This second edition of Mathematical Geosciences book adds five new topics: Solution equations with uncertainty, which proposes two novel methods for solving nonlinear geodetic equations as stochastic variables when the parameters of these equations have uncertainty characterized by probability distribution. The first method, an algebraic technique, partly employs symbolic computations and is applicable to polynomial systems having different uncertainty distributions of the parameters. The second method, a numerical technique, uses stochastic differential equation in Ito form; Nature Inspired Global Optimization where Meta-heuristic algorithms are based on natural phenomenon such as Particle Swarm Optimization. This approach simulates, e.g., schools of fish or flocks of birds, and is extended through discussion of geodetic applications. Black Hole Algorithm, which is based on the black hole phenomena is added and a new variant of the algorithm code is introduced and illustrated based on examples; The application of the Gröbner Basis to integer programming based on numeric symbolic computation is introduced and illustrated by solving some standard problems; An extension of the applications of integer programming solving phase ambiguity in Global Navigation Satellite Systems (GNSSs) is considered as a global quadratic mixed integer programming task, which can be transformed into a pure integer problem with a given digit of accuracy. Three alternative algorithms are suggested, two of which are based on local and global linearization via McCormic Envelopes; and Machine learning techniques (MLT) that offer effective tools for stochastic process modelling. The Stochastic Modelling section is extended by the stochastic modelling via MLT and their effectiveness is compared with that of the modelling via stochastic differential equations (SDE). Mixing MLT with SDE also known as frequently Neural Differential Equations is also introduced and illustrated by an image classification via a regression problem.

Professor Joseph Awange joined Spatial Sciences (School of Earth and Planetary Sciences, Curtin University, Australia) in 2006 under a Curtin Research Fellowship and concurrently undertook the prestigious Alexander von Humboldt (AvH) Fellowship at the Geodetic Institute (Karlsruhe Institute of Technology, Germany) having been awarded the Australian 2008-2011 Ludwig Leichhardt Memorial Fellowship for experienced researchers. In 2015, he won all the three major Fellowship Awards: Alexander von Humboldt (Germany), Japan Society of Promotion of Science (Japan) and Brazil Frontier of Science (Brazil) to carry out research in those countries. At Curtin University, he is currently a Professor of Environmental Geoinformatics engaged in teaching and research having attracted more than $2.5M worth of research grants. He obtained his BSc and MSc degrees in Surveying from the University of Nairobi (Kenya), and was also awarded a merit scholarship by the German Academic Exchange Program (DAAD), which facilitated his obtaining a second MSc degree and PhD in Geodesy at Stuttgart University (Germany). In 2002-2004, he was awarded the prestigious Japan Society for Promotion of Science (JSPS) Fellowship to pursue postdoctoral research at Kyoto University (Japan). Prof Awange attained International Editorial role in Springer Earth Science Books and has authored more than 20 scholarly books with the prestigious Springer International publishers and more than 200 peer-reviewed high impact journal publications (in e.g., Remote Sensing of Environment, Journal of Climate, Climatic Change, Advances in Water Resources, International Journal of Climatology, and Journal of Hydrology among others). His main research areas that have attracted media coverage (e.g., Environmental Monitor) are in the fields of (i) Environmental Geoinformatics: Satellite Environmental Sensing (e.g., changes in global and regional stored water (surface, underground, ice, and soil moisture) using GRACE/GRACE-FO and TRMM satellites; Climate Change using GNSS and altimetry satellites), which is employed to face the emerging challenges of the 21st century posed by increased extreme hydroclimatic conditions, e.g., severity and frequency of droughts in Australia and Greater Horn of Africa (GHA), and the changing monsoon characteristics in Asia and Africa leading to floods, and (ii) Mathematical Geosciences: Hybrid-symbolic solutions that delivers hybrid symbolic-numeric computations (HSNC), which is a large and growing area at the boundary of mathematics and computer science and currently an active area of research.

This book provides updated forms of the algorithms and new examples with high-quality visualizations added in every chapter. In addition, the five new following chapters have been added:

(1) Solution Equations with Uncertainty: (i) This chapter proposes two novel methods for computing the solutions of nonlinear geodetic equations as stochastic variables when the parameters of these equations have uncertainty characterized by probability distribution. The first method is based on an algebraic technique employing partly symbolic computations and can be employed for polynomial systems even with different uncertainty distributions of the parameters. The second method is a numerical one using stochastic differential equation in Ito form.

(2) Nature Inspired Global Optimization: Meta-heuristic algorithms are based on natural phenomenon such as Particle Swarm Optimization, which simulates schools of fish or flocks of birds, the Firefly Algorithm, Artificial Bee or Ant Colony, etc. Other aspects discussed include the Harmony Search, Black Hole Algorithm, Immunized Evolutionary Programming, Cuckoo Search, Levy flights, and random walks.

(3) Employing Gröbner Basis Theory: Gröbner bases are an important theoretical building block in modern (polynomial) ring theory. In this chapter, four practical applications of Gröbner basis theory are considered.

(4) Discrete Optimization in the Geosciences: Solving phase ambiguity in Global Navigation Satellite Systems (GNSSs) is considered as a global quadratic mixed integer programming task, which can be transformed into a pure integer problem with a given digit of accuracy. In this chapter, three alternative algorithms are suggested. Two of them are based on local and global linearization via McCormic Envelopes, respectively.

(5) Neural Networks: Neural Networks as a machine learning technique offer an effective tool for data processing (i.e., classification) as well as function approximation methods (i.e., regression). Following a brief introduction, this chapter demonstrates the most effective network algorithms, e.g., Hopfield network for image reconstruction, unsupervised net for fitting sphere to point cloud data, and recurrent network for time series prediction. Deep learning and convolutional neural networks algorithms and their applications are discussed, after which stochastic approximation methods like drop out and assembly learning techniques are demonstrated.

Solution of algebraic polynomial systems.- Homotopy solution of nonlinear systems.- Over and underdeterminated systems.- Simulated annealing.- Genetic algorithm.- Particle swarm optimization.- Integer programming.- Multiobjective optimization.- Approximation with radial bases functions.- Support vector machines (SVM).- Symbolic regression.- Quantile regression.- Robust regression.- Stochastic modeling.- Parallel computations.