Pointwise Observations Research Articles

Covariance models and Gaussian process regression for the wave equation. Application to related inverse problems

In this article, we consider the general task of performing Gaussian process regression (GPR) on pointwise observations of solutions of the 3 dimensional homogeneous free space wave equation. In a recent article, we obtained promising covariance expressions tailored to this equation: we now explore the potential applications of these formulas. We first study the particular cases of stationarity and radial symmetry, for which significant simplifications arise. We next show that the true-angle multilateration method for point source localization, as used in GPS systems, is naturally recovered by our GPR formulas in the limit of the small source radius. Additionally, we show that this GPR framework provides a new answer to the ill-posed inverse problem of reconstructing initial conditions for the wave equation from a limited number of sensors, and simultaneously enables the inference of physical parameters from these data. We finish by illustrating this “physics informed” GPR on a number of practical examples.

Existence and Uniqueness Theorems in the Inverse Problem of Recovering Surface Fluxes from Pointwise Measurements

Inverse problems of recovering surface fluxes on the boundary of a domain from pointwise observations are considered. Sharp conditions on the data ensuring existence and uniqueness of solutions in Sobolev classes are exposed. They are smoothness conditions on the data, geometric conditions on the location of measurement points, and the boundary of a domain. The proof relies on asymptotics of fundamental solutions to the corresponding elliptic problems and the Laplace transform. The inverse problem is reduced to a linear algebraic system with a nondegerate matrix.

Imprint of chaotic ocean variability on transports in the southwestern Pacific at interannual timescales

Abstract. The southwestern Pacific Ocean sits at a bifurcation where southern subtropical waters are redistributed equatorward and poleward by different ocean currents. The processes governing the interannual variability of these currents are not completely understood. This issue is investigated using a probabilistic modeling strategy that allows disentangling the atmospherically forced deterministic ocean variability and the chaotic intrinsic ocean variability. A large ensemble of 50 simulations performed with the same ocean general circulation model (OGCM) driven by the same realistic atmospheric forcing and only differing by a small initial perturbation is analyzed over 1980–2015. Our results show that, in the southwestern Pacific, the interannual variability of the transports is strongly dominated by chaotic ocean variability south of 20∘ S. In the tropics, while the interannual variability of transports and eddy kinetic energy modulation are largely deterministic and explained by the El Niño–Southern Oscillation (ENSO), ocean nonlinear processes still explain 10 % to 20 % of their interannual variance at large scale. Regions of strong chaotic variance generally coincide with regions of high mesoscale activity, suggesting that a spontaneous inverse cascade is at work from the mesoscale toward lower frequencies and larger scales. The spatiotemporal features of the low-frequency oceanic chaotic variability are complex but spatially coherent within certain regions. In the Subtropical Countercurrent area, they appear as interannually varying, zonally elongated alternating current structures, while in the EAC (East Australian Current) region, they are eddy-shaped. Given this strong imprint of large-scale chaotic oceanic fluctuations, our results question the attribution of interannual variability to the atmospheric forcing in the region from pointwise observations and one-member simulations.

Finite element modelling of geophysical electromagnetic data with goal-oriented hr-adaptivity

Large discontinuities (jumps) in coefficients often appear when modelling physical problems involving inhomogeneous media. An example of this is the geophysical electromagnetic (EM) problem, where these jumps occur at interfaces which separate regions of (high) conductivity contrasts. These interfaces, along with other problem features, such as singular EM sources, motivate the use of adaptive mesh refinement to improve the efficiency of the solution of forward problems. Also, pointwise observations are made in geophysical EM surveys, which motivates the use of goal-oriented mesh adaptivity. In this work, we study the combined application of h- and r-refinements for the modelling of geophysical EM data. The proposed hr-adaptivity algorithm is thoroughly investigated in 1D, and then extended, and numerically validated, in 2D, using simple examples as well as a realistic model with irregular interfaces. In 1D, the steady-state diffusion and Helmholtz equations, which are commonly solved for the EM scalar and vector potentials, respectively [16], constitute the physical partial differential equations (PPDEs). In 2D, the Helmholtz equation is used as the PPDE. Additionally, a real 2D problem with a benchmark model is considered where the transverse electric (TE) mode of Maxwell’s equations is used as the PPDE. The r-refinements are based on an equidistribution principle and variational methods, for the 1D and 2D cases, respectively, which lead to mesh PDEs (MPDEs). The coupled PPDE and MPDE are solved in an iterative manner to enhance the accuracy of the PPDE solution by improving the equidistribution of a monitor function based on a posteriori error estimates. The results, in 1D, with two hierarchical error–based monitor functions and a residual error–based monitor function display the similarity between these functions in terms of the accuracy that could be achieved. In both 1D and 2D, comparisons are made between global and goal-oriented adaptivity which show the advantage of goal-oriented error estimates in gaining higher accuracy at a target, compared to global error estimates. In 2D, the results also demonstrate the higher efficiency of hr-adaptivity compared with pure h-refinement, in terms of computation time.

Impact of bias correction and downscaling through quantile mapping on simulated climate change signal: a case study over Central Italy

Quantile mapping (QM) represents a common post-processing technique used to connect climate simulations to impact studies at different spatial scales. Depending on the simulation-observation spatial scale mismatch, QM can be used for two different applications. The first application uses only the bias correction component, establishing transfer functions between observations and simulations at similar spatial scales. The second application includes a statistical downscaling component when point-scale observations are considered. However, knowledge of alterations to climate change signal (CCS) resulting from these two applications is limited. This study investigates QM impacts on the original temperature and precipitation CCSs when applied according to a bias correction only (BC-only) and a bias correction plus downscaling (BC + DS) application over reference stations in Central Italy. BC-only application is used to adjust regional climate model (RCM) simulations having the same resolution as the observation grid. QM BC + DS application adjusts the same simulations to point-wise observations. QM applications alter CCS mainly for temperature. BC-only application produces a CCS of the median ~ 1 °C lower than the original (~ 4.5 °C). BC + DS application produces CCS closer to the original, except over the summer 95th percentile, where substantial amplification of the original CCS resulted. The impacts of the two applications are connected to the ratio between the observed and the simulated standard deviation (STD) of the calibration period. For the precipitation, original CCS is essentially preserved in both applications. Yet, calibration period STD ratio cannot predict QM impact on the precipitation CCS when simulated STD and mean are similarly misrepresented.

The role of land surface fluxes in Saudi-KAU AGCM: Temperature climatology over the Arabian Peninsula for the period 1981–2010

A new version of the Community Land Model (CLM) was introduced to the Saudi King Abdulaziz University Atmospheric Global Climate Model (Saudi-KAU AGCM) for better land surface component representation, and so to enhance climate simulation. CLM replaced the original land surface model (LSM) in Saudi-KAU AGCM, with the aim of simulating more accurate land surface fluxes globally, but especially over the Arabian Peninsula. To evaluate the performance of Saudi-KAU AGCM, simulations were completed with CLM and LSM for the period 1981–2010. In comparison with LSM, CLM generates surface air temperature values that are closer to National Centre for Environmental Prediction (NCEP) observations. The global annual averages of land surface air temperature are 9.51, 9.52, and 9.57°C for NCEP, CLM, and LSM respectively, although the same atmospheric radiative and surface forcing from Saudi-KAU AGCM are provided to both LSM and CLM at every time step. The better temperature simulations when using CLM can be attributed to the more comprehensive plant functional type and hierarchical tile approach to the land cover type in CLM, along with better parameterization of upward land surface fluxes compared to LSM. At global scale, CLM exhibits smaller annual and seasonal mean biases of temperature with respect to NCEP data. Moreover, at regional scale, CLM demonstrates reasonable seasonal and annual mean temperature over the Arabian Peninsula as compared to the Climatic Research Unit (CRU) data. Finally, CLM generated better matches to single point-wise observations of surface air temperature and surface fluxes for some case studies.

Some applications of weighted norm inequalities to the error analysis of PDE-constrained optimization problems

The purpose of this work is to illustrate how the theory of Muckenhoupt weights, Muckenhoupt weighted Sobolev spaces and the corresponding weighted norm inequalities can be used in the analysis and discretization of PDE constrained optimization problems. We consider: a linear quadratic constrained optimization problem where the state solves a nonuniformly elliptic equation; a problem where the cost involves pointwise observations of the state and one where the state has singular sources, e.g. point masses. For all three examples we propose and analyze numerical schemes and provide error estimates in two and three dimensions. While some of these problems might have been considered before in the literature, our approach allows for a simpler, Hilbert space-based, analysis and discretization and further generalizations.

Determination of the initial condition in parabolic equations from integral observations

We study the problem of determining the initial condition in parabolic equations with time-dependent coefficients from integral observations which can be regarded as generalizations of point-wise interior observations. Our approach is new in the sense that for determining the initial condition we do not assume that the data available in the whole space domain at the final moment or in a subset of the space domain during a certain time interval, but some integral observations during a time interval. We propose a variational method in combination with Tikhonov regularization for solving the problem and then discretize it by finite difference splitting methods. The discretized minimization problem is solved by the conjugate gradient method and tested on computer to show its efficiency. Also as a by-product of the variational method, we propose a numerical scheme for estimating the degree of ill-posedness of the problem.

How well would modern‐day oceanic property distributions be known with paleoceanographic‐like observations?

Author(s): Gebbie, G; Streletz, GJ; Spero, HJ | Abstract: Compilations of paleoceanographic observations for the deep sea now contain a few hundred points along the oceanic margins, mid-ocean ridges, and bathymetric highs, where seawater conditions are indirectly recorded in the chemistry of buried benthic foraminiferal shells. Here we design an idealized experiment to test our predictive ability to reconstruct modern-day seawater properties by considering paleoceanographic-like data. We attempt to reconstruct the known, modern-day global distributions by using a state estimation method that combines a kinematic tracer transport model with observations that have paleoceanographic characteristics. When a modern-like suite of observations (Θ, practical salinity, seawater δ18O, δ13CDIC, PO4, NO3, and O2) is used from the sparse paleolocations, the state estimate is consistent with the withheld data at all depths below 1500 m, suggesting that the observational sparsity can be overcome. Physical features, such as the interbasin gradients in deep δ13CDIC and the vertical structure of Atlantic δ13CDIC, are accurately reconstructed. The state estimation method extracts useful information from the pointwise observations to infer distributions at the largest oceanic scales (at least 10,000 km horizontally and 1500 m vertically) and outperforms a standard optimal interpolation technique even though neither dynamical constraints nor constraints from surface boundary fluxes are used. When the sparse observations are more realistically restricted to the paleoceanographic proxy observations of δ13C, δ18O, and Cd/Ca, however, the large-scale property distributions are no longer recovered coherently. At least three more water mass tracers are likely needed at the core sites in order to accurately reconstruct the large-scale property distributions of the Last Glacial Maximum.

Abstract State-Space Models for a Class of Linear Hyperbolic Systems of Balance Laws

The paper discusses and compares different abstract state-space representations for a class of linear hyperbolic systems defined on a one-dimensional spatial domain. It starts with their PDE representation in both weakly and strongly coupled forms. Next, the homogeneous state equation including the unbounded formal state operator is presented. Based on the semigroup approach, some results of well-posedness and internal stability are given. The boundary and observation operators are introduced, assuming a typical configuration of boundary inputs as well as pointwise observations of the state variables. Consequently, the homogeneous state equation is extended to the so-called boundary control state/signal form. Next, the classical additive statespace representation involving (A, B, C)-triple of state, input and output operators is considered. After short discussion on the appropriate Hilbert spaces, state-space equation in the so-called factor form is also presented. Finally, the resolvent of the system state operator A is discussed.

Characterisation and predictability of a strong and a weak forcing severe convective event – a multi-data approach

Two severe summer-time convective events in Germany are investigated which can be classified by the prevailing synoptic conditions into a strong and a weak forcing case. The strong forcing case exhibits a larger scale precipitation pattern caused by frontal ascent whereas scattered convection is dominating the convective activity in the weak forcing case. Other distinguished differences between the cases are faster movement of convective cells and larger regions with significant loss mainly due to severe gusts in the strong forcing case. A comprehensive set of various observations is used to characterise the two different events. The observations include measurements from a lightning detection network, precipitation radar, geostationary satellite and weather stations, as well as information from an automated cell detection algorithm based on radar reflectivity which is combined with severe weather reports, and damage data from insurances. Forecast performance at various time scales is analysed ranging from nowcasting and warning to short-range forecasting. Various methods and models are examined, including human warnings, observation-based nowcasting algorithms and high-resolution ensemble prediction systems. The analysis shows the advantages of a multi-sensor and multi-source approach in characterising convective events and their impacts. Using data from various sources allows to combine the different strengths of observational data sets, especially in terms of spatial coverage or data accuracy, e.g. damage data from insurances provide good spatial coverage with little meteorological information while measurements at weather stations provide accurate but pointwise observations. Furthermore, using data from multiple sources allow for a better understanding of the convective life cycle. Several parameters from different instruments are shown to have a predictive skill for convective development, these include satellite-based cloud-top cooling rates as measure for intensive convective growth, 3D-radar reflectivity, mesocyclone detection from doppler radar, overshooting top detection or lightning jumps to evaluate storm intensification and formation of severe weather. This synergetic approach can help to improve nowcasting algorihtms and thus the warning process. The predictability of the analysed severe convective events differs with different types of forcing which is reflected in both, convective-scale ensemble prediction system forecasts and human weather warnings. Human warnings show larger false alarm rates in the weak forcing case. Ensemble predictions are able to capture the characteristics of the convective precipitation. The forecast skill is connected strongly to the synoptic situation and the presence of large-scale forcing increases the forecast skill. This has to be considered for potential future warn-on-forecast strategies.

Why Did Large Differences Arise in the Sea Surface Temperature Datasets across the Tropical Pacific during 2012?

AbstractDuring June–November 2012, pronounced differences in tropical Pacific sea surface temperature (SST) anomalies were observed between three widely used SST products: the extended reconstructed SST version 3b (ERSSTv3b), and the optimum interpolation SST version 2 analyses (OISST), produced weekly (OISSTwk) and daily (OISSTdy). During June–August 2012, the Niño-3.4 SST anomaly (SSTA) index was 0.2°–0.3°C lower in ERSSTv3b than in OISSTwk and OISSTdy, while it was 0.3°–0.4°C higher from September to November 2012. Such differences in the Niño-3.4 SSTA index can impact the assessment of the status of the El Niño–Southern Oscillation, which is determined using a threshold of ±0.5°C in the Niño-3.4 SSTA index.To investigate the reasons for the differences between ERSSTv3b and OISSTdy/OISSTwk, an experimental analysis (called ERSSTsat) is created that is similar to ERSSTv3b but includes satellite-derived SSTs. However, significant differences in the Niño-3.4 SSTA index remained between ERSSTsat and OISSTdy/OISSTwk. Comparisons of ERSSTsat and OISSTdy indicate that their differences are mostly associated with the different schemes for bias adjustment applied to the satellite-based SSTs. It is therefore suggested that the differences in the Niño-3.4 SSTA index between ERSSTv3b and OISSTdy cannot be solely due to the inclusion of but by the bias adjustment methodology of satellite data in OISSTdy.Finally, the SST products are compared with observations from ships, buoys, and satellites. On the monthly time scale, the area-averaged Niño-3.4 SSTA index in the tropical Pacific is more consistent with in situ observations in ERSSTv3b than in OISSTdy. In contrast, pointwise observations across the tropical Pacific are more consistent with OISSTdy than ERSSTv3b. It is therefore suggested that the differences among SST products are partially due to a structural uncertainty of various SST estimates.

Numerical analysis for the approximation of optimal control problems with pointwise observations

In this paper, we study the numerical methods for optimal control problems governed by elliptic PDEs with pointwise observations of the state. The first order optimality conditions as well as regularities of the solutions are derived. The optimal control and adjoint state have low regularities due to the pointwise observations. For the finite dimensional approximation, we use the standard conforming piecewise linear finite elements to approximate the state and adjoint state variables, whereas variational discretization is applied to the discretization of the control. A priori and a posteriori error estimates for the optimal control, the state and adjoint state are obtained. Numerical experiments are also provided to confirm our theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.

Force traction microscopy: An inverse problem with pointwise observations

Force traction microscopy is an inversion method that allows one to obtain the stress field applied by a living cell on the environment on the basis of a pointwise knowledge of the displacement produced by the cell itself. This classical biophysical problem, usually addressed in terms of Green functions, can be alternatively tackled using a variational framework and then a finite elements discretization. In such a case, a variation of the error functional under suitable regularization is operated in view of its minimization. This setting naturally suggests the introduction of a new equation, based on the adjoint operator of the elasticity problem. In this paper we illustrate the rigorous theory of the two-dimensional and three-dimensional problems, involving in the former case a distributed control and in the latter case a surface control. The pointwise observations require one to exploit the theory of elasticity extended to forcing terms that are Borel measures.

Uniqueness from pointwise observations in a multi-parameter inverse problem

In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. When the reaction term contains an unknown polynomial part of degree $N,$ with non-constant coefficients $\mu_k(x),$ our result gives a sufficient condition for the uniqueness of the determination of this polynomial part. This sufficient condition only involves pointwise measurements of the solution $u$ of the reaction-diffusion equation and of its spatial derivative $\partial u / \partial x$ at a single point $x_0,$ during a time interval $(0,\varepsilon).$ In addition to this uniqueness result, we give several counter-examples to uniqueness, which emphasize the optimality of our assumptions. Finally, in the particular cases $N=2$ and $N=3,$ we show that such pointwise measurements can allow an efficient numerical determination of the unknown polynomial reaction term.

An augmented BV setting for feedback switching control

This paper considers dynamical systems under feedback with control actions limited to switching. The authors wish to understand the closed-loop systems as approximating multi-scale problems in which the implementation of switching merely acts on a fast scale. Such hybrid dynamical systems are extensively studied in the literature, but not much so far for feedback with partial state observation. This becomes in particular relevant when the dynamical systems are governed by partial differential equations. The authors introduce an augmented BV setting which permits recognition of certain fast scale effects and give a corresponding well-posedness result for observations with such minimal regularity. As an application for this setting, the authors show existence of solutions for systems of semilinear hyperbolic equations under such feedback with pointwise observations.

Detection of Snowmelt Using Spaceborne Microwave Radiometer Data in Eurasia From 1979 to 2007

Determining the date of snowmelt clearance is an important issue for hydrological and climate research. Spaceborne radiometers are ideally suited for global snowmelt monitoring. In this paper, four different algorithms are used to determine the snowmelt date from Scanning Multichannel Microwave Radiometer and Special Sensor Microwave/Imager data for a nearly 30-year period. Algorithms are based on thresholding channel differences, on applying neural networks, and on time series analysis. The results are compared with ground-based observations of snow depth and snowmelt status available through the Russian INTAS-SSCONE observation database. Analysis based on Moderate Resolution Imaging Spectroradiometer data indicates that these pointwise observations are applicable as reference data. The obtained error estimates indicate that the algorithm based on time series analysis has the highest performance. Using this algorithm, a time series of the snowmelt from 1979 to 2007 is calculated for the whole Eurasia showing a trend of an earlier snow clearance. The trend is statistically significant. The results agree with earlier research. The novelty here is the demonstration and validation of estimates for a large continental scale (for areas dominated by boreal forests) using extensive reference data sets.

The impact of spatial correlation and incommensurability on model evaluation

Standard evaluations of air quality models rely heavily on a direct comparison of monitoring data matched with the model output for the grid cell containing the monitor's location. While such techniques may be adequate for some applications, conclusions are limited by such factors as the sparseness of the available observations (limiting the number of grid cells at which the model can be evaluated) and the incommensurability between volume-averages and pointwise observations. We examine several sets of simulations to illustrate the effect of incommensurability in a variety of cases distinguished by the type and extent of spatial correlation present. Block kriging, a statistical method which can be used to address the issue, is then demonstrated using the simulations. Lastly, we apply this method to actual data and discuss the practical importance of understanding the impact of spatial correlation structure and incommensurability.

A hierarchical Bayesian approach to the spatio-temporal modeling of air quality data

The statistical evaluation of an air quality model is part of a broader process, generally referred to as ‘model assessment’, including sensitivity analysis and other tools. The evaluation process is usually implemented through the comparison of model predicted data with point-wise observations. However, this analysis is based on several (implicit) assumptions which are difficult, if not impossible, to assess: e.g. unbiased observations, measurements errors small enough in comparison to the typical usage of observed data, observations representative of the true area-averaged values within each computational cell, numerical model errors small enough in comparison to mis/un-represented physics/chemistry, and so on. In this work we address the problem of the comparison between point measured data and cell-averaged model values. We present a Bayesian approach for the space-time interpolation of measured data and the prediction of cell-averaged values. We used cell-averaged observations to validate the results from the CAMx air quality model. We found that a relevant fraction of the model bias can be explained by the subgrid spatial variability. This analysis may be important in all cases in which one is interested in a model and/or process comparison exercise.

ε-Insensitizing controls for pointwise observations of the heat equation

In this paper, we study the ε-insensitizing controllability for the functional given by integrating in time the square of the solution of the heat equation in a finite number of points of the domain Ω⊂ R N , i.e., when the observation set is reduced to a finite set of points. We reduce the controllability problem to a unique continuation property for a cascade system of linear heat equations and we obtain a positive result in the case N⩽3.