and there we will demonstrate its cyclo-stationary nature. ...
... Since, the idea of cyclostationarity requires the knowledge of the signal content in data, we will demonstrate the signal covariance modelling in chapter 6. For computing the noise covariance matrix we will follow the method of Lorenz [2009], and for the regularization parameter we will use the approach of [King et al., 2006] by comparing regularized solutions of different γ values with deformation time-series to find out the optimum value. ...
Geodesists employ signal processing techniques on the sphere to analyse gravity field data, and the primary mathematical tool of choice has been the spherical harmonics. Harmonic analysis and synthesis were the predominant signal processing techniques that were employed. However, with the launch of the Gravity Recovery And Climate Experiment (GRACE) satellite mission, there was a strong need for low-pass filtering techniques as the grace data is heavily contaminated with noise. Now, after a decade since the launch there is a garden of filters that have been proposed, which has brought with it the problem of filter choice. It is in this context that this study would like to understand the anatomy of low-pass linear filters, their mechanics of filtering, and measure their performance that will enable consistency in the choice of a filter for the problem in hand. When applying filters there is always a question of choice, and from the experiences in filtering GRACE, it can be said that the output is heavily influenced by the chosen filter. Irrespective of the filter chosen, filtering smudges part of the signal in addition to smoothing out noise, and the amount of signal lost depends on the filter. In order to assess the suitability of a filter and to understand its behaviour, a framework has been developed. The framework consists of a set of metrics designed on the basis of the energy of the filters and log-normal of the filter weights. This thesis elucidates a number of attributes of the filters and filtering on the sphere. It makes positive strides in the direction of understanding the mechanics of linear low-pass filtering on the sphere, especially with respect to resolution and leakage. Further, it also puts forth a set of metrics that provide a generic understanding of the filter in hand, enabling appropriate filter choice.
... Therefore, to eliminate the noise, we need to apply a filter to the data. A regularization filter, which is described in (Lorenz, 2009) and (Sneeuw et al., 2014), is applied to decrease the high noise content in the higher frequencies of the GRACE data. ...
Since 2002, the Gravity Recovery and Climate Experiment (GRACE) provides monthly snapshots of the Earth's time variable gravity field. The GRACE-Follow-On (GRACE-FO) Mission and future satellite gravity missions are intended to continue and improve the measurements of the original GRACE mission. Their primary objective is to obtain highly accurate global estimates for both the static and the time variable part of the Earth's gravity field. However, especially the higher-degree spherical harmonic coefficients are contaminated with noise. It is thus still mandatory to apply state-of-the-art filtering techniques to the provided spectral data in order to reveal reasonable gravity field estimates. In this study, we analyzed the potential of stochastic Copula-based filtering methods for gravity field solutions from GRACE-FO and other future satellite gravity missions. Our approach exploits linear and non-linear relations between two or more variables (e.g. filtered and unfiltered spherical harmonics) by fitting a theoretical Copula function into an empirical bivariate or multivariate distribution function. The fitted Copula then contains the complete information about the statistical dependency between the variables. As a result, noisy data can be filtered consistently with the captured statistical structure. In this study, we used simulated a potential future double-pair missions gravity recoveries, filtered with different known techniques, e.g. regularization or empirical orthogonal functions. The dependency structure between the unfiltered and filtered spectrum is then derived within a short training period. Minimizing the duration of the training period is of utmost importance to the viability of the Copula-based filter approach. Based on the derived dependency structure, we then generate random data, which is (statistically) consistent with the filtered of our double pair mission. The Copula-based random data are generated out of the derived dependency structure between the unfiltered and filtered one year time series of 11-day gravity solutions of a double pair inline satellite mission as a potential future gravity mission (taken from ESA SC4MGV project). Comparing the simulated data from the Copula-based approach with the originally filtered spherical harmonic coefficients reveals only minor differences. Copula–based approach Sklar (1959) introduced Copula functions as a powerful tool to model the dependency between variables. The Copula–based approach exploits linear and non-linear relationships between two or more variables by fitting a theoretical Copula function into an empirical bivariate or multivariate distribution function. Finally, new data which is now consistent with the previously derived dependency structure can be simulated by evaluating the conditional distribution function which is given by the theoretical Copula. Therefore, it can model complex dependency structures. Nelsen (2010) defined several bivariate Archimedean Copulas with parameter í µí»³. We assimilate data using Copula-based approach by the following steps (Vogl et al., 2012).
... Therefore, to eliminate the noise, we need to apply a filter to the data. A regularization filter, which is described in (Lorenz, 2009) and (Sneeuw et al., 2014), is applied to decrease the high noise content in the higher frequencies of the GRACE data. The degree 0 and 1 coefficients were removed throughout our computations, as GRACE is blind towards these coefficients. ...
Data from the Gravity Recovery and Climate Experiment (GRACE) has significantly improved our knowledge of the terrestrial water cycle. With the availability of GRACE data from 2002, we are now able to perform even climate change studies with respect to water storage variations. However, as GRACE is already after its expected lifetime, we have to find methods for filling the missing months in the past data and to possibly bridge the gap until GRACE Follow On.
In this study, we, therefore, analyze the potential of Copula-based methods for simulating GRACE coefficients data from other hydrological data sources. The method exploits linear and non-linear relationships between two or more variables by fitting a theoretical Copula function into an empirical bivariate or multivariate distribution function. Finally, new data, which is then consistent with the previously derived dependence structure, can be simulated by evaluating the conditional distribution function given by the theoretical Copula.
First, we want to analyze the applicability of the proposed method to spherical harmonic coefficients data from GRACE. As the approach involves several drawings of random data, we are interested if this random nature has any impact on the results. We, therefore, generate filtered out of unfiltered GRACE coefficients, based on the previously derived dependence structure. The comparison between the simulated and filtered data shows a very good agreement with negligible differences in both of the spatial and spectral domain.
We also want to evaluate if Copula-based methods are able to estimate reliable water storage changes from the independent hydrological data. Therefore, we derive the dependence structure between filtered water storage changes from GRACE and global gridded precipitation data from the Global Precipitation Climatology Center GPCC. Based on the fitted theoretical Copula, we then simulate water storage changes from precipitation data. The Copula-based estimates are compared with filtered GRACE coefficients data in both of the spectral and spatial domain. We also perform a catchment-based analysis between area-aggregated time-series of simulated and GRACE-derived water storage change. The analysis shows that our estimates and the original filtered GRACE coefficients data are in very good agreement. Thus, we conclude that the proposed method is indeed able to fill the missing months in the GRACE-dataset and to extend even the time-series until the launch of GRACE Follow On.
... The regularization filter requires signal and noise covariance matrices, and the modifications we make pertain to these covariance matrices. Kusche (2007) computes the signal covariance from a combination of different models, while we compute the signal covariance from the GRACE data itself as described by Lorenz (2009). The signal covariance so calculated is a power-law ...
Given the continuous decline in global runoff data availability over the past decades, alternative approaches for runoff determination are gaining importance. When aiming for global scale runoff at a sufficient temporal resolution and with homogeneous accuracy, the choice to use spaceborne sensors is only a logical step. In this respect, we take water storage changes from Gravity Recovery And Climate Explorer (grace) results and water level measurements from satellite altimetry, and present a comprehensive assessment of five different approaches for river runoff estimation: hydrological balance equation, hydro-meteorological balance equation, satellite altimetry with quantile function-based stage-discharge relationships, a rudimentary instantaneous runoff-precipitation relationship, and a runoff-storage relationship that takes time lag into account. As a common property, these approaches do not rely on hydrological modeling; they are either purely data driven or make additional use of atmospheric reanalyses. Further, these methods, except runoff-precipitation ratio, use geodetic observables as one of their inputs and, therefore, they are termed hydro-geodetic approaches. The runoff prediction skill of these approaches is validated against in situ runoff and compared to hydrological model predictions. Our results show that catchment-specific methods (altimetry and runoff-storage relationship) clearly outperform the global methods (hydrological and hydro-meteorological approaches) in the six study regions we considered. The global methods have the potential to provide runoff over all landmasses, which implies gauged and ungauged basins alike, but are still limited due to inconsistencies in the global hydrological and hydro-meteorological datasets that they use.
... One can determine the a priori variance through geophysical model. Lorenz (2009) solved this problem by a simulation for a full covariance matrix of time-variable GRACE coefficients. This simulated full covariance matrix is adopted for the Wiener filtering in the thesis, and the filtered GRACE coefficients are then provided by Mr. Devaraju 1 . ...
Mass transport within the Earth system over time (e.g., hydrological circulation) induces the mass redistribution on the surface. The temporal variation of mass load on the surface consequently leads to elastic deformation of Earths surface (van Dam et al., 2001; Ilk et al., 2005; De Linage et al., 2007). The surface deformation could be derived from GRACE through time-variable gravity field and also be observed by IGS stations in GPS 3D coordinates. The surface deformations derived from GRACE are spatially smoothed with about 350 km resolution. However, the deformations of IGS stations observed by GPS are discrete point measurements on the globe. Therefore, a validation of the consistency between the deformations from GRACE and GPS is necessary to be done, which would benefit the further research on mass transport and climate change. In this study, using the data from GFZ, the deformations from GRACE are theoretically calculated in vertical and horizontal directions (Wahr et al., 1998; Kusche and Schrama, 2005). To investigate the disagreement between GPS and GRACE, a number of IGS stations in three regions are selected (i.e., Tibetan plateau, Danube basin and Great Lakes area) with period of 8 years (2003 2011). For a proper comparison, the spatial and temporal reference of GRACE and GPS need to be unified. For validation, the correlation coefficient, the Nash-Sutcliffe efficiency, and WRMS reduction are estimated. After comparisons of deformation time series, almost all the stations in those regions show good consistency between GRACE and GPS in vertical component. There is distinct disagreement in horizontal component, probably due to the weak loading signals and strong local effects. Thus, several representative stations in those regions would be discussed and analysed in detail. Furthermore, to detect an optimal filter for GRACE, 40 IGS stations in Europe are involved to evaluate the filter performance. As a result, 52.5% stations filtered by the stochastic filter (i.e., Wiener filter) show better results, which indicates the optimal choice.
... (ii) the methods which use the error-covariance information (Koch and Kusche, 2002;Klees et al., 2008;Save, 2009;Lorenz, 2009;Kurtenbach, 2011). ...
... However, the power law behavior of the Kaula's rule allows a simple estimation of a fitting power law, i.e. a degree variance model, for other kinds of spherical harmonic coefficients. The basic equation for such a power law reads as (Lorenz, 2009) σ 2 l = 10 a l b (4.23) ...
... In this study, the signal variance of the combined input model AOHIS (see Figure 4.2) is utilized to estimate the unknown parameters, a and b, of the power law equation (4.24), where they are used for filling the regularization matrix K (Lorenz, 2009). ...
The launch of the GRACE mission has generated a broad interest within the geophysical community in the detection of temporal gravity fields and their applications, e.g. the detection of ice mass loss over Greenland and Antarctica, the hydrological cycle over Amazon and central Africa and the estimation of sea level rise. However the spatio-temporal resolution of GRACE solutions is limited by a restricted sensitivity of the metrology system, the reduced isotropy of the inline leader-follower formation (which mainly manifests itself in a North-South striped error pattern) and the temporal aliasing of high frequency time variable geophysical signals into the long time-interval solutions.
When using high quality sensors in future gravity missions, aliasing of the high frequency (short period) geophysical signals to the lower frequency (longer period) signals is one of the most challenging obstacles.
Two sampling theorems mainly govern the space-time sampling of a satellite-mission:
(i) a Heisenberg-type principle in satellite geodesy which states that the product of spatial resolution and time resolution is constant, and (ii) the Colombo-Nyquist rule (CNR), which requires the number of satellite revolutions in the full repeat-cycle to be equal at least twice the maximum spherical harmonic degree to be detected. The latter rule, therefore, limits the spatial resolution of the solution.
This study investigates the quality of sub-Nyquist recoveries, i.e. solutions from time intervals shorter than required by CNR, of different orbit configurations and satellite formations. In particular, the dependence of such quality on the measurement duration and ground-track patterns is investigated. It is shown that (i) the number of observations with specific coverage of the Earth by a satellite configuration (as indicated by a modified Colombo-Nyquist rule), (ii) the mission altitude and (iii) avoidance of large unobserved gaps by satellite ground-track patterns have the most important effect on the quality of the recoveries. The sub-cycle concept apparently does not play an important role in assessing the quality.
Moreover, the study investigates the modified Colombo-Nyquist rule for two pairs of satellites, where the number of revolutions by both satellite pairs is taken into account. It is also found that sub-Nyquist recoveries by such double pair scenarios outperform the ones from single inline satellite missions with twice the size of time intervals. It is indeed expected that using an inclined satellite mission, together with a near-polar mission, adds East-West measurement component to the North-South component of the near-polar satellite mission. Furthermore, the short time interval recoveries suffer less from temporal aliasing of certain time-variable gravity field components. Consequently, it means that the recovery also benefits from higher time resolution.
The gravity recovery simulations of this study are based on a quick-look tool, developed at the Institute of Geodesy, University of Stuttgart. The closed-loop simulation tool assumes a nominal repeat orbit for a satellite mission. Based on the quality assessment of the recoveries and the technical concerns with the implementation of formation flights, a near-polar moderate pendulum formation with an opening angle of less than 10°, approximately 300 km altitude and almost homogeneous gap evolution is suggested for a next generation of single pair gravity mission. For double pair satellite missions, a combination of a near-polar inline or moderate pendulum and a 72° inclined inline pair is recommended. The suggested optimal scenarios of this study are selected through the quality assessment of sub-Nyquist gravity recoveries of different configurations.
It is also shown that the quality of the sub-Nyquist gravity recoveries can be improved by employing post processing tools. The post-processing tools of this research study include a white noise filter, based on EOF+KS-Test analysis and a regularization method which can handle all kinds of noise. The tools are employed to deal with the poorer quality of short-time interval recoveries due to the spatial aliasing, although it is almost impossible to remove all noise without diminishing some of the real signals.
... Furthermore there are many other filters such as the EOF filter (Wouters and Schrama, 2007;Bentel, 2009), Regularization filter (Lorenz, 2009), Optimal filter (Klees et al., 2008) and Han's filter (Han et al., 2005) which can be used alternatively. ...
The Gravity Recovery And Climate Experiment (GRACE) mission was launched on Mar. 17, 2002 and has provided the scientists with the gravity data for nearly ten years. The time variable gravity field provided by the GRACE has improved our knowledge of the earth in many fields such as hydrology, oceanography and glaciology.
But compared to those hot fields, the publications of GRACE in seismology is considerably less. However, GRACE can provide scientists with an independent observation of the earthquake process. Coincidentally, some of the largest earthquakes are within GRACEs life span - Sumatra-Andaman Earthquake (Indonesia) 2004, Maule Earthquake (Chile) 2010 and Tohoku Earthquake (Japan) 2011. Furthermore, a smaller earthquake - Sichuan Earthquake (China) 2008 has also been examined to test whether the GRACE can detect earthquakes smaller than Mw = 8.0. Different from the traditional methods of the earthquake researches, the gravity method has its advantages: 1. Massive: global scale; 2. Insight: gravity changes can reveal the underground mass changes which do not cause so much motion on the earth surface; 3. Convenient: superior to the traditional methods, the spaceborne gravimetry can get the data from the ocean and glacier parts.
The conditions of the data are different among these four earthquakes. The procedures to eliminate the GRACE observation errors and unwanted geophysical data are necessary. First, the C20 term should be replaced by the Satellite Laser Ranging (SLR) data. Second, the hydrology signal especially in the regions of Chile and Sichuan should be eliminated by the Global Land Data Assimilation System (GLDAS) model. Third, Fan filter or Gauss filter 350 km should be applied.
Time series analysis by the two-phase changepoint detection and hypothesis testing are applied for each earthquake which is a point-wise analysis. Least squares adjustment is performed on each point to display the coseismic and postseismic signals. Meanwhile, the surface analysis is done by the Empirical Orthogonal Functions (EOF) as it has a flexible base which can suit the data automatically.
Although the observation errors have been removed as much as possible, the limited spatial and time resolutions of the GRACE satellite and to retrieve relatively weak earthquake signal among the strong hydrological signals are still problems in the analysis.
GRACE can detect some of the large earthquakes, but it depends on the earthquake type, area and the length of the time-series before and after the earthquake. Both coseismic signal and postseismic signal are detected in Sumatra-Andaman Earthquake. Meanwhile, there is no significant coseismic signal in the time series of Sichuan Earthquake, but the EOF detects suspicious earthquake signal in mode 2 with the magnitude less than 1 µGal. For Maule Earthquake, only the coseismic signal is detected. Due to the limited dataset, the detection of the coseismic signal is successful but the postseismic signal is not long enough to be detected in Tohoku Earthquake. However, the different filters will affect the magnitude of the gravity change, so the real gravity changes of those four areas are still under debate. Last, EOF can be used for the separation of the earthquake signals.
Compared to other geodetic technics the gravity method can detect the signals underground and in the ocean areas. The coseismic and postseismic signals detected by GRACE show underground processes of the earthquakes which can help scientists better understand the earthquake mechanism and will contribute to the earthquake prediction in the future.
Drawing on experience from Gravity Recovery and Climate Experiment (GRACE) data analysis, the scientific challenges were already identified in several studies. Any future mission should focus on improvement in both precision and resolution in space and time. For future gravity missions which use high quality sensors, aliasing of high frequency time-variable geophysical signals to the lower frequency signals is one of the most serious problems. The aliasing problem and the spatio-temporal resolution are mainly restricted by two sampling theorems describing the space-time sampling of satellite missions: (i) a Heisenberg-like uncertainty theorem which states that the product of spatial resolution and time resolution is constant, and (ii) the Colombo–Nyquist rule (CNR), which requires the number of satellite revolutions in a repeat period to be at least twice a given maximum spherical harmonic degree. The CNR holds under the assumption of equal ground-track spacing, and limits the spatial resolution of the gravity solution.
