T. van Leeuwen

Response of the Amazon carbon balance to the 2010 drought derived with CarbonTracker South America

Two major droughts in the past decade had large impacts on carbon exchange in the Amazon. Recent analysis of vertical profile measurements of atmospheric CO2 and CO by Gatti et al. (2014) suggests that the 2010 drought turned the normally close-to-neutral annual Amazon carbon balance into a substantial source of nearly 0.5 PgC/yr, revealing a strong drought response. In this study, we revisit this hypothesis and interpret not only the same CO2/CO vertical profile measurements but also additional constraints on carbon exchange such as satellite observations of CO, burned area, and fire hot spots. The results from our CarbonTracker South America data assimilation system suggest that carbon uptake by vegetation was indeed reduced in 2010 but that the magnitude of the decrease strongly depends on the estimated 2010 and 2011 biomass burning emissions. We have used fire products based on burned area (Global Fire Emissions Database version 4), satellite-observed CO columns (Infrared Atmospheric Sounding Interferometer), fire radiative power (Global Fire Assimilation System version 1), and fire hot spots (Fire Inventory from NCAR version 1), and found an increase in biomass burning emissions in 2010 compared to 2011 of 0.16 to 0.24 PgC/yr. We derived a decrease of biospheric uptake ranging from 0.08 to 0.26 PgC/yr, with the range determined from a set of alternative inversions using different biomass burning estimates. Our numerical analysis of the 2010 Amazon drought results in a total reduction of carbon uptake of 0.24 to 0.50 PgC/yr and turns the balance from carbon sink to source. Our findings support the suggestion that the hydrological cycle will be an important driver of future changes in Amazonian carbon exchange.

A penalty method for PDE-constrained optimization in inverse problems

Many inverse and parameter estimation problems can be written as PDE-constrained optimization problems. The goal is to infer the parameters, typically coefficients of the PDE, from partial measurements of the solutions of the PDE for several right-hand sides. Such PDE-constrained problems can be solved by finding a stationary point of the Lagrangian, which entails simultaneously updating the parameters and the (adjoint) state variables. For large-scale problems, such an all-at-once approach is not feasible as it requires storing all the state variables. In this case one usually resorts to a reduced approach where the constraints are explicitly eliminated (at each iteration) by solving the PDEs. These two approaches, and variations thereof, are the main workhorses for solving PDE-constrained optimization problems arising from inverse problems. In this paper, we present an alternative method that aims to combine the advantages of both approaches. Our method is based on a quadratic penalty formulation of the constrained optimization problem. By eliminating the state variable, we develop an efficient algorithm that has roughly the same computational complexity as the conventional reduced approach while exploiting a larger search space. Numerical results show that this method indeed reduces some of the nonlinearity of the problem and is less sensitive to the initial iterate.

Anomalous carbon uptake in Australia as seen by GOSAT

One of the unanswered questions of climate change is how the biospheric uptake of carbon responds to events such as droughts and floods. Especially, semiarid regions have received interest recently, as they can respond very rapidly to changing environmental conditions. Here we report on a large enhanced carbon sink over Australia from the end of 2010 to early 2012 detected using the Greenhouse Gases Observing SATellite (GOSAT). This enhanced sink coincides with the strong La Nina episode, accompanied by record-breaking amounts of precipitation. This precipitation led to an enhanced growth of vegetation, resulting in large increases in biospheric carbon uptake in line with increased levels of vegetation fluorescence. An inversion based on the satellite retrievals confirms this strong anomaly in the sink of roughly 0.77 0.10PgCyr(-1) or 1.5 0.2PgC in total for the April 2010 to December 2011 period, which corresponds to 25% of the multiyear annual average gross primary production of the Australian biosphere.

Source Estimation for Frequency-domain FWI with Robust Penalties

Source estimation is an essential component of full waveform inversion. In the standard frequency- domain formulation, there is closed form solution for the the optimal source weights, which can thus be cheaply estimated on the fly. A growing body of work underscores the importance of robust modeling for data with large outliers or artifacts that are not captured by the forward model. Ef- fectively, the least-squares penalty on the residual is replaced by a robust penalty, such as Huber, Hybrid l1-l2 or Student’s t. As we will demonstrate, it is essential to use the same robust penalty for source estimation. In this abstract, we present a general approach to robust waveform inversion with robust source estimation. In this general formulation, there is no closed form solution for the optimal source weights so we need to solve a scalar optimization problem to obtain these weights. We can efficiently solve this optimization problem with a Newton-like method in a few iterations. The computational cost involved is of the same order as the usual least-squares source estimation procedure. We show numerical examples illustrating robust source estimation and robust waveform inversion on synthetic data with outliers.

A Hybrid Stochastic-deterministic Optimization Method for Waveform Inversion

Present-day high quality 3D acquisition can give us lower frequencies at longer offsets with which to invert. However, the computational costs involved in handling this data explosion are tremendous. Therefore, recent developments in full-waveform inversion have been geared towards reducing the computational costs involved. Recent attention has been drawn towards reducing the number of sources by randomly combining the sources in to a few supershots, but other strategies are also possible. In all cases, the full data misfit, which involves all the sequential sources, is replaced by a reduced misfit that is much cheaper to evaluate, but also less accurate. The optimization of such an inaccurate, or noisy, misfit is the topic of stochastic optimization. In this paper, we propose an optimization strategy that borrows ideas from this field. The strategy consists of starting with very few sources (low cost) and gradually increasing the accuracy of the misfit as the iterations proceed. We test the proposed strategy on a synthetic dataset. We achieve a very reasonable inversion result at the cost of roughly 13 evaluations of the full misfit and observe a speed-up of roughly a factor 20.

Fast Least-Squares Migration with Multiples and Source Estimation

The advent of modern computing has made it possible to do seismic imaging using least-squares reverse-time migration. We obtain superior images by solving an optimization problem that recovers the true-amplitude images. However, its success hinges on overcoming several issues, including overwhelming problem size, unknown source wavelet, and interfering coherent events like multiples. In this abstract, we reduce the problem size by using ideas from compressive sensing, and estimate source wavelet by generalized variable projection. We also demonstrate how to invert for subsurface information encoded in surface-related multiples by incorporating the free-surface operator as an areal source in reverse-time migration. With multiples we can remove the amplitude ambiguity in wavelet estimation. We demonstrate the efficacy of the proposed method with synthetic examples.

Wave-equation Based Inversion with the Penalty Method - Adjoint-state Versus Wavefield-reconstruction Inversion

In this paper we make a comparison between wave-equation based inversions based on the adjoint-state and penalty methods. While the adjoint-state method involves the minimization of a data-misfit and exact solutions of the wave-equation for the current velocity model, the penalty-method aims to first find a wavefield that jointly fits the data and honours the physics, in a least-squares sense. Given this reconstructed wavefield, which is a proxy for the true wavefield in the true model, we calculate updates for the velocity model. Aside from being less nonlinear--the acoustic wave equation is linear in the wavefield and model parameters but not in both--the inversion is carried out over a solution space that includes both the model and the wavefield. This larger search space allows the algorithm to circumnavigate local minima, very much in the same way as recently proposed model extensions try to accomplish. We include examples for low frequencies, where we compare full-waveform inversion results for both methods, for good and bad starting models, and for high frequencies where we compare reverse-time migration with linearized imaging based on wavefield-reconstruction inversion. The examples confirm the expected benefits of the proposed method.

Preconditioning the Helmholtz Equation via Row-projections

3D frequency-domain full waveform inversion relies on being able to efficiently solve the 3D Helmholtz equation. Iterative methods require sophisticated preconditioners because the Helmholtz matrix is typically indefinite. We review a preconditioning technique that is based on row-projections. Notable advantages of this preconditioner over existing ones are that it has low algorithmic complexity, is easily parallelizable and extendable to time-harmonic vector equations.

Affordable Full Subsurface Image Volume - An Application to WEMVA

Common image gathers are used in building velocity models, inverting for anisotropy parameters, and analyzing reservoir attributes. In this paper, we offer a new perspective on image gathers, where we glean information from the image volume via efficient matrix-vector products. The proposed formulation make the computation of full subsurface image volume feasible. We illustrate how this matrix-vector product can be used to construct objective functions for automatic MVA.

In which Domain Should We Measure the Misfit for Robust Full Waveform Inversion

Full-waveform inversion relies on minimizing the difference between observed and modeled data, as measured by some penalty function. A popular choice, of course, is the least-squares penalty. However, when outliers are present in the data, the use of robust penalties such as the Huber or Student’s t may significantly improve the results since they put relatively less weight on large residuals. In order for robust penalties to be effective, the outliers must be somehow localized and distinguishable from the good data. We propose to first transform the residual into a domain where the outliers are localized before measuring the misfit with a robust penalty. This is exactly how one would normally devise filters to remove the noise before applying conventional FWI. We propose to merge the two steps and let the inversion process implicitly filter out the noise. Results on a synthetic dataset show the effectiveness of the approach.