**8. Acknowledgment**

This work was supported by the Spanish Ministerio de Ciencia e Innovación, Project TEC2008-06598.

### **9. Nomenclature**



**Υ** Perturbation matrix employed to obtain **C**<sup>χ</sup>.

### **10. References**

16 Numerical Simulations / Book 1

accurate approximation of 'm' equal power log-normal scintillation sequences with any CC coefficient is proposed in this paper, overcoming the restrictions of a Cholesky decomposition. Hence, the AR-model proposed in (Jurado-Navas & Puerta-Notario, 2009) and its inherent numerically ill-conditioned covariance matrix (Baddour & Beaulieu, 2002) may be avoided in many cases when calculating burst error rate curves due to the difference between the two extreme scenarios studied in this chapter is usually limited to approximately 2 dB at a burst rate of 10−6. In this sense, the space-time separable statistics model proposed here can be used to consider spatial correlations among scintillation sequences without fear of making big mistakes and with the advantage of a reduced computational time. Thus, such separable statistics model may be seen as a highly accurate upper error bound of the whole model

This work was supported by the Spanish Ministerio de Ciencia e Innovación, Project

**A**[*k*] *m* × *m* matrices containing the multichannel AR model coefficients.

*fχ*(*χ*) Probability density function of random log-amplitude scintillation. *fI*(*I*) Probability density function of intensity fluctuations (=*fαsc* (*αsc*)).

**L** Lower triangular matrix obtained after applying a Cholesky decomposition.

*<sup>u</sup>*<sup>⊥</sup> Component of the wind velocity transverse to the propagation direction.

*I*<sup>0</sup> Level of irradiance fluctuation in the absence of air turbulence.

**C**<sup>w</sup> Covariance matrix of the driving noise vector process of an AR model.

**C***χ*[*j*] Covariance matrix of the log-amplitude scintillation evaluated in the *j*-time

detailed in (Jurado-Navas & Puerta-Notario, 2009).

*<sup>n</sup>* Refractive-index structure parameter.

**C**<sup>χ</sup> Positive semi-definite approximation of **C***χ*. *d*<sup>0</sup> Correlation length of intensity fluctuations.

**E** Vector amplitude of the electric field.

*k* Wave number of beam wave (=2*π*/*λ*).

*n*<sup>0</sup> Average value of index of refraction. *n*<sup>1</sup> Fluctuations of the refractive index.

**w**[*n*] Coloring Gaussian vector (= **Kz**[*n*]).

**r** Transverse position of observation point.

*U*(**r**, *z*) Complex amplitude of the field in random medium.

*I* Irradiance of the random field.

**Im** *m*-element identity matrix.

*L* Propagation path length.

*l*<sup>0</sup> Inner scale of turbulence. *n*(**r**) Index of refraction.

**Q** Orthogonal matrix.

**K** Coloring matrix.

instant.

**C***<sup>χ</sup>* Covariance matrix of the log-amplitude scintillation.

*dij* Distance between points *i* and *j* in the receiver plane (m).

**8. Acknowledgment**

TEC2008-06598.

*C*2

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**0**

**21**

*Venezuela*

**Decision-Making**

**Climate System Simulations: An Integrated,**

Climate physicists define the Climate System in terms of the different interactions taking place between hydrosphere, atmosphere, cryosphere, biosphere and lithosphere. Being the associated general thermo-hydrodynamic partial differential equation system so complex for analytical solving, it is usually integrated numerically involving a good deal of computational

Moreover, today different spatial scales involve different physical parametrisations, and each forecast horizon (few days, seasonal, annual, decadal and climate change periods) of interest deserves a special treatment, mostly defined by the predictability of the Climate System and the characteristic response time of the interacting components at the corresponding temporal scale. Thus, it is presently customary to distinguish between large-, meso- and micro-scale numerical models, their descriptions ranging from global climate state to basin availability of hydrological resources. An integrated, multi-scale approach considering the output of the various models is of vital to understand at the different levels the behaviour of the Climate

In this chapter, after introducing a few fundamental concepts and equations in Section 2, and a hierarchy flux of information among the models for different scales in Section 3, the methodologies involving downscaling executions are discussed in detail, regarding both scientific research and policy-making applications. In section 4 we explain the usefulness of conducting several simulations (realisations) to partially reduce the inherent uncertainties. Later, in Section 5 we offer a plausible way to put into operation all the components, presenting several examples based on the experience acquired through a regional initiative known as the *Latin-American Observatory for Climate Events*. Some concluding remarks and

In this section several fundamental aspects related to the definition of the Climate System, its temporal scales and the governing equations of the atmosphere and the oceans are discussed. While the formal definition in terms of the Climate Subsystems is standard, it is important to remark that different models tend to write the corresponding system of thermo-hydrodynamical equations in different ways, due to the employment of particular numerical methods, simplifications, coordinate systems or physical assumptions.

System, and thus offer useful tools for decision makers and stake holders.

suggestions for future research are presented in Section 6.

**1. Introduction**

**2. Fundamentals**

resources.

**Multi-Scale Approach for Research and**

Ángel G. Muñoz, Alfredo Nuñez and Ramón J. Cova *Centro de Modelado Científico (CMC). Universidad del Zulia.*

