Towards a reliable implementation of least-squares collocation for higher index differential-algebraic equations—Part 2: the discrete least-squares problem

In the two parts of the present note we discuss questions concerning the implementation of overdetermined least-squares collocation methods for higher index differential-algebraic equations (DAEs). Since higher index DAEs lead to ill-posed problems in natural settings, the discrete counterparts are expected to be very sensitive, which attaches particular importance to their implementation. We provide in Part 1 a robust selection of basis functions and collocation points to design the discrete problem whereas we analyze the discrete least-squares problem and substantiate a procedure for its numerical solution in Part 2.

An application of the Marchenko internal multiple elimination scheme formulated as a least-squares problem

Seismic images produced by migration of seismic data related to complex geologies, suchas pre-salt environments, are often contaminated by artifacts due to the presence of multipleinternal reflections. These reflections are created when the seismic wave is reflected morethan once in a source-receiver path and can be interpreted as the main coherent noise inseismic data. Several schemes have been developed to predict and subtract internal multiplereflections from measured data, such as the Marchenko multiple elimination (MME) scheme,which eliminates the referred events without requiring a subsurface model or an adaptivesubtraction approach. The MME scheme is data-driven, can remove or attenuate mostof these internal multiples, and was originally based on the Neumann series solution ofMarchenko’s projected equations. However, the Neumann series approximate solution isconditioned to a convergence criterion. In this work, we propose to formulate the MMEas a least-squares problem (LSMME) in such a way that it can provide an alternative thatavoids a convergence condition as required in the Neumann series approach. To demonstratethe LSMME scheme performance, we apply it to 2D numerical examples and compare theresults with those obtained by the conventional MME scheme. Additionally, we evaluatethe successful application of our method through the generation of in-depth seismic images,by applying the reverse-time migration (RTM) algorithm on the original data set and tothose obtained through MME and LSMME schemes. From the RTM results, we show thatthe application of both schemes on seismic data allows the construction of seismic imageswithout artifacts related to internal multiple events.

A General Solution to the Rank Deficient Integer Least Squares and its Application to GNSS Positioning

<p>The contribution will present a general solution for the estimation of rank deficient integer parameters. A procedure will be presented that allows the computation of integer estimable function for any integer rank deficient least squares problem. The procedure is then applied to GNSS estimation problems. In the framework of undifferenced and uncombined GNSS models, the specific solution to some rank deficient integer least squares model will be presented, namely: the choice of pivot ambiguities in a network of receivers, GLONASS positioning, codeless positioning in the presence of ionospheric delay, satellite specific pseudorange biases estimation in the presence of ionospheric delay. It will been shown how the developed theory generalize previous results and ad hoc solutions present in the literature. Numerical results from real GNSS data will be presented too.</p>