В работе представлен результат численного эксперимента по оценке источников и стоков атмосферных примесей на основе данных измерений концентраций и решения обратной задачи для модели адвекции-диффузии-реакции. В качестве данных взяты результаты измерений содержания в воздухе химических веществ, полученные во время корабельной экспедиции по озеру Байкал. Для задания процессов трансформации примесей в атмосфере использована система реакции на основе соотношений Лейтона. Оценка источников проводится с помощью алгоритма, основанного на операторах чувствительности и ансамблей решений сопряжённых уравнений. Постоянные по времени поля источников оксидов азота оцениваются по данным измерений озона. Вместе с оценками распределения источников получены оценки распределения полей концентрации.

With the help of a transport model, the issues of the distribution of neutrally floating particles simulating microplastic (MP) in Lake Baikal are studied. A three-dimensional model of the lake hydro-thermodynamics in the non-hydrostatic formulation is used to produce the fields of currents and the other necessary parameters for the transport model. To give generality to the conclusions, we use a set of parameters that are conditionally combined into hypothetical scenarios. In them, we design the "climatic" scenarios of external influences, in accordance to which the fields of currents and temperature in the hydrodynamic model are calculated. Depending of the goals of the study, we also formulate the specific tasks for giving the impurity sources in the transport model. The results of calculations are presented, in which some possible locations of an increased concentration of impurities in the lake were revealed.

В транспортной модели в Лагранжевой формулировке рассматривается распространение в озере Байкал нейтрально плавучих частиц, имитирующих микропластик. Трехмерная модель гидротермодинамики озера в негидростатическом приближении используется для задания необходимых параметров для транспортной модели. Для придания общности выводам мы используем набор параметров, которые условно объединены в «климатические» сценарии, по которым рассчитывается гидродинамический фон. Приведены примеры сценарных расчетов в Баргузинском заливе.

The joint use of atmospheric chemistry transport and transformation models and observational data makes it possible to solve a wide range of environment protection tasks, including pollution sources identification and reconstruction of the pollution fields in unobserved areas. Seamless usage of different measurement data types can improve the accuracy of air quality forecasting systems. The approach considered is based on sensitivity operators and adjoint equations solutions ensembles. The ensemble construction allows for the natural combination of various measurement data types in one operator equation. In the paper, we consider combining image-type, integral-type, pointwise, and time series-type measurement data for the air pollution source identification. The synergy effect is numerically illustrated in the inverse modeling scenario for the Baikal region.

Air quality monitoring systems differ in composition and accuracy of observations and their temporal and spatial coverage. A monitoring system’s performance can be assessed by evaluating the accuracy of the emission sources identified by its data. In the considered inverse modeling approach, a source identification problem is transformed to a quasi-linear operator equation with the sensitivity operator. The sensitivity operator is composed of the sensitivity functions evaluated on the adjoint ensemble members. The members correspond to the measurement data element aggregates. Such ensemble construction allows working in a unified way with heterogeneous measurement data in a single-operator equation. The quasi-linear structure of the resulting operator equation allows both solving and predicting solutions of the inverse problem. Numerical experiments for the Baikal region scenario were carried out to compare different types of inverse problem solution accuracy estimates. In the considered scenario, the projection to the orthogonal complement of the sensitivity operator’s kernel allowed predicting the source identification results with the best accuracy compared to the other estimate types. Our contribution is the development and testing of a sensitivity-operator-based set of tools for analyzing heterogeneous air quality monitoring systems. We propose them for assessing and optimizing observational systems and experiments.

A three-dimensional model of the hydrothermodynamics of a lake in a non-hydrostatic approximation and an advectiondiffusion model are used to describe the processes of distribution of impurities in the Barguzin Bay of Lake Baikal. To give generality to the conclusions, we use a set of parameters, which are conventionally combined into "climatic" scenarios, according to which the hydrodynamic background is calculated. The distribution of impurities in the bay, as in the entire lake, is completely determined by hydrodynamic processes. The main natural source of impurities in the bay, which also plays an important role in hydrodynamics, is the Barguzin River. In addition to this source, a set of point sources located on the shore of the bay is specified that simulate objects of the tourism industry. The results of the hypothetic scenarios on the propagation of impurities from a given configuration of impurity sources and their mode of operation are presented.

Air quality monitoring systems vary in temporal and spatial coverage, the composition of the observed chemicals, and the data's accuracy. The developed inverse modeling approach [1] is based on sensitivity operators and ensembles of adjoint equations solutions. An inverse problem is transformed to a quasi-linear operator equation with the sensitivity operator. The sensitivity operator is composed of the sensitivity functions, which are evaluated on the adjoint ensemble members. The members correspond to the measurement data elements.This ensemble construction allows working in a unified way with heterogeneous measurement data in a single operator equation. The quasi-linear structure of the resulting operator equation allows both solving and analyzing the inverse problem. More specifically, by analyzing the sensitivity operator's singular structure, we can estimate the informational content in the measurement data with respect to the considered process model. This type of analysis can estimate the inverse problem solution before its actual solution and evaluate the monitoring system efficiency with respect to the considered inverse modeling task [1,2].Numerical experiments with the emission source identification problem for air pollution transport and transformation model were carried out to illustrate the developed framework. In the numerical experiments, we considered in-situ, image-type, and integral-type measurement data.The work was supported by the grant №075-15-2020-787 in the form of a subsidy for a Major scientific project from Ministry of Science and Higher Education of Russia (project "Fundamentals, methods and technologies for digital monitoring and forecasting of the environmental situation on the Baikal natural territory").References[1] Penenko, A. Convergence analysis of the adjoint ensemble method in inverse source problems for advection-diffusion-reaction models with image-type measurements // Inverse Problems & Imaging, American Institute of Mathematical Sciences (AIMS), 2020, 14, 757-782 doi: 10.3934/ipi.2020035[2] Penenko, A.; Gochakov, A. & Penenko, V. Algorithms based on sensitivity operators for analyzing and solving inverse modeling problems of transport and transformation of atmospheric pollutants // IOP Conference Series: Earth and Environmental Science, IOP Publishing, 2020, 611, 012032 doi: 10.1088/1755-1315/611/1/012032

Using mathematical modeling methods, we study the possibility of spreading green filamentous algae Spirogyra cells from their original locations throughout the lake by currents. The simulation is performed using a three-dimensional model of lake hydrothermodynamics in a non-hydrostatic approximation and a model of impurity propagation against the background of hydrodynamic processes. Due to the significant uncertainties in the problem statement, a scenario approach is used to obtain possible solutions in the answers to the questions raised. An example of a hypothetical simulation scenario is given in which the impurity is distributed from sources located at four initial occurrence sites selected in different parts of the lake over two summer months.

Within the framework of many-year studies of the carbon cycle in the water-atmosphere system and in order to determine high-priority measures for conservation of the unique ecosystem of Lake Baikal, the specialized combined expedition was carried out in August 2019 in Barguzin and Chivyrkuy Bays of Lake Baikal. With the unique onboard instrumentation system, we have measured the gas content in the surface water and the near-water atmosphere, as well as concentrations of biogenic elements and organic matter at the sampling stations. The spatial distribution of the increased methane concentrations in the surface water of Barguzin Bay has been analyzed to determine the main direction of the Barguzin River’s water inflow into the bay. The observational data were compared with the results of scenario calculations by a large-scale model of currents in the lake. This model gives us some understanding of a general nature of water mass circulation in the bay. For modelling small-scale manifestations in the near shore zones, it is necessary to use higher resolution models for nested regions.

Variational approach and sensitivity theory methods are used to construct algorithms for solving the problems of environmental forecast and design. When studying the behavior of the model in the parameter space, sensitivity functions are calculated as partial derivatives of the target functionals with respect to the model parameters. The sensitivity functions are used to investigate the properties of mathematical models and solve inverse problems. Using the proposed approach, which involves a model of air quality of the Novosibirsk agglomeration and an algorithm based on an ensemble of sensitivity functions, the inverse problem of estimating the position and intensity of pollution sources is solved.

The inverse source problems for nonlinear chemical transport models with image-type measurement data are considered. The use of the sensitivity operators, constructed of the ensemble of adjoint problem solutions, allows transforming the inverse problems stated as the systems of nonlinear ODE or PDE to a family of operator equations depending on the given set of functions in the space of measurement results. In the paper, the set of consistent discrete analytical schemes for 1D diffusion-reaction model is presented. The operator equations are solved with the relevant methods for nonlinear operator equations.

Abstract This paper presents a description of algorithms for solving direct and inverse climatic and environmental problems based on a variational principle with weak constraints. The initial first-order system is supplemented by a system of equations in variations. It provides a calculation of all necessary components of the modeling system by combining the model and observation data and including first- and second-order functionals of sensitivity to variations in the model parameters, input data, and observation results.

Possible formulations of mathematical modeling problems intended for estimating transboundary transport are considered. Two approaches are possible. The first approach is related to direct modeling methods. It actually consists in obtaining forecasts of impurity distribution with all the necessary data to solve such problems. The second approach, using the methods of inverse modeling, makes it possible to obtain estimates of some functionals characterizing the desired solutions, for example, related to the search for possible sources of disturbances that led to transboundary transport. We develop both of these approaches based on the variational principle. Some remarks on both approaches are discussed.

A model of hydrodynamics of a two-phase liquid-solid system is formulated, in which forces of interphase dynamic interactions are taken into account. A scenario on distributions of insoluble solid particles emerging from an area source at the bottom of a lake in a stratified medium is considered. A calculation is performed in a three-dimensional statement taking into account the basic parameters of Lake Baikal. The results of the numerical simulation under winter conditions have shown that the two-phase system has a complex behaviour in space and time. The presence of solid particles strongly affects the hydrodynamics of the system and serves as a trigger for deep convection.

Formulations of mathematical models and statements of problems for investigating the behavior of multiphase systems under natural conditions are discussed. In particular, here we consider the hydrodynamics of the liquid-solid system in the context of possible surges of gas hydrates from the bottom of Lake Baikal. The results of numerical experiments on the simulation of hydrodynamic processes in a system where the source of disturbances is the income of solid particles from the bottom of the lake are presented. Parametrizations of the interphase dynamic exchange are studied.

A variational data assimilation algorithm is studied numerically. In situ concentration measurement data are assimilated into transport and transformation model of atmospheric chemistry. The algorithm is based on decomposition and splitting methods with solution of variational data assimilation problems for separate splitting stages. A direct algorithm without iterations is used for the linear transport stage. An iterative gradient algorithm is applied for data assimilation at the non-linear chemical transformation stage. In a realistic numerical experiment, the contributions of data assimilation algorithms for the different splitting stages are compared.

In continuation of the work on creating models of multiphase systems applied to the problems of Lake Baikal, a two-phase liquid-solid system is considered. The results of a numerical experiment of simulation of the ascent of solid particles from an area source at the lake bottom are analyzed.

Hydrophysical studies and mathematical modeling of ring structures during ice cover on Lake Baikal have shown that their existence at the stage of ice cover degradation is due to anticyclonic currents. Such currents can be generated as a result of local upwelling, which we associate with the rise of methane hydrates from the top layer of bottom sediments and their dissociation. Analysis of satellite images shows that the radii of ice rings range from 1300 to 2400 m, which is close to the baroclinic Rossby radius. The measured ice thicknesses in the area of the rings are in agreement with model calculations. Deep water renewal in Lake Baikal can also be associated with the rise of hydrates.