Surveys in Geophysics

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Imaging Spectroscopy for Soil Mapping and Monitoring

Wed, 03/20/2019 - 00:00
Abstract

There is a renewed awareness of the finite nature of the world’s soil resources, growing concern about soil security and significant uncertainties about the carrying capacity of the planet. Regular assessments of soil conditions from local through to global scales are requested, and there is a clear demand for accurate, up-to-date and spatially referenced soil information by the modelling scientific community, farmers and land users, and policy- and decision-makers. Soil and imaging spectroscopy, based on visible–near-infrared and shortwave infrared (400–2500 nm) spectral reflectance, has been shown to be a proven method for the quantitative prediction of key soil surface properties. With the upcoming launch of the next generation of hyperspectral satellite sensors in the next years, a high potential to meet the demand for global soil mapping and monitoring is appearing. In this paper, we briefly review the basic concepts of soil spectroscopy with a special attention to the effects of soil roughness on reflectance and then provide a review of state of the art, achievements and perspectives in soil mapping and monitoring based on imaging spectroscopy from air- and spaceborne sensors. Selected application cases are presented for the modelling of soil organic carbon, mineralogical composition, topsoil water content and characterization of soil crust, soil erosion and soil degradation stages based on airborne and simulated spaceborne imaging spectroscopy data. Further, current challenges, gaps and new directions toward enhanced soil properties modelling are presented. Overall, this paper highlights the potential and limitations of multiscale imaging spectroscopy nowadays for soil mapping and monitoring, and capabilities and requirements of upcoming spaceborne sensors as support for a more informed and sustainable use of our world’s soil resources.

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Imaging Spectroscopy for the Detection, Assessment and Monitoring of Natural and Anthropogenic Hazards

Mon, 03/18/2019 - 00:00
Abstract

Natural and anthropogenic hazards have the potential to impact all aspects of society including its economy and the environment. Diagnostic data to inform decision-making are critical for hazard management whether for emergency response, routine monitoring or assessments of potential risks. Imaging spectroscopy (IS) has unique contributions to make via the ability to provide some key quantitative diagnostic information. In this paper, we examine a selection of key case histories representing the state of the art to gain an insight into the achievements and perspectives in the use of visible to shortwave infrared IS for the detection, assessment and monitoring of a selection of significant natural and anthropogenic hazards. The selected key case studies examined provide compelling evidence for the use of the IS technology and its ability to contribute diagnostic information currently unattainable from operational spaceborne Earth observation systems. User requirements for the applications were also evaluated. The evaluation showed that the projected launch of spaceborne IS sensors in the near-, mid and long term future, together with the increasing availability, quality and moderate cost of off the shelf sensors, the possibilities to couple unmanned autonomous systems with miniaturized sensors, should be able to meet these requirements. The challenges and opportunities for the scientific community in the future when such data become available will then be ensuring consistency between data from different sensors, developing techniques to efficiently handle, process, integrate and deliver the large volumes of data, and most importantly translating the data to information that meets specific needs of the user community in a form that can be digested/understood by them. The latter is especially important to transforming the technology from a scientific to an operational tool. Additionally, the information must be independently validated using current trusted practices and uncertainties quantified before IS derived measurement can be integrated into operational monitoring services.

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Assessment of Modern Roadways Using Non-destructive Geophysical Surveying Techniques

Wed, 03/06/2019 - 00:00
Abstract

The main purpose of modern roadways is to provide roadway users with both a comfortable and safe ride to their destinations. As such, they need pavements in good physical conditions to ensure safe and uninterrupted transportation of the public. During the previous decades, roadway engineers’ interests have shifted towards maintenance and rehabilitation of existing pavement structures, rather than the construction of new structures. Nevertheless, pavement condition assessment (PCA) remains imperative both during construction for quality assurance purposes and during roadways’ service life for efficient maintenance planning. Research and current practices have shifted towards a broadened utilization of advanced non-destructive testing systems that enable non-invasive PCA. The current investigation aims to provide a comprehensive overview of the geophysical methods available for modern roadways’ assessment. Geophysical surveying techniques including ground penetrating radar (GPR) and those based on stress waves theory can substantially improve PCA. They cover roadway applications including layer thicknesses determination, stiffness estimation of asphalt and concrete pavements, as well as the determination of physical properties, subsurface defects detection and most recently density monitoring. In particular, it is demonstrated that GPR can assist pavement engineers at all stages of PCA from the construction process through density control and compaction monitoring. Furthermore, throughout a roadway’s service life, GPR can be effectively incorporated as a supplementary tool for monitoring and evaluation within a pavement management system, contributing to optimizing roadways design and maintenance, preserving durable and sustainable structures, ensuring cost savings for road authorities and highway operators through enhanced decision-making processes.

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The Relevance of Forest Structure for Biomass and Productivity in Temperate Forests: New Perspectives for Remote Sensing

Mon, 03/04/2019 - 00:00
Abstract

Forests provide important ecosystem services such as carbon sequestration. Forest landscapes are intrinsically heterogeneous—a problem for biomass and productivity assessment using remote sensing. Forest structure constitutes valuable additional information for the improved estimation of these variables. However, survey of forest structure by remote sensing remains a challenge which results mainly from the differences in forest structure metrics derived by using remote sensing compared to classical structural metrics from field data. To understand these differences, remote sensing measurements were linked with an individual-based forest model. Forest structure was analyzed by lidar remote sensing using metrics for the horizontal and vertical structures. To investigate the role of forest structure for biomass and productivity estimations in temperate forests, 25 lidar metrics of 375,000 simulated forest stands were analyzed. For the lidar-based metrics, top-of-canopy height arose as the best predictor for describing horizontal forest structure. The standard deviation of the vertical foliage profile was the best predictor for the vertical heterogeneity of a forest. Forest structure was also an important factor for the determination of forest biomass and aboveground wood productivity. In particular, horizontal structure was essential for forest biomass estimation. Predicting aboveground wood productivity must take into account both horizontal and vertical structures. In a case study based on these findings, forest structure, biomass and aboveground wood productivity are mapped for whole of Germany. The dominant type of forest in Germany is dense but less vertically structured forest stands. The total biomass of all German forests is 2.3 Gt, and the total aboveground woody productivity is 43 Mt/year. Future remote sensing missions will have the capability to provide information on forest structure (e.g., from lidar or radar). This will lead to more accurate assessments of forest biomass and productivity. These estimations can be used to evaluate forest ecosystems related to climate regulation and biodiversity protection.

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Earth Observation Imaging Spectroscopy for Terrestrial Systems: An Overview of Its History, Techniques, and Applications of Its Missions

Sat, 03/02/2019 - 00:00
Abstract

Imaging spectroscopy in the visible-to-shortwave infrared wavelength range (VSWIR), or nowadays more commonly known as ‘hyperspectral imaging’, for terrestrial Earth Observation remote sensing, dates back to the early 1980s when its development started with mainly airborne demonstrations. From its initial use as a research tool, imaging spectroscopy encompassing the VSWIR spectral range has gradually evolved towards operational and commercial applications. Today, it is one of the fastest growing research areas in remote sensing owing to its diagnostic power by means of discrete spectral bands that are contiguously sampled over the spectral range with which a target is observed. The main principles of imaging spectroscopy rely on the exploitation of light dispersion technologies to split the incoming light through a telescope before being projected onto detector arrays. The light dispersion can be achieved by using prism or diffractive grating optical systems, perpetually aiming for improved performances in terms of efficiency, straylight rejection, and polarization sensitivity. The sensor technique has been first used in airborne imaging spectroscopy since the early 1980s and later in spaceborne hyperspectral missions from the end of the 1990s onwards. Currently, several hyperspectral spaceborne systems are under development and in preparation to be launched within the next few years. Through hyperspectral remote sensing, physical, chemical, and biological components of the observed matter can be separated and resolved thus providing a spectral ‘fingerprint’. The analyses of the spectral absorptions often give rise to quantitative retrievals of components of the observed target. The derived information is vital for the generation of a wide variety of new quantitative products and services in the domain of agriculture, food security, raw materials, soils, biodiversity, environmental degradation and hazards, inland and coastal waters, snow hydrology and forestry. Many of these are relevant to various international policies and conventions. Originally developed as a powerful detection and analysis tool for applications predominantly related to planetary exploration and non-renewable resources, imaging spectroscopy now covers many disciplines in atmospheric, terrestrial vegetation, cryosphere, and marine research and application fields. There is an increasing number of visible/near-infrared (VNIR) imaging spectrometers emerging also as small payloads on small satellites and cubesats, built and launched by small-medium enterprises. These are targeted to address commercial applications mainly in agriculture, resources and environmental management, and hazard observations.

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Linear Vary-Chap Topside Electron Density Model with Topside Sounder and Radio-Occultation Data

Fri, 03/01/2019 - 00:00
Abstract

The Linear Vary-Chap function has received increased attention in describing the topside ionosphere due to its good performance for predicting and extrapolating radio-occultation (RO) electron density ionospheric profiles. The systematic increase in the scale height is consistent with first principles corresponding to the increase in the electron temperature; however, the altitude where the linear scale height approximation does not stay valid has not been explicitly discussed in the literature. In order to demonstrate up to what extent the linear behavior of the scale height is still valid, this work analyzes more than 50,000 manually scaled ionospheric profiles measured by topside sounders on board Alouette and International Satellites for Ionospheric Studies satellites. Based on this initial analysis, a new topside model is proposed to take into consideration the nonlinear behavior of the topside scale height. The proposed climatological model, a fit of spherical harmonics to parameters derived from topside RO profiles, is used to predict topside sounder measurements. An assessment of the predicted, RO-derived, topside is carried out and the experimental results are discussed in order to show the viability of extrapolating RO ionospheric profiles for altitudes above the low Earth orbit.

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Diffusion and Thermodiffusion of Atmospheric Neutral Gases: A Review

Fri, 03/01/2019 - 00:00
Abstract

The current state of knowledge on diffusion and thermodiffusion of neutral gases that govern the composition of the greater part of the Earth’s thermosphere is reviewed. Hydrodynamic equations determining diffusion velocities of neutral gases of the multicomponent atmosphere are considered in the first-order approximation of the velocity distribution functions of the Chapman–Enskog method using the first- and second-order approximations of the Sonine polynomial expansion to transport multicomponent coefficients of neutral gases. The interrelations among various definitions for the multicomponent diffusivities are given. The thermodiffusion factors and binary diffusion coefficients of N2, O2, O, He, H, Ar, N, NO, and H2 used in atmospheric studies to calculate the diffusion velocities and number densities of these atmospheric gases are discussed. The recommended temperature dependencies of the binary diffusion coefficients and the recommended average multicomponent diffusion correction and thermodiffusion factors of the atmospheric neutral gases under consideration are presented.

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Spherical Harmonic Expansions for the Gravitational Field of a Polyhedral Body with Polynomial Density Contrast

Fri, 03/01/2019 - 00:00
Abstract

We provide the spherical harmonic solutions to evaluate the external gravitational field of a general polyhedral body with arbitrary polynomial density contrast, including the gravitational potential and its arbitrary-order derivatives. The linear recursive algorithm for computation of the spherical harmonic coefficients of the potential is derived by using the Gauss divergence theorem and the Stokes theorem, and the computations are performed on the basis of the division of general polygonal pyramid of the polyhedron instead of the division of tetrahedron. The algorithm of this paper can handle the density contrast in both horizontal and vertical directions and the polynomial function of the density at arbitrary degree. Both the conversion relations of the density function and the arbitrary-order derivatives of the spherical harmonic potential between the initial and rotated reference frames are given in tensor product forms, which assist the calculations. The space-domain method for evaluating the gravity field of the polyhedral body may leads to numerical problems at a remote observation point. However, the spherical harmonic method is numerically stable at arbitrary observation points outside the smallest enclosed sphere. The numerical experiments for three actual and synthetic polyhedral models including a right rectangular prism with cubic density varying with depth, the asteroid 433 EROS with cubic polynomial density and a right rectangular prism with quartic polynomial density, are implemented to test the accuracy, convergence, and numerical stability of the spherical harmonic algorithm.

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A Review on Scaling of Earthquake Source Spectra

Fri, 03/01/2019 - 00:00
Abstract

In this review paper, the theoretical and observational studies of scaling of earthquake source displacement spectra (abbreviated as source spectra) are compiled and discussed. The earlier studies, including the kinetic models proposed by several authors [including Haskell (Bull Seismol Soc Am 56:125–140, 1966) and Aki (J Geophys Res 72:1217–1231, 1967), provided the so-called ω−1, ω-square, and ω-cube models. Aki (1967) favored the ω-square model and also assumed constant stress drop, Δσ, and self-similarity of earthquakes. Some observations agree to one of the two models, but others do not. Hence, numerous alternative forms of the three scaling models to interpret the observations have been made by seismologists. For the corner frequencies, fc, the seismic moment Mo scales with fc in the form of Mo ~ f c −3 . For fcP of the P-waves and fcS of the S-waves, fcP > fcS is more reasonable than fcP < fcS. Mo scales as T D 3 , where TD is the duration time of source rupture and inversely related to fc, for both small and large events. The second corner frequency or cut-off frequency, fmax, at higher ω, that exists in the source spectra could be yielded by source, path, and site effects. The mechanisms to cause the patch corner frequency are still open. Analytical and numerical studies of the scaling laws based on the dislocation (e.g., Aki 1967), cracks (e.g., Walter and Brune in J Geophys Res 98(B3):4449–4459, 1993), spring-slider (e.g., Shaw in Geophys Res Lett 20:643–646, 1993), and statistical physics models (e.g., Hanks in J Geophys Res 84:2235–2242, 1997), including self-organized criticality (e.g., Bak et al. in Phys Rev Lett 59:381–384, 1987), show different scaling laws under various physical conditions.

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Acknowledgement of Reviewers for 2018

Fri, 03/01/2019 - 00:00
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High-Resolution Characterization of Near-Surface Structures by Surface-Wave Inversions: From Dispersion Curve to Full Waveform

Fri, 03/01/2019 - 00:00
Abstract

Surface waves are widely used in near-surface geophysics and provide a noninvasive way to determine near-surface structures. By extracting and inverting dispersion curves to obtain local 1D S-wave velocity profiles, multichannel analysis of surface waves (MASW) has been proven as an efficient way to analyze shallow-seismic surface waves. By directly inverting the observed waveforms, full-waveform inversion (FWI) provides another feasible way to use surface waves in reconstructing near-surface structures. This paper provides a state of the art review of MASW and shallow-seismic FWI and a comparison of both methods. A two-parameter numerical test is performed to analyze the nonlinearity of MASW and FWI, including the classical, the multiscale, the envelope-based, and the amplitude-spectrum-based FWI approaches. A checkerboard model is used to compare the resolution of MASW and FWI. These numerical examples show that classical FWI has the highest nonlinearity and resolution among these methods, while MASW has the lowest nonlinearity and resolution. The modified FWI approaches have an intermediate nonlinearity and resolution between classical FWI and MASW. These features suggest that a sequential application of MASW and FWI could provide an efficient hierarchical way to delineate near-surface structures. We apply the sequential-inversion strategy to two field data sets acquired in Olathe, Kansas, USA, and Rheinstetten, Germany, respectively. We build a 1D initial model by using MASW and then apply the multiscale FWI to the data. High-resolution 2D S-wave velocity images are obtained in both cases, whose reliabilities are proven by borehole data and a GPR profile, respectively. It demonstrates the effectiveness of combining MASW and FWI for high-resolution imaging of near-surface structures.

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Assessing Vegetation Function with Imaging Spectroscopy

Fri, 02/15/2019 - 00:00
Abstract

Healthy vegetation function supports diverse biological communities and ecosystem processes, and provides crops, forest products, forage, and countless other benefits. Vegetation function can be assessed by examining dynamic processes and by evaluating plant traits, which themselves are dynamic. Using both trait-based and process-based approaches, spectroscopy can assess vegetation function at multiple scales using a variety of sensors and platforms ranging from proximal to airborne and satellite measurements. Since spectroscopic data are defined by the instruments and platforms available, along with their corresponding spatial, temporal and spectral scales, and since these scales may not always match those of the function of interest, consideration of scale is a necessary focus. For a full understanding of vegetation processes, combined (multi-scale) sampling methods using empirical and theoretical approaches are required, along with improved informatics.

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The Role and Need for Space-Based Forest Biomass-Related Measurements in Environmental Management and Policy

Mon, 02/11/2019 - 00:00
Abstract

The achievement of international goals and national commitments related to forest conservation and management, climate change, and sustainable development requires credible, accurate, and reliable monitoring of stocks and changes in forest biomass and carbon. Most prominently, the Paris Agreement on Climate Change and the United Nations’ Sustainable Development Goals in particular require data on biomass to monitor progress. Unprecedented opportunities to provide forest biomass data are created by a series of upcoming space-based missions, many of which provide open data targeted at large areas and better spatial resolution biomass monitoring than has previously been achieved. We assess various policy needs for biomass data and recommend a long-term collaborative effort among forest biomass data producers and users to meet these needs. A gap remains, however, between what can be achieved in the research domain and what is required to support policy making and meet reporting requirements. There is no single biomass dataset that serves all users in terms of definition and type of biomass measurement, geographic area, and uncertainty requirements, and whether there is need for the most recent up-to-date biomass estimate or a long-term biomass trend. The research and user communities should embrace the potential strength of the multitude of upcoming missions in combination to provide for these varying needs and to ensure continuity for long-term data provision which one-off research missions cannot provide. International coordination bodies such as Global Forest Observations Initiative (GFOI), Committee on Earth Observation Satellites (CEOS), and Global Observation of Forest Cover and Land Dynamics (GOFC‐GOLD) will be integral in addressing these issues in a way that fulfils these needs in a timely fashion. Further coordination work should particularly look into how space-based data can be better linked with field reference data sources such as forest plot networks, and there is also a need to ensure that reference data cover a range of forest types, management regimes, and disturbance regimes worldwide.

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Imaging Spectroscopy of Forest Ecosystems: Perspectives for the Use of Space-borne Hyperspectral Earth Observation Systems

Sun, 02/10/2019 - 00:00
Abstract

The emerging challenges in preserving and managing forest ecosystems are multiscale in terms of space and time, and therefore require spatially and temporally contiguous information sources. Imaging spectroscopy has the potential to contribute information that cannot be raised by other Earth Observation Systems. In particular, the spectral capacity to monitor the distributions of chemical traits, such as canopy foliar nitrogen distribution, and to track changes in water content or the percentage water in plants, has already opened novel pathways toward assessing the global variability of ecosystem functions and services. However, there is an ongoing debate on how to best extract this type of information from the spectral measurements. Empirical approaches have demonstrated their efficiency in a multitude of local studies, but are criticized with respect to poor generalization capacities. Alternative strategies, such as the use of physically based models of leaf or canopy reflectance, or hybrid approaches, have the potential advantage to be more widely applicable. This paper attempts to assess achievements and shortcomings of these strategies and finds that the often-cited disadvantages of using empirical approaches are becoming less pronounced in the light of recent research results. While retrievals based on physically based models on leaf/needle level are close to laboratory quality, results on canopy level available to date still have considerable deficits. Owing to improved instrumental designs, better data calibration, new approaches for compensating canopy effects, and the use of increasingly efficient methods for establishing data-driven models, the scope of empirical approaches has considerably widened and they have been successfully applied to large areas. The future availability of regularly acquired hyperspectral imagery from Earth orbits will substantially contribute to their generalizability.

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Using a Finer Resolution Biomass Map to Assess the Accuracy of a Regional, Map-Based Estimate of Forest Biomass

Fri, 02/08/2019 - 00:00
Abstract

National greenhouse gas inventories often use variations of the gainloss approach whereby emissions are estimated as the products of estimates of areas of land-use change characterized as activity data and estimates of emissions per unit area characterized as emission factors. Although the term emissions is often intuitively understood to mean release of greenhouse gases from terrestrial sources to the atmosphere, in fact, emission factors can also be negative, meaning removal of the gases from the atmosphere to terrestrial sinks. For remote and inaccessible forests for which ground sampling is difficult if not impossible, emission factors may be based on map-based estimates of biomass or biomass change obtained from regional maps. For the special case of complete deforestation, the emission factor for the aboveground biomass pool is simply mean aboveground, live-tree, biomass per unit area prior to the deforestation. If biomass maps are used for these purposes, estimates must still comply with the first IPCC good practice guideline regarding accuracy relative to the true value and the second guideline regarding uncertainty. Accuracy assessment for a map-based estimate entails comparison of the estimate to a second estimate obtained using independent reference data. Assuming ground sampling is not feasible, a map of greater quality than the regional map may be considered as a source of reference data where greater quality connotes attributes such as finer resolution and/or greater accuracy. For a local, sub-regional study area in Minnesota in the USA, the accuracy of an estimate of mean aboveground, live-tree biomass per unit area (AGB, Mg/ha) obtained from a coarser resolution, regional, MODIS-based biomass map was assessed using reference data sampled from a finer resolution, local, airborne laser scanning (ALS)-based biomass map. The rationale for a local assessment of a regional map is that, although assessment of a regional map would be difficult for the entire extent of the map, it can likely be assessed for multiple local sub-regions in which case expected local regional accuracy for the entire map can perhaps be inferred. For this study, the local assessment was in the form of a test of the hypothesis that the local sub-regional estimate from the regional map did not deviate from the local true value. A hybrid approach to inference was used whereby design-based inferential techniques were used to estimate uncertainty due to sampling from the finer resolution map, and model-based inferential techniques were used to estimate uncertainty resulting from using the finer resolution map unit values which were subject to prediction error as reference data. The test revealed no statistically significant difference between the MODIS-based and ALS-based map estimates, thereby indicating that for the local sub-region, the regional, MODIS-based estimate complied with the first IPCC good practice guideline for accuracy.

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Understanding the Land Carbon Cycle with Space Data: Current Status and Prospects

Wed, 02/06/2019 - 00:00
Abstract

Our understanding of the terrestrial carbon cycle has been greatly enhanced since satellite observations of the land surface started. The advantage of remote sensing is that it provides wall-to-wall observations including in regions where in situ monitoring is challenging. This paper reviews how satellite observations of the biosphere have helped improve our understanding of the terrestrial carbon cycle. First, it details how remotely sensed information of the land surface has provided new means to monitor vegetation dynamics and estimate carbon fluxes and stocks. Second, we present examples of studies which have used satellite products to evaluate and improve simulations from global vegetation models. Third, we focus on model data integration approaches ranging from bottom-up extrapolation of single variables to carbon cycle data assimilation system able to ingest multiple types of observations. Finally, we present an overview of upcoming satellite missions which are likely to further improve our understanding of the terrestrial carbon cycle and its response to climate change and extremes.

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Coastal Sea Level and Related Fields from Existing Observing Systems

Tue, 02/05/2019 - 00:00
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We review the status of current sea-level observing systems with a focus on the coastal zone. Tide gauges are the major source of coastal sea-level observations monitoring most of the world coastlines, although with limited extent in Africa and part of South America. The longest tide gauge records, however, are unevenly distributed and mostly concentrated along the European and North American coasts. Tide gauges measure relative sea level but the monitoring of vertical land motion through high-precision GNSS, despite being essential to disentangle land and ocean contributions in tide gauge records, is only available in a limited number of stations. (25% of tide gauges have a GNSS station at less than 10 km.) Other data sources are new in situ observing systems fostered by recent progress in GNSS data processing (e.g., GPS reflectometry, GNSS-towed platforms) and coastal altimetry currently measuring sea level as close as 5 km from the coastline. Understanding observed coastal sea level also requires information on various contributing processes, and we provide an overview of some other relevant observing systems, including those on (offshore and coastal) wind waves and water density and mass changes.

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Wave Polarization Analyzed by Singular Value Decomposition of the Spectral Matrix in the Presence of Noise

Tue, 01/01/2019 - 00:00
Abstract

Analysis of wave polarization provides wave propagation parameters and enables an identification of modes in space plasmas. It is based on measurements of several components of fluctuating electromagnetic fields. This technique has become a conventional part of modern instrumentation onboard scientific spacecraft. A definition of the degree of polarization can be reduced to a very basic form, i.e., the ratio of a signal’s polarized power to its total power. However, this simple definition can have several different realizations which depend mainly on the underlying assumptions about separating the polarized (coherent) part from the unpolarized part (noise). After reviewing polarization of a plane wave in two and three dimensions, we examine the singular value decomposition technique for a complex spectral matrix as well as for a real spectral matrix. The meaning of singular values is explained, and we show to what extent the singular values are able to contribute to a separation between polarized signal and noise. Finally, our theoretical findings are verified with synthetic data as well as with whistler-mode chorus wave observations from the THEMIS spacecraft.

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Correction to: Mathematical Simulation of the Ionospheric Electric Field as a Part of the Global Electric Circuit

Tue, 01/01/2019 - 00:00

Our colleague A. P. Nickolaenko has pointed out an unfortunate, and obvious, misprint in our paper. The error is in line 5 of page 12 showing values for s0, which corresponds to the value of the surface conductivity in units of S/m. These values should not be negative.

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Correction to: Wave Polarization Analyzed by Singular Value Decomposition of the Spectral Matrix in the Presence of Noise

Tue, 01/01/2019 - 00:00

The article Wave Polarization Analyzed by Singular Value Decomposition of the Spectral Matrix in the Presence of Noise, written by Ulrich Taubenschuss and Ondřej Santolík, was originally published electronically on the publisher’s internet portal (currently SpringerLink) on 19 August 2018 without open access.

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