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Nonparametric estimations of the sea state bias for a radar altimeter
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Hongli MIAO1, *, Yujie JING1, Yongjun JIA2, Mingsen LIN2, Guoshou ZHANG1, Guizhong WANG1
Acta Oceanologica Sinica | 2017, 36(9) : 108 - 113
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Acta Oceanologica Sinica | 2017, 36(9): 108-113
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Nonparametric estimations of the sea state bias for a radar altimeter
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Hongli MIAO1, *, Yujie JING1, Yongjun JIA2, Mingsen LIN2, Guoshou ZHANG1, Guizhong WANG1
Affiliations
  • 1 College of Information Science and Engineering, Ocean University of China, Qingdao 266100, China
  • 2 National Satellite Ocean Application Service, State Oceanic Administration, Beijing 100081, China
Published: 2017-09-01 doi: 10.1007/s13131-017-1116-x
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To estimate the sea state bias (SSB) for radar altimeter, two nonparametric models, including a Nadaraya-Watson (NW) kernel estimator and a local linear regression (LLR) estimator, are studied based on the Jason-2 altimeter data. Selecting from different combinations of the Gaussian kernel function, spherical Epanechnikov kernel function, a fixed bandwidth and a local adjustable bandwidth, it is observed that the LLR method with the spherical Epanechnikov kernel function and the local adjustable bandwidth is the optimal nonparametric model for the SSB estimation. The comparisons between the nonparametric and parametric models are conducted and the results show that the nonparametric model performs relatively better at high-latitudes of the Northern Hemisphere. This method has been applied to the HY-2A altimeter as well and the same conclusion can be obtained.

radar altimeter  /  sea state bias  /  significant wave height  /  wind speed  /  nonparametric model  /  parametric model
Hongli MIAO, Yujie JING, Yongjun JIA, Mingsen LIN, Guoshou ZHANG, Guizhong WANG. Nonparametric estimations of the sea state bias for a radar altimeter[J]. Acta Oceanologica Sinica, 2017 , 36 (9) : 108 -113 . DOI: 10.1007/s13131-017-1116-x
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1 Introduction
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The radar altimeter produces various errors during the height measurements. The sea state bias (SSB) is generally accounted for 1/3 of the total error in the Jason-1 altimeter, and has taken place of the orbital error as the largest error source due to the development of precise orbit determination technologies (Fu and Cazenave, 2001; Wang et al., 2014a, b). When the radar altimeter transmits pulses toward a nadir point, the curvature of wave crests is usually larger than that of wave troughs. The results in the backscattered power measured by the altimeter from the wave troughs are greater than that from the wave crests. The mean scattering surface biases toward the wave troughs from a mean sea level, so that the mean sea level measured by altimeters is lower than the real one. The difference between the measured and real mean sea surfaces is called the electromagnetic bias. In addition, there are a skewness bias, which is caused by mismatching between the mean scattering surface and the median scattering surface because specular scattering in sea level does not follow the Gaussian distribution, and a tracker bias in determination of the median height by tracker. These three errors are all known as the sea state bias. At present, the SSB estimation still relies on empirical methods, including both parametric models and nonparametric models (Li et al., 2013).
The parametric model for the SSB estimation based on the Taylor expansion of the significant wave height (SWH) and a wind speed is simple, intuitive, easy to use and its extensionality is good (Miao et al., 2015; Wang et al., 2014a, b). Due to the lack of true SSB values, the model coefficients are determined by the sea surface height (SSH) differences, the SWH differences and the wind speed differences at crossover points or along collinear tracks. However, when calculating the SSB, only the SWH and the wind speed can be used. Therefore, the parametric model is not based on a true least square method, which may cause an error in the model coefficients learning (Vandemark et al., 2002).
The nonparametric models based on the technology of kernel smoothing can avoid the problems existing in the parametric models. The nonparametric model characterized in the form of regression function has no constraints on the distribution of the estimated variables. Furthermore, the curve fitting using the nonparametric model meets the principle of least squares very well and can describe subtle changes between the variables. Therefore, the SSB estimation accuracy at the northern latitudes, where the sea state is complicated due to the influence of the land, is significantly improved over the parametric model (Gaspar et al., 2002; Labroue et al., 2004; Scharoo and Lillibridge, 2004; Ye et al., 2015).
The nonparametric estimations mainly include the Nadaraya-Watson kernel estimation (NW), a local linear regression estimation (LLR), a K-neighbor estimation, an orthogonal sequence estimation and a polynomial spline estimation, etc (Tran et al., 2010a, b ; Feng et al., 2010; Rami et al., 2011). In this paper, we focus on the first two methods, and the results are compared with the parametric models.
2 Modeling principle
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An uncorrected sea surface height ${\rm{SSH'}}$ ($h'$) can be expressed as
$h' = b + {h_{\rm{g}}} + {h_{\rm{d}}} + \varepsilon ',$
where b is the SSB, hg is the geoid, hd is the dynamic topography, and $\varepsilon '$ is the sum of all height measurement errors except the SSB (Li et al., 2013).
The geoid is the largest signal with an order of meters to tens of meters. The geoid is a time-invariant signal which can be eliminated by carrying out the difference between measurements at crossover points. Adopting the discrepancy between individual measurements at different time, Eq. (1) can be expressed directly as
${h_2}^\prime - {h_1}^\prime = ({b_2} - {b_1}) + \varepsilon ,$
where the subscripts 1 and 2 denote measurements at the crossover point when the altimeter ascending and descending, respectively. Parameter $\varepsilon$ is the mean error (or residual) which includes the time-variation of the dynamic topography and other measurement errors except the SSB (Gasper et al., 1994).
The research shows that the SSB is closely related to the SWH and the wind speed. Hence, if x is a binary variable of the SWH and the wind speed (U), ${\text{x}} = ({h_{\rm{s}}},U)$, where hs is the SWH, the SSB can be determined as
$b = \varphi ({\text{x}}).$
Setting $y = {h_2}^\prime - {h_1}^\prime $, Eq. (2) can be rewritten as
$y = \varphi ({{\text{x}}_2}) - \varphi ({{\text{x}}_1}) + \varepsilon .$
Because the mean error is a zero-mean noise, the conditional expectation of y is
$E[y{\left| {\text{x}} \right._2} = {\text{x}}] = \varphi ({\text{x}}) - E[\varphi ({{\text{x}}_1}){\left| {\text{x}} \right._2} = {\text{x}}].$
For kernel functions, the conditional expectation $E[\zeta_i \left| {\text{x}} \right.]$ of the kernel estimator $\hat r({\text{x}})$ is a weighted mean of the measurements ζi, and the weight is a decreasing function of a distance between x and xi (Gaspar and Florens, 1998),
$\hat r({\text{x}}) = \sum\limits_{i = 1}^n {{\zeta _i}} {\alpha _n}({\text{x}} - {{\text{x}}_i}),$
where αn is a weighted function which is associated with n and satisfies the following condition:
$\sum\limits_{i = 1}^n {{\alpha _n}} ({\text{x}} - {{\text{x}}_i}) = 1.$
By using Eq. (6) and a set of n altimeter crossover measurements, Eq. (5) can be rewritten as
$\begin{array}{l}\varphi ({\text{x}}) = \sum\limits_{i = 1}^n {{y_i}{\alpha _n}({\text{x}} - {{\text{x}}_{2i}}) + } \\\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{i = 1}^n {\varphi ({{\text{x}}_{1i}}){\alpha _n}({\text{x}} - {{\text{x}}_{2i}}).} \end{array}$
Equation (8) provides a practical method to estimate $\varphi ({\text{x}})$ for any vector $\mathop x\limits^{\rightharpoonup}$, provided that the value of $\varphi ({{\text{x}}_1})$ is known, where, $\varphi ({{\text{x}}_1})$ is the estimated value of the SSB at the crossover point when the altimeter ascending.
The following two methods can be used to determine the weighted function αn (Gaspar et al., 2002).
(1)the Nadaraya-Watson kernel estimation (NW)
${α_n}\left( {{x} - {x_i}} \right) =\displaystyle \frac{{K\left( \displaystyle {\frac{{{x} - {x_i}}}{h_n}} \right)}}{{\sum\limits_{i = 1}^n {K\left(\displaystyle {\frac{{x} - {x_i}}{{h}_n}} \right)} }},$
where $K( \cdot )$ is a kernel function; ${α_n}({\text{x}} - {{\text{x}}_i})$ is the (1×n) weighted vector, n is the number of crossover points, and hn is the bandwidth vector; and the index n indicates that the bandwidth depends on the number of measurements.
(2) The local linear regression estimation (LLR)
${α_n}({\text{x}} - {x_i}) = e_{\rm{1}}^{\rm{T}}{({\text{X}}_{\rm{D}}^{\rm{T}}{\text{W}}{{\text{X}}_{\rm{D}}})^{ - 1}}{\text{X}}_{\rm{D}}^{\rm{T}}{\text{W}},$
where ${\text{e}}_{\rm{1}}^{\rm{T}} = (1,0,0)$, is the (1×3) unit vector; ${{\text{X}}_{\rm{D}}} = (1,{\text{x}} - {{\text{x}}_i})$, is the (n×3) matrix; and ${\text{W}} = {{M}}\left\{ {K\left(\frac{{{\text{x}} - {{\text{x}}_i}}}{{{{\text{h}}_n}}}\right)} \right\}$, is the (n×n) diagonal matrix, M is diagonal matrix operator. Thus ${α_n}(x - {{\text{x}}_i})$ is the (1×n) weighted vector.
The above two methods both involve the problems how to select the kernel function $K( \cdot )$ and bandwidth hn. The kernel function $K( \cdot )$ is a symmetric scalar function with $\int {K({\text{x}})} {\rm{d}}{\text{x}} = 1$, which is usually selected as a probability density function.
In this paper, for ${\text{x}} = (U,{h_{\rm{s}}})$ in which hs is the SWH, hn= $ ({h_U},{h_{{h_{\rm s}}}})$ is the bandwidth vector. Two forms of $K( \cdot )$ are selected:
(1) the Gaussian kernel function. In two-dimensional case, the function is
$\begin{array}{l}K\left( {\frac{{{\text{x}} - {{\text{x}}_i}}}{{{{\text{h}}_n}}}} \right) = \frac{1}{{2{{\text{π}}}{h_{{U}}}{h_{{{{h}}_s}}}}}\exp \left[ {\frac{{ - {{(U - {U_i})}^2}}}{{2h_{{U}}^{\rm{2}}}}} \right] \times \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\exp \left[ {\frac{{ - {{({h_{\rm{s}}} - {h_{\rm{s}}}_i)}^2}}}{{2h_{{{{h}}_{\rm{s}}}}^{\rm{2}}}}} \right];\end{array}$
(2) the spherical Epanechnikov kernel function. In two-dimensional case, the function is
$\begin{array}{l}K\left( {\frac{{{\text{x}} - {{\text{x}}_i}}}{{{{\text{h}}_n}}}} \right) = \max \left\{ {0,\frac{2}{{\text{π}{h_{{U}}}{h_{{{{h}}_{\rm{s}}}}}}} \times } \right.\left[ {1 - {{\left( {\frac{{U - {U_i}}}{{{h_{{U}}}}}} \right)}^2} - } \right.\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. {{{\left( {\frac{{{h_{\rm{s}}} - {h_{\rm{s}}}_i}}{{{h_{{{{h}}_{\rm{s}}}}}}}} \right)}^2}} \right]} \right\}.\end{array}$
There are two choices for the bandwidth hn. One is the global bandwidth (h_Global):
${{\text{h}}_n} = C{{\text{σ}}_x}{n^{ - 1/5}},$
where σx is the standard deviation of the wind speed or SWH; and C is an empirical value of 1.06 (Gaspar and Florens, 1998). The other one is the locally variable bandwidth (h_Local),
${{\text{h}}_n} = {{\text{h}}_{x0}}{[n({\text{x}})/\overline {n({\text{x}})} ]^{ - 1/6}}.$
where hx0 is a reference bandwidth of the wind speed or SWH. All the wind speed and the SWH at the crossover points are divided into groups according to a certain interval, n(x) is the number of measurements in the group of x. $\overline {n({\text{x}})} $ is the mean value of n(x) in each group. The group bandwidth is variable and negatively related to the number of measurements (Gaspar et al., 2002).
3 Model establishment
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3.1 Data set determination
The nonparametric models have large computational costs because of complex matrix operations. So, proper modeling data volume cannot only ensure the accuracy but also improve the efficiency. The segmented fitting method of track was adopted to extract crossovers data from Cycles 207, 218, 225 and 234 of Jason-2. The nonparametric model with the NW estimation, the Gaussian kernel function and the global bandwidth was established and compared based on these data sets. The explained variance (D) changes little even the data set increasing to four cycles, as shown in Table 1. There is no significant improvement for the accuracy of the SSB estimation with increasing data volume. Therefore, only the Cycle 207 crossovers measurements were selected for modeling in this paper.
The explained variance is defined as the difference between the variance of SSH discrepancy without SSB correction and that with the SSB correction which can be understood as the part of the variance of the SSH discrepancy that can be explained by the SSB. The higher the explained variance value is, the more effective the model is (Miao et al., 2015).
3.2 Determination of SSB estimated value $\varphi ({{\text{x}}_1})$ at the ascending arc of crossover points
In order to obtain the estimation $\varphi ({\text{x}})$ of any measurement x by Eq. (8), the value of $\varphi ({{\text{x}}_1})$ should be estimated first (Gaspar and Florens, 1998; Gaspar et al., 2002).
Setting ${\text{x}} = {{\text{x}}_{1j}}$, Eq. (8) can be rewritten as
$\begin{array}{l}\varphi ({{\text{x}}_{1j}}) = \sum\limits_{i = 1}^n {[{y_i}} {\alpha _n}({{\text{x}}_{1j}} - {{\text{x}}_{2i}})] + \\\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{i = 1}^n {[\varphi ({{\text{x}}_{1i}})} {\alpha _n}({{\text{x}}_{1j}} - {{\text{x}}_{2i}})],\\\;\;\;\;\;\;\;\;\;\;\;\forall j = 1, \cdot \cdot \cdot ,n.\end{array}$
Equation (15) can be changed to a matrix form
$({\text{I}} -{\text{A}}){\text{φ}}({{\text{x}}_1}) = {\text{Ay}},$
where I is the n×n identity matrix; and A is the n×n matrix with elements ${\alpha _{ij}}(x_{1j} - {x_{2i}})$, $\text{φ}{({{\text{x}}_1})^{\rm T}} = [\varphi ({{\text{x}}_{{\rm{1}}1}}), \ldots ,\varphi ({{\text{x}}_{1n}})]$, and ${{\text{y}}^{\rm T}} = [{y_1}, \ldots ,{y_n}]$.
Equation (16) cannot be solved for ${\text{φ}}({{\text{x}}_1})$ because ${\text{I}} - {\text{A}}$ is a singular matrix. The definition of the weights [Eq. (7)] implies that the sum of elements is 0 in each line of this matrix. We have to impose (rather than calculate) one value of ${\text{φ}}({{\text{x}}_1})$, that is, φ11 is the first element in the data set
$\varphi ({x_{11}}) = {\varphi _0}.$
So Eq. (16) can be rewritten as
${{\text{B}}_1}{\text{φ}} = {\text{Ay}} - {{\text{B}}_0}{\varphi _0},$
where φ is the vector made by the n–1 components of ${\text{φ}}({{\text{x}}_1})$; ${{\text{φ}}^{\rm{T}}} = [\varphi ({{\text{x}}_{12}}), \ldots ,\varphi ({{\text{x}}_{1n}})]$, and $\left( {{{\text{B}}_0},{{\text{B}}_1}} \right)$ is a partitioned matrix of ${\text{I}} - {\text{A}}$. We now have a system of n equations with n–1 unknowns. A linear least squares matrix is readily obtained
${\hat{\text{φ} }} = {({\text{B}}_1^{\rm{T}}{{\text{B}}_1})^{ - 1}}{\text{B}}_1^{\rm{T}}({\text{Ay}} - {{\text{B}}_0}{\varphi _0}).$
This provides the values of $\text{φ} ({{{x}}_{1i}})$ with i=2, …, n and we can then get the SSB estimates $\text{φ} ({{{x}}_1})$ at the ascending arc with $\text{φ} ({x_{11}})$ (generally it takes any reasonable value, such as –0.05 m for the first element value).
4 Results and discussion
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The ascending arc estimates of the SSB at the crossover of Jason-2 Cycle 207 was obtained firstly, then the corresponding descending arc estimates of the SSB can be obtained by Eq. (8). The efficiency of nonparametric model can be verified by the explained variance, the correlation between the SSB and the SWH or wind speed and the residual of the model.
The SSB is a function of the SWH and the wind speed, such that the model is effective only when the correlation between the SSB and the SWH or wind speed is high. This paper analyzed the correlation degree (r2) between the SSB and the SWH or wind speed to validate the proposed model.
From Eq. (2), the residual ($\text{ε} $) is the difference between the SSH discrepancy and the SSB discrepancy at the crossover points. The SWH discrepancy was sorted with 0.5 m interval and the wind speed discrepancy with 0.5 m/s interval. The mean value ($\bar \varepsilon $) of the residual was calculated and the correlation with ∆hs or ∆U was analyzed. The model will be more effective if the correlation between $\bar \varepsilon $ and ∆hs or ∆U is weak and the absolute deviation and standard deviation (std) of $\bar \varepsilon $ are smaller in each section.
4.1 Optimization selection of different nonparametric SSB estimated models
Two nonparametric methods, NW or LLR, both can be used to estimate the SSB. The global bandwidth, hU=0.7 m/s and $ {{h_h}_{_{\rm{s}}}} $= 0.3 m, can be calculated by Eq. (13). We can selected ${h_U} = 2.0\;{\rm{m}}/{\rm{s}}$ and ${h_{{h_{\rm{s}}},\,{0}}} = 0.9\;{\rm{m}}$ as the initial bandwidth which are the locally variable bandwidths and can be calculated by Eq. (14). By using the Gaussian kernel function or the spherical Epanechnikov kernel function, and selecting different forms of bandwidth, eight nonparametric models of estimating SSB can be established. The results are shown in Table 2. As shown in Table 2, the SSB estimates are different corresponding to different estimation methods. According to the explained variance (D), the LLR estimation is better than the NW estimation and the locally variable bandwidth is better than the global bandwidth. The nonparametric model which is based on the LLR estimation, the spherical Epanechnikov kernel function and the locally variable bandwidth (hereafter referred to as LLR-E-L) lead to significant improved results. In this model, the r2 has higher values (0.93, 0.80) and the mean and standard deviations of ε with respect to ∆hs and ∆U are relatively smaller.
Further analysis of the different nonparametric methods from theory was carried out by Labroue et al. (2004) and Scharoo and Lillibridge (2004), the same conclusions can be obtained with the above results.
(1) The NW estimation is a local constant estimation. The rate of convergence is slow at a boundary point than that at an interior point which is called boundary effect. While the LLR estimation is a local function estimation which can effectively eliminate the boundary errors.
(2) By using the Epanechnikov kernel function to replace the Gaussian kernel function, we can transform the huge linear matrix into a sparse matrix which can greatly improve the computation efficiency.
(3) Bandwidth is the main parameter that affects the accuracy of the nonparametric estimation. Too large bandwidth leads to a smooth regression curve, and the results close to the parameters. Too small bandwidth leads to a rough regression curve, which means that random errors are hard to be excluded. The selection of bandwidth must be appropriate. By using the locally variable bandwidth to replace the global bandwidth, we can improve the smoothing based on the data density which has obvious advantages in the data-sparse regions.
4.2 Comparisons between LLR-E-L and PM at different latitudes
4.2.1 Analysis of sea state at different latitudes
We have divided the data of Cycle 207 into three latitude ranges, the north latitudes [+20°, 66°), the mid-latitudes [–20°, +20°) and the south latitudes (–66°, –20°). The average and standard deviations of the SWH and the wind speed were calculated respectively in each latitude ranges as shown in Table 3.
As shown in Table 3, the sea states are different at different latitudes. Both SWH and wind speed are large at north latitudes while the SWH and the wind speed are small and steady at mid-latitudes. These will cause large differences of the SSB at different latitudes.
4.2.2 Analysis for SSB estimated models at different latitudes
We selected the following six-parameter form as the parametric model (PM) (Gasper et al., 1994):
$b = {h_{\rm{s}}}[{a_1} + {a_2}{h_{\rm{s}}} + {a_3}U + {a_4}{h_{\rm{s}}}^2 + {a_5}{U^2} + {a_6}{h_{\rm{s}}}U].$
Based on the data set $\left[ {\Delta h',\;\Delta {h_{\rm{s}}},\;\Delta ({h_{\rm{s}}}^2),\;\Delta ({h_{\rm{s}}}U),\;\Delta ({h_{\rm{s}}}^3),\;}\right.$$\left. {\Delta ({h_{\rm{s}}}{U^2}),\;\Delta ({h_{\rm{s}}}^2U)} \right]$, the coefficients of the PM can be calculated by a least square fit as shown in Table 4.
Both the LLR-E-L and the PM were established based on full latitude data set which then were used to estimate the SSB in the above three latitude ranges. The results are shown in Table 5. From Table 5, the explained variances are significant different at different latitudes for either LLR-E-L or PM model because of different sea-state effects. At the mid-latitudes, the SWH and the wind speed are smaller and the induced SSB is smaller. The SWH and the wind speed at the south latitudes are larger but not complicated because of the vast ocean. But at the north latitudes, the sea state is relative complicated due to the land distribution. Accordingly, the nonparametric model LLR-E-L performs well at north latitudes. Its explained variance value (33.25 cm2) is the largest in all. The correlations between the SSB and the SWH or wind speed in the LLR-E-L are higher than those in the PM, especially the correlation between the SSB and the wind speed and the residual distribution is also better than that of the PM. Compared with the PM, the LLR-E-L has no significant advantages at the mid-latitudes and south latitudes. So the PM should be used to estimate the SSB in these two regions considering the computation efficiency.
Scatter plots and residual plots between the SSB and the SWH or wind speed for two models at north latitudes are shown in Figs 1–4. The results show that the nonparametric model LLR-E-L performs better at north latitudes. Especially, the SSB and the wind speed variabilities have been improved significantly.
5 Application of LLR-E-L in HY-2A altimeter
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Both the nonparametric model LLR-E-L (NP) and the parametric model (PM) are used to estimate the SSB of Cycles 70 and 71 (June, 2014) for HY-2A altimeter in the above three latitude ranges. The ascending arc estimates of the SSB at the crossover (n=6 000) were obtained firstly, then the corresponding descending arc estimates of the SSB can be obtained by Eq. (8). The coefficients of the PM can be fitted by crossovers data set of Cycles 70 and 71. The results of two models are shown in Table 6. From Table 6, the explained variance are significantly different at different latitudes because of different sea-state effects. At the mid-latitudes, the SWH and the wind speed are smaller and the induced SSB is smaller. The nonparametric model performs well at the north latitudes. Its explained variance value (32.12 cm2) is largest in all, the correlations between the SSB and the SWH or wind speed are higher (0.88, 0.87), and the mean and standard deviations of residual distributions ($\text{ε} $-∆hs, $\text{ε} $-∆U) are smallest in the three different latitudes. At the mid-latitudes and the south latitudes, the nonparametric model has no significant advantages except the correlation between the SSB and the wind speed. So the nonparametric model is more effective at the north latitudes.
6 Conclusions
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(1) In several nonparametric SSB estimation models, the LLR estimation model with the spherical Epanechnikov kernel function and the locally variable bandwidth has significant advantages.
(2) Compared with the parametric model, the nonparametric model significantly improves the correlation between the SSB and the wind speed which makes the wind speed and the SWH have the same effects on the SSB.
(3) The nonparametric model is more effective than the parametric model above 20° N. Considering an estimation accuracy and a computation efficiency, both models have advantages at different latitudes. Thus, different models should be used to estimate the SSB at different latitudes.
Funding
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  • The National Key R&D Program of China under contract No. 2016YFC1401004; the National Natural Science Foundation of China under contract Nos 41406207, 41176157 and 41406197.
References
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Feng Hui, Yao Shan, Li Linyuan, et al. 2010. Spline-based nonparametric estimation of the altimeter sea-state bias correction. IEEE Geoscience and Remote Sensing Letters, 7(3): 577–581
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Rami A, Kahlouche S, Haddad M, et al. 2011. Nonparametric estimation of the sea state bias in Jason-1 measurements and their effect on Mediterranean mean sea surface height. International Journal of Academic Research, 3(2): 1024–1028
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Appendix
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Year 2017 volume 36 Issue 9
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Article Info
doi: 10.1007/s13131-017-1116-x
  • Receive Date：2016-05-23
  • Online Date：2026-04-16
  • Published：2017-09-01
Article Data
Affiliations
History
  • Received：2016-05-23
  • Accepted：2017-04-25
Funding
The National Key R&D Program of China under contract No. 2016YFC1401004; the National Natural Science Foundation of China under contract Nos 41406207, 41176157 and 41406197.
Affiliations
    1 College of Information Science and Engineering, Ocean University of China, Qingdao 266100, China
    2 National Satellite Ocean Application Service, State Oceanic Administration, Beijing 100081, China

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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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