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Sensitivity Analysis of the Row Models Input Parameters

Peng Zhang, Feng Zhao*, Yiqing Guo, Huijie Zhao School of Instrumentation Science and Opto-electronics

Engineering Beihang University

Beijing 100191, P.R. China zhaofeng@buaa.edu.cn (F. Zhao)

Yanhua Zhao Beijing Institute of Space Mechanics & Electricity

Beijing 100094, P.R. China

Lan Dong School of Computer and Information Technology

Beijing Jiaotong University Beijing 100044, P.R. China

Abstract In this study, we use the variance based sensitivity

method to analyze the sensitivities of the row crop models input parameters. This method consists of three steps: sample generation, model execution and the calculation of sensitivity indices. The results of sensitivity analysis of row crop models input parameters indicate that the sensitivities of the row models parameters are different under different viewing angles. We also find that the row structure of canopy can affect the sensitivities of models inputs parameters. LAI is found to be generally the most sensitive parameter in three typical spectral bands. a and h are also identified as sensitive parameters in the along-row or near-along row directions in the VNIR bands. LIDF.a is relatively sensitive at some viewing angles and LIDF.b is insensitive in three typical spectral bands. k has marginal influence on the model outputs in all of the viewing angles and spectral bands, except around hotspot directions in the red band. The main contribution of this work is that the technique of global sensitivity analysis is applied to the row model and the results are informative for the retrieval of the parameters during the inversion.

KeywordsSensitivity analysis; Row model; Variance Based Sensitivity Method

I. INTRODUCTION Sensitivity analysis is the study of how uncertainty of the

model outputs can be apportioned to different sources of uncertainty in the models input parameters [1]. The essence of sensitivity analysis is to vary the models parameters one by one to study the rule of the model outputs affected by the variance of the models parameters. The importance of the parameters to the model can be identified in the process of sensitivity analysis [2].

There are two different schools for the sensitivity analysis, the local sensitivity analysis and the global sensitivity analysis [3]. Local sensitivity analysis studies the sensitivities of parameters by varying parameters one at a time and holding the others fixed to the central values. This method neglects the influence of the interactions among all the parameters on the model outputs. So the result of this sensitivity analysis is not comprehensive. In contrast, global sensitivity analysis takes

into account the interactions among all of the models parameters. Different methods of global sensitivity analysis are currently in use, such as Sobols method [3], Extend FAST (Fourier Amplitude Sensitivity Test) method [4], and variance based sensitivity method [2].

Sensitivity analysis for parameters of models is a prerequisite for the successful inversion of them. The reason is that the number of the parameters may be substantial and the sensitivities of parameters are different. Sensitivity analysis can help to construct an efficient inversion strategy to determine which parameter should be retrieved or fixed based on the sequence of parameters by importance. In the process of model inversion, we should treat the sensitive parameters with more attention, because the uncertainties of these parameters will greatly affect the model outputs.

Zhao et al. [5] developed a row crop model, which is able to simulate the characteristic distribution of directional reflectance factors (DRFs) of row planted canopies with a rather satisfactory accuracy. The row crop model captures the basic characteristics of a row canopy's DRF distributions such as the row effect and the hotspot effect. Up to now, no research has been reported to analyze the sensitivity of the row models parameters. In this study, the method of variance based sensitivity analysis is applied to analyze the sensitivities of the row models parameters.

This paper is organized as follows: Section 2 briefly describes the basic concepts of variance based sensitivity method and the row model; Section 3 discusses the global sensitivity analysis of the row models parameters in three typical spectral bands; Section 4 summarizes the results.

II. METHODS AND MODEL

A. The Variance Based Sensitivity Method The variance based sensitivity method is versatile and

effective among the various available techniques for global sensitivity analysis of models parameters. It is based on the decomposition of the total variance of model outputs into

This work is jointly supported by the Chinese Natural Science Foundation under Project 41371325 and 40901156, the Fundamental Research Funds for the Central Universities (Contract No. 2012JBM030), and the Civil Aerospace Technology Pre-research Project of China (Grant No. D040201-03).

components related to all the models parameters and their interactions. This analysis relies on the law of total variance or variance decomposition formula, and the Sobol decomposition of function [6, 7].

Given a model of the form Y = f (x1, x2, ..., xk), the model output Y is decomposed through functional analysis of variance into increasing order terms [3]:

f (x1, x2, , xk) = f0+i (xi)+ij (xi, xj) ++ 12k (x1, x2,, xk)

1

Where i (xi) is a function dependent on xi, ij (xi, xj) is a function dependent on xi and xj, and so on up to 12k (x1, x2,, xk), which is a function dependent on x1, x2,, xk.

By using the theorem of orthogonality in (1), the variance V of Y can be expressed as the sum of the partial variances, which include Vi for i, Vij for ij and so on up to V12k for 12k:

Where Vi=V(E(Y|xi)) is the contribution of xi to V(Y), Vij=V(E(Y|xi , xj) ViVj is the contribution of the interactions of xi and xj to V(Y), and so on up to V12k, which is the contribution of the interactions of all the parameters to V(Y).

Equation (2) decomposes the total variance into distinct components related to each parameter and parameters interactions. E(Y|xi) denotes the expectation of Y under the condition of xi, and V(E(Y|xi)) is the variance for all the possible values of xi; Similarly, V(E(Y|xi , xj)) is the variance for all the possible values of xi and xj. Variance based sensitivity indices are estimated as ratios between the partial variances and the total variance:

Si = Vi / V 3

Sij = Vij / V 4

Where Si is the main (first-order) effect of parameter xi and Sij is the second-order effect of parameter xi [1]. The main effect Si indicates the relative importance and influence of xi on the model output. The effect of Si is additive on the model output and independent of the other parameters [8]. The measure of the total-order effect can be interpreted as the effect of parameter xi on the variance of model output with interactions among all models parameters being taken into account [9]. The total-order effect

iTS can be calculated by:

~ ~~ ~( ( | ))/ ( ) 1 ( ( | )) / ( )

i i i i iT x x i x x iS E V Y x V Y V E Y x V Y= = 5

Where ~( | )ix iV Y x denotes the variance of Y under the

condition that only xi is changed and the other parameters are fixed, and the expectation

~ ~( ( | ))

i ix x iE V Y x is computed over

all the parameters except xi; Similarly, ~( | )ix iE Y x denotes the

expectation of Y under the condition that only xi is changed and other parameters are fixed, and

~ ~( ( | ))

i ix x iV E Y x is the

variance of all the parameters except xi.

The total-order effect consists of the first-order sensitivity index, the second-order sensitivity index and so on. Assuming a model with k parameters, the total-order effect of the parameter xi is calculated by a sequence of sensitivity indices [10, 11, 12]:

iTS = Si + Sij + Sijm ++ S12ik 6

The variance based sensitivity method is used here to study the impact of the variations in parameters on the variations of row models outputs.

B. The Row Model

Agricultural crops are important target for remote sensing. During the early growing stages, the row planted canopies have an obvious row structure. The assumption that the crop canopy being horizontally homogeneous made in the one-dimensional radiative transfer models has been demonstrated problematic for row planted canopies. A radiative transfer model for row crop [5] has been proposed to study row planted canopys radiative characteristics in the visible near-infrared (VNIR) and thermal infra-red (TIR) spectral domains. Based on the intrinsic relation of radiation transfer of vegetative canopies, the row model is able to describe the radiative characteristics of the row model in the different spectral domains.

The parameters of the row model are grouped as follows:

1) Solar and viewing geometry: solar zenith angle (SZA), solar azimuth angle (SAA), viewing zenith angle (VZA), and viewing azimuth angle (VAA).

2) Optical parameters in the VNIR band: leaf hemispherical reflectance (rl), leaf hemispherical transmittance (tl), soil hemispherical reflectance (rs), and the ratio of direct to total irradiance at the given wavelength at the top of the canopy.

3) Spectral parameters in the TIR band: leaf emissivity (v), soil emissivity (s), thermodynamic temperature of sunlit leaf (Th), shade leaf (Tc), sunlit soil (Ts), and shade soil (Td).

4) Canopy structural parameters: Leaf Area Index (LAI), row width (a), row height (h), row distanced (L), leaf inclination distribution parameters (LIDF.a and LIDF.b), and hotspots parameter (k). (In the

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