Sensitivity Analysis of the Row Model’s 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 model’s 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 model’s input parameters indicate that the sensitivities of the row model’s parameters are different under different viewing angles. We also find that the row structure of canopy can affect the sensitivities of model’s 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.

Keywords—Sensitivity 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 model’s input parameters [1]. The essence of sensitivity analysis is to vary the model’s parameters one by one to study the rule of the model outputs affected by the variance of the model’s 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 model’s parameters. Different methods of global sensitivity analysis are currently in use, such as Sobol’s 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 model’s parameters. In this study, the method of variance based sensitivity analysis is applied to analyze the sensitivities of the row model’s 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 model’s 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 model’s 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 model’s 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+ΣΦi (xi)+ΣΦij (xi, xj) +…+ Φ12…k (x1, x2,…, xk)

（1）

Where Φi (xi) is a function dependent on xi, Φij (xi, xj) is a function dependent on xi and xj, and so on up to Φ12…k (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 Φi, Vij for Φij and so on up to V12…k for Φ12…k:

Where Vi=V(E(Y|xi)) is the contribution of xi to V(Y), Vij=V(E(Y|xi , xj) −Vi−Vj is the contribution of the interactions of xi and xj to V(Y), and so on up to V12…k, 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 model’s 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 +…+ S12…i…k （6）

The variance based sensitivity method is used here to study the impact of the variations in parameters on the variations of row model’s 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 canopy’s 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 (εv), soil emissivity (εs), 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 implementation of the row model, L is set to 1. a and h are normalized by L).

With a given group of model’s above parameters, the row model can calculate the canopy’s DRF (VNIR band) and directional brightness temperature (DBT, TIR band).

C. Scheme of Sensitivity Analysis of Row Model

V (Y) = ΣVi + ΣVij +…+ V12…k （2）

The variance based sensitivity method consists of three steps: sample generation, model execution and the calculation of sensitivity indices [1]. The first step for the sensitivity analysis is to generate sample points. The selected model’s parameters of the row model to analyze their sensitivities include LAI, a, h, LIDF.a, LIDF.b, and k (TABLE. I).

According to the probability distribution functions (PDF) of the model’s parameters, 10000 groups of sample points are generated using the Sobol’s sequence generator from the parameter space. Other parameters of the row model are fixed to their default values (TABLE. II).

When generating the values of LIDF.a and LIDF.b, the restriction | LIDF.a | + | LIDF.b | ≤ 1 is applied. During the process of row canopy growth, canopy structure changes proportionally. In practice, we set reasonable correlation coefficients among the row structure parameters to generate sample points. TABLE. III gives the correlation coefficients, which are based on the experimental data of winter wheat collected in the Shunyi experimental area, near Beijing, China [13].

The second step of the variance based sensitivity method is the execution of the row model. At certain viewing angles, DRFs in the red band (666 nm) and NIR band (850 nm), DBTs in the TIR band (12 µm) are calculated by the row model. The last step is to calculate the sensitivity indices of the row model’s parameters, first-order sensitivity indices Si and total-effect sensitivity indices

iTS by (3) and (5), respectively.

III. RESULTS AND DISCUSSION In order to study the impact of the row model’s parameters

to the model outputs under different viewing angles, we use the polar plots (Fig. 1) to analyze the sensitivity of model’s parameters. Fig. 1 shows the distributions of the total-effect

sensitivity indices for LAI, a, h, LIDF.a, LIDF.b, and k in three spectral bands. In the polar plot, the center stands for the nadir viewing direction. And outward from the center, the numbers above the contours stand for the VZAs. The original direction is north with a VAA of 0°, and in clockwise direction angles for east, south and west are 90°, 180° and 270°, respectively. The row direction is south and north. Based on this polar coordinate system, the sampling strategy of the DRFs and DBTs for row model to plot the polar maps was chosen in this way: from 0° to 350° with a step of 10° for VAA, and from 0° to 60° with a step of 5° for VZA. We define the direction opposite to the sun as the forward direction, and the direction on the same side of the sun position as the backward direction.

A. Red Band The distributions of the total-effect sensitivity indices for

LAI, a, h, LIDF.a, LIDF.b, and k are different in the red band. In Fig. 1a, there is a high value stripe nearly parallel to the row orientation for parameter LAI. And Figs. 1b and 1c show a high value stripe approximately parallel to the row orientation for parameter a and h, respectively.

These phenomena are mainly caused by the row structure of canopy. In the along-row or near-along row directions with high VZAs, more leaves are sunlit and visible, which greatly affect the distribution of the DRFs. The sensitivity indices of LAI are also affected by the other parameters’ sensitivity indices, which result in the high value stripe nearly parallel to the row orientation.

For the along-row or near-along row directions, more sunlit soil between rows, and more sunlit leaves at all height levels are visible through the void space between rows. These correspond to the single scattering contributions from the soil and leaf, which eventually affect the distribution of the DRFs in the row model. The distributions of sensitivity indices for a and h are influenced by the hotspot effect.

Figs. 1d and 1e show the distributions of the sensitivity indices of LIDF.a and LIDF.b in the red band. The sensitivity indices of LIDF.a are relatively high at some viewing angles and the sensitivity indices of LIDF.b are relatively low. The ranges of sensitivity indices for LIDF.a and LIDF.b are [0.001, 0.410] and [0.001, 0.175], respectively.

Fig. 1f shows the sensitivity indices of k in the red band, its values are relatively small, except around hotspot directions.

B. NIR Band Figs. 1g-i show the distributions of sensitivity indices for

LAI, a and h, respectively in the NIR band. For LAI, the sensitivity indices on the right side of the row are higher than

TABLE I . THE SELECTED PARAMETERS OF THE ROW MODEL TOANALYZE THEIR SENSITIVITIES Parameters LAI a h LIDF.a LIDF.b k Distribution

Function U* U* U* U* U* U*

Range (0, 7] [0.1, 0.9] [0.1, 3.9] [-1, 1] [-1, 1] (10,40)

*U stands for the uniform distribution.

TABLE. II. THE FIXED PARAMETERS OF THE ROW MODEL

Parameters SZA SAA L εv εs Ts Td Th Tc rl tl rs

Red NIR Red NIR Red NIR

Value 40° 140° 1 0.98 0.95 45℃ 36℃ 29℃ 24℃ 0.1584 0.6100 0.0450 0.2939 0.1338 0.3249

TABLE. Ⅲ. THE CORRELATION COEFFICIENTS OF ROW MODELPARAMETERS Parameters a and h k and h a and k a and

LAI h and LAI

k and LAI

The Correlation Coefficients

0.9651 0.98 0.9532 0.9502 0.9657 0.9588

those on the other side. For parameters a and h, there are high value stripes approximately parallel to the row orientation.

The distributions of sensitivity indices are conditioned by the row structure of canopy and the spectral difference between the leaf and soil. In the NIR band, the leaf reflectance and transmittance are larger than the reflectance of the soil. There are more leaves are sunlit and visible in the backward directions, which can greatly affect the distribution of the DRFs in the row model.

Similarly, the sensitivity indices of LIDF.a (Fig. 1j) are high at some viewing angles and the sensitivity indices of LIDF.b (Fig. 1k) are relatively low in the NIR band. The ranges of sensitivity indices for LIDF.a and LIDF.b are [0.001, 0.410] and [0.001, 0.015], respectively.

Fig. 1l shows the sensitivity indices of k in the NIR band, its values are relatively small.

C. TIR Band The polar plot of LAI (Fig. 1m) shows two high value

stripes approximately parallel to the row orientation at high VZAs, and a low value stripe approximately parallel to the row orientation with its center around the nadir viewing direction. For a and h, the sensitivity indices display a high value stripe approximately parallel to the row orientation in Figs. 1n and 1o, respectively.

From the polar plots of LIDF.a and LIDF.b in the TIR band (Figs. 1p and 1q), we can find that LIDF.a is sensitive around the nadir viewing direction and LIDF.b is relatively insensitive.

The sensitivity indices of k in the TIR band (Fig. 1r) are relatively small, which indicates that k is relatively insensitive in the TIR band.

The distributions of the first-order sensitivity indices Si for LAI, a, h, LIDF.a, LIDF.b, and k in the three spectral bands are similar to these of the total-order effect

iTS as shown in Fig. 1 and not shown here.

IV. SUMMARY AND CONCLUSION To quantitatively analyze the sensitivities of the row

model’s parameters, the variance based sensitivity method is applied in this study. This method is powerful in quantifying the relative importance of row model’s parameters and interaction effects among parameters on the row model outputs. Based on the results of sensitivity analysis, we can identify the properties of the sensitivity of the row model’s parameters and construct an efficient inversion strategy to retrieve them. The main conclusions of this study are as follows.

1) The variance based sensitivity method can quantitatively analyze the sensitivities of the row structure parameters.

2) In three typical spectral bands, the sensitivities of the row model’s parameters are different under different viewing angles.

3) 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 results presented in this study are informative for the retrieval of the parameters of the row model in the future work.

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Fig. 1. Polar plots of the row model parameters’ total-effect sensitivity indices in the three bands. The left six panels (a, b, c, d, e, f) correspond to LAI, a, h, LIDF.a, LIDF.b, and k in the red band respectively. The middle six panels (g, h, i, j, k, l) correspond to LAI, a, h, LIDF.a, LIDF.b, and k in the NIR band respectively. The right six

panels (m, n, o, p, q, r) correspond to LAI, a, h, LIDF.a, LIDF.b, and k in the TIR band respectively.

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