A Particle Filter Approach for Solar Radiation Estimate Using Satellite Image and in Situ Data

The estimation of global solar radiation incident on the earth’s surface is an important issue for several solar-based applications. From a signal processing point of view, it falls within nonstationary, non-linear/non-Gaussian dynamical inverse problems. In this paper, we propose a sequential Monte Carlo state space approach combining satellite images and in situ data. We propose original observation and transition functions taking advantages of the characteristics of both the involved type of data. A simulation study is carried along with a comparison with the state of the art established method, Heliosat. INTRODUCTION The knowledge of the global solar radiation incident on the earth’s surface and its geographical distribution is of prime importance for numerous solar-based applications (Climate change assessment, solar renewable energy systems). Hourly and daily values of solar radiation measurements with high spatial resolution that are always necessary for these applications imply unacceptable cost if provided by a high density ground based radiometric network. Besides interpolation techniques applied to radiation measurements become ineffective when the distance between the meteorological stations is greater than 34-50 km. Satellite sensors can provide an alternative to the sparse coverage of radiometric networks since they can produce database over large regions on a high spatial resolution grid (1 by 1 km in visible range). However the computation of solar radiation by means of satellite images is unfortunately not straightforward. The satellite image is a top-of the-atmosphere (TOA) observation. The pixel value represents the flux density of the upward solar radiation emerging from the atmosphere, and the solar radiation absorbed by the ground is the fraction of the flux density of the downward solar radiation incident on the atmosphere. Determination of models capable of deriving global solar radiation at ground level from satellite images at high spatio-temporal resolution is an open issue in environmental research and solar applications that we are proposing to tackle in this paper. Several mathematical models were studied, in order to estimate solar radiation from satellite images. Two different approaches to this subject were developed. Statistical models from the one hand, physical models from the other hand (1, 2). Statistical models (3) have evolved toward complex hybrid models by incorporating additional observation data and both empirical and physical information ((4, 5) among others). Direct information are given by meteorological satellites and indirect information such as transmittance are obtained by radiative transfert equation. This led to better spatial distribution of the models response. However, despite their increasing complexity and an improved usage of numerous available data, recurring obstacles (e.g., the difference in spatiotemporal scale between the model and measurements; measurement errors; or the simplification of physical processes) still introduce a significant amount of uncertainty into the model predictions. In this paper we consider a Bayesian filtering approach to the dynamical estimation of the global solar radiation at ground level from satellite images. We defend the idea that an inverse approach based on sequential Monte Carlo filtering (6) helps to relax several assumptions and constraints while keeping estimations results in accordance with those of existing methods. Among these 1 EARSeL Workshop on Temporal Analysis of Satellite Images 209 Mykonos, Greece, 23 – 25 May, 2012 constraints, one can note the physical model and its parameter estimation. In fact, using a stochastic model allows, if the amount of data samples is sufficient, to associate (in a statistical sense) the satellite data to their corresponding in situ samples. This conducts to infer the radiation measure in a continuous way of a geographic map with a precision comparable to the well established methods. The paper is organized as follows: after presenting the stochastic model along with the observation and transition laws in Section 2, we explain the sequential Monte Carlo sampling in Section 3. Section 4 presents our experiments and results and compares them to those obtained with a traditional model. Section 5 concludes and presents future directions. METHODS Stochastic models are commonly used to describe the behavior of many processes. Model variables can be divided into hidden variables (that are not measured) as solar radiation estimates at surface, and observed variables as satellite image pixels. A combination of hidden and measured variables can be used to represent the dynamic behavior of the nonlinear process as described before. The data and notations A common assumption underlying solar irradiance signal is that it can’t be regarded as a stationary process due to the diurnal and annual variation related to the sun’s changing angle. To remove these effects and obtain a weekly stationary stochastic process solar irradiance is often normalized by dividing solar radiation at the earth surface by the extraterrestrial solar irradiance. The result is defined as the clearness index. The horizontal irradiance outside the atmosphere is determined using: G0(i,j)=IscE0coss(i,j) (1) where Isc= 1367 W/m2 is the solar constant, the extraterrestrial irradiance normal to the solar beam; E0 is the excentricity correction factor and s(i,j) is the sun zenithal angle at pixel (i,j). E0 and s depend on astronomical relationships and can analytically be determined for each instant k. Thus the knowledge of the clearness index allows the calculation of solar radiation at the earth’s surface and inversely. Let xk denote the clearness index at time k : xk = Gk(i,j) / G0k(i,j) (2) where Gk(i,j), is the horizontal global irradiance at ground level for the time k and the pixel (i, j) and G0k(i,j) is the horizontal irradiance outside the atmosphere for the time k and the pixel (i, j). They are expressed in W.m. Observations of our model refer to the apparent albedo (i,j) observed by the satellite sensor for the pixel (i, j) (containing the ground location). (i,j) has no unit and is equal to the bidimensional reflectance. (i,j) = L(i,j) / (IscE0coss(i,j)) (3) L(i,j) is the is the observed radiance. xk∈ R n is a state vector evolving according to the following equation: xk+1 =fk(xk, vk) where vk is i.i.d. random noise with unknown probability distribution function (pdf). At discrete times, observations zk ∈ become available and are related to the state vector via the observation equation: zk= hk(xk,wk) The filtering problem can be formulated as: xk=fk(xk-1)+vk-1 (4 ) zk=hk(xk)+wk (5) 1 EARSeL Workshop on Temporal Analysis of Satellite Images 210 Mykonos, Greece, 23 – 25 May, 2012 v and w are the process noise and the observation noise. The state transition density is fully specified by fk and the process noise distribution and the observation likelihood are fully specified by hk and the observation noise distribution. The transition law (process law) The first part of the stochastic model (eq. 4) is the transition law. In this work we use a transition law based on the ARMA (Auto-Regressive Moving Average Model) process called TAG (Timedependant Autoregressive, Gaussian model) developped by Aguiar and Collares-Pereira (7) and designed to be independent of location and time of the year. This model generates synthetic daily sequences of the hourly clearness index xk as a Markov chain. To takes into account seasonal phenomena the variable xk is normalized (centrered and reduced):