Inverse Days 2013

Tuesday, December 10, 2013 to Friday, December 13, 2013

10-13 December 2013

ban math2013 neutre


The 19th Inverse Days comprises of two events:

1. A thematic day on mathematics with applications in geospace and atmospheric research will be held in Sodankylä, Finland, 10 December.

2. Traditional Inverse Days will be held in Inari, Finland, 11-13 December. The conference venue is Sámi Cultural Centre Sajos in Inari. 

Inverse Days is the annual meeting of the Finnish Inverse Problems Society. It is part of the activities of Finnish Centre of Excellence in Inverse Problems Research.

A special theme of Inverse Days 2013 is the mathematics of Planet Earth, a project managed by UNESCO.


Examples of Finnish Inverse Problems Research

Matti Lassas and Samuli Siltanen, University of Helsinki, Finland

In collaboration with Maarten de Hoop of Purdue University and Gunther Uhlmann of University of Washington

Nonlinear seismic inversion 

The deep structure of Earth cannot be measured directly. Earthquake data is currently the best available source of information about the inner structure, but interpreting such data is a nonlinear and ill-posed inverse problem. While plenty of useful knowledge have been extracted already, there is room for improvement for example in determining the shape of the Earth's core. Our team approaches the problem by using adaptive Fourier transform methods based on so-called complex geometric optics solutions. 


Johanna Tamminen, Finnish Meteorological Institute

Satellite remote sensing has become an important method of studying the planet Earth. Remote sensing measurements are by definition indirect and the interpretation of the measurements requires solving an inverse problem, which is typically ill-posed. At the Finnish Meteorological Institute we are developing inversion algorithms for several satellite instruments, which are targeted for studying the composition of the atmosphere. One of the instruments is European Space Agency’s GOMOS instrument (2002-2012) that measures the composition of the middle atmosphere and the ozone layer in particular by observing stellar light coming through the atmosphere.

The satellite measurements include noise and other systematic errors and the inverse problem is challenging due to uncertainties in the physical modeling of the measured signal.  Bayesian formulation has become a powerful method for taking these uncertainties into account.  While the operational algorithms of the satellites need to be fast and robust, more detailed algorithms can be used for research purposes. In GOMOS inverse problem the modeling uncertainties have been studied using adaptive Markov chain Monte Carlo algorithms.


Mikko Kaasalainen, Tampere University of Technology, Finland

Forests are important for the modern world for more reasons than earlier. Today, information on forest resources and functions is crucial to ecological, economical, societal and climatological assessments. The current methodologies of analysing forests meet the growing information needs only partially.

Our aim is to redefine the current methodology of collecting and representing essential information on forest resources and functioning. We will implement the recently developed techniques on tree measurements to produce a completely new, functional approach to forest modelling. This research is based on a new concept of synergy among the latest developments in 1) in situ remote sensing technologies and 2) mathematical modelling, combined with understanding on 3) tree eco-physiology and 4) forest soil science.

This synergistic combination of expertise provides means to revolutionize our knowledge on forests. The new high-tech remote sensing technologies can measure features that could not be measured earlier, replace earlier destructive measurements with non-destructive ones, and increase the old measurement rates by orders of magnitude. We will use such measurements, together with a priori information on trees and soil, as input to mathematical modelling employing the most advanced methods of inverse problems and dynamic systems.

We will demonstrate the value of the methodology by applications, such as the first 3D estimates of photosynthesis of tree canopies over a growing season and 3D growth and mortality of tree biomass in response to forest management practices. We will disseminate the methodology as measurement and modelling procedures and databases. This research will have a significant impact and open up new avenues of research in any field where forest structure or functions related to it are important, such as most forest sciences, ecology, eco-physiology, forestry, forest monitoring and inventory, forest carbon sequestration, and climate change studies.

Highlights: With the new analysis methods for TLS scans, there will be a growing and improving database of 3D descriptions of trees and forest stands. The attributes determining these descriptions can be represented as Bayesian probability distributions, with functional-structural models providing the a priori information. These distributions can then be used to create versions of new realistic Bayes forests, where none of the trees are copied from data, but the structure of each is drawn from the data-based distributions. Repeated TLS measurements add a fourth dimension, time, to the mathematical modelling; in this way, we can simulate functional 4D Bayes forests. As in the modelling of the 3D structure, forest models and regularities of growth and mortality are used as a priori information, and the accumulating modelling results improve the measurements and the a priori information. We formulate the theoretical framework of the Bayes forest method and present some examples.


Mikko Salo, University of Jyväskylä, Finland 

The inverse problems group at the University of Jyväskylä focuses on fundamental theoretical aspects of inverse problems such as the Calderón problem in electrical imaging and travel time tomography in seismic imaging. Both methods are related to the study of Planet Earth: electrical impedance tomography is used in geophysical prospection, and travel time tomography in the imaging of deep Earth. In particular, the Jyväskylä group has contributed to the theory of electrical imaging in anisotropic media, and has provided new methods for tensor tomography problems arising in travel time imaging.


Timo Lähivaara, University of Eastern Finland

In collaboration with Tomi Huttunen, Jari P. Kaipio, Nicholas F. Dudley Ward

New approaches for estimating groundwater resources

Despite the importance of groundwater resources, knowledge about groundwater reservoir is usually very poor. Traditional approaches to aquifer (porous layer that stores water) mapping and characterization are too time consuming and costly for large scale application. The aim of this research is to develop innovative numerical techniques that can be applied for  monitoring  the state of a groundwater reservoir. The new method uses seismic data  recorded from microearthquakes (i.e. passive seismic monitoring). Research focuses on developing  efficient computational techniques  (such as the discontinuous Galerkin method and spectral element method) for solving time-depended seismic wave fields in porous media (forward problem) and the Bayesian framework for solving the inverse problem associated with the seismic mapping of the aquifer.


Aku Seppänen, University of Eastern Finland

Bayesian Inversion Approach to Tree Detection Based on Airborne Laser Scanning Data

Global and local environmental changes are modifying forest ecosystems rapidly. The consequences of these changes on biodiversity, functioning of forests, and provision of ecosystem services are unclear and remain as major scientific challenges. Recently, airborne laser scanning (ALS) has become the most accurate remote sensing (RS) method for forest applications. Especially, in the area of forest inventory, the development of the ALS based methods has been very active. We develop new computational methods for interpreting the ALS data. Especially, we focus on automatic detection and shape estimation of individual trees. The ALS-based estimates for tree locations, heights and shapes can further be utilized for the inference of other quantities, such as total trunk volume and biomass of the forest. Our approach to tree detection problem is to fit automatically multiple 3D crown height models to ALS data. The estimation problems are considered in Bayesian inversion framework, which allows for modeling of uncertainties and utilizing prior information on tree shape in the estimation. Our recent results have indicated that the use of the developed computational method in the data interpretation can improve the reliability of the tree detection significantly: in our studies, the success rate with the Bayesian inversion approach was about 20 percent units higher than with a conventional computational method used for interpreting the ALS data.


Valery Serov, University of Oulu, Finland

Scattering theory and wave propagation for nonlinear PDEs

Our main attention is the direct and inverse scattering theory for general nonlinear Schrödinger equations.  We study also nonlinear Maxwell's equations with absorption. Both subjects are related to many practical problems such as geophysical prospection, where one wants to detect and locate resistive areas in a conductive background. Here, the justification of the Born approximation is of central importance. Another application concerns nonlinear optics. This part of our research is done in collaboration with H.-W. Schuermann from the University of Osnabrueck, Germany.


Jouni Susiluoto, Finnish Meteorological Institute

Parameter estimation in climate models with Markov chain Monte Carlo (McMC) methods

Climate models, being complex, strongly non-linear, and computationally extremely demanding, contain simplifications of processes for generally two reasons: a) we don't have the computational capacity to model everything, and b) even if we did, we really do not know all the processes on earth. This leads scientists to formulate parametrizations of e.g. cloud or hydrology-related phenomena.

Being simplifications, these parametrizations do not represent any physical thing, rather they represent something model-internal and non-physical, of which we do not have measurements. For instance, we might have one parameter for some physical process in a cloud, yet all clouds are different and should have different parameter values for some processes. Hence validating values cannot be done in a straightforward manner.

Using McMC along with data assimilation techniques, we aim to improve hydrological cycle-related parametrizations in ECHAM climate model and its land component JSBACH. The techniques have the potential to systematically improve climate models' predictive skill.


Heikki Haario, Lappeenranta University of Technology

Antti Solonen, Lappeenranta University of Technology and Finnish Meteorological Institute

Novel estimation methods for weather and climate models

Climate models and numerical weather prediction (NWP) models are central tools for understanding the behavior of the Earth's atmosphere. Both models are governed by the equations of atmospheric flow, but the models differ in the regimes they are used. NWP models are used to make real-time short-term predictions of the weather, while climate models are used to analyze the overall behavior of the climate over long time periods to study, for instance, climate change.

There are several sources of uncertainty in climate and NWP models. Both models are described as partial differential equations, which are solved numerically in a discretized form using a computational grid. However, many important processes, such as cloud formation, operate in much smaller scales than the grid size. These processes are included in the models as so called closure parameters. Defining the numerical values for them is difficult, and is bound to introduce parametric uncertainties. The LUT group develops algorithms for tuning these parameters and evaluating the uncertainty related to them. The methods include adaptive parallel MCMC with early and delayed rejection, as well as special focus on formulations of the likelihoods arising from chaotic dynamics.  Another approach is to tune weather model parameters on-line, by monitoring the performance of operational ensemble predictions. We work in tight collaboration with the Finnish Meteorological Institute (FMI). Our algorithms have also been implemented in the operational forecasting system of one of the leading global NWP centers, the European Center of Medium Range Weather Forecasts (ECMWF).

In NWP, the goal is to predict the near future weather. For this purpose, the state of the current weather, which acts as the initial value for the prediction, needs to be known as accurately as possible. In NWP, this is done via dynamical state estimation methods, also known as data assimilation. A state-of-art approach is to use optimization based algorithms (4D-VAR), while the computational requirements of standard Kalman filtering methods have been prohibitively high for the large-scale systems of NWP models. We develop approximative low memory filtering  approaches that are applicable for such high-dimensional problems. The developed algorithms are currently being scaled up towards realistic NWP systems, in collaboration with FMI and ECMWF.

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Last updated: 26.10.2016