Supplement of Biogeosciences, 13, 3991–4004, 2016
http://www.biogeosciences.net/13/3991/2016/
doi:10.5194/bg-13-3991-2016-supplement
© Author(s) 2016. CC Attribution 3.0 License.
Supplement of
Analysing the uncertainty of estimating forest carbon stocks in China
Tian Xiang Yue et al.
Correspondence to: Tian Xiang Yue ([email protected]) and Yi Fu Wang ([email protected])
The copyright of individual parts of the supplement might differ from the CC-BY 3.0 licence.
Supplement 1: HASM-SOA
In terms of the fundamental theorem of surfaces, a surface is uniquely defined by the
first and the second fundamental coefficients. The first fundamental coefficients are
used to express the intrinsic geometric properties that do not depend on the shape of
the surface, but only on measurements that we can carry out while on the surface itself.
The second fundamental coefficients reflect the local warping of the surface, which
can be observed from outside the surface (Yue et al. 2016).
If
{( x , y )}
i
j
is an orthogonal division of a computational domain and h
represents the simulation step length, the central point of the lattice ( xi , y j ) could be
expressed as (0.5h + (i − 1)h,0.5h + ( j − 1)h) , in =
which i 0,1, 2,..., I , I + 1 and
j 0,1, 2,..., J , J + 1 . If fi,j(
=
n)
( n ≥ 0 ) represents the iterants of f ( x, y ) at ( xi , y j )
in the nth iterative step, in which
{f ( ) }
0
i, j
are results from SOA, the iterative
formulation of the master equation set of the method for high accuracy surface
modeling (HASM) can be formulated as (Yue et al. 2013b; Zhao and Yue 2014 a, b),
(𝑛𝑛+1)
(𝑛𝑛+1)
(𝑛𝑛+1)
−𝑓𝑓𝑖𝑖+2,𝑗𝑗 + 16𝑓𝑓𝑖𝑖+1,𝑗𝑗 − 30𝑓𝑓𝑖𝑖,𝑗𝑗
=
(𝑛𝑛)
(𝑛𝑛)
−𝑓𝑓
𝑓𝑓
1 (𝑛𝑛) 𝑖𝑖+1,𝑗𝑗 𝑖𝑖−1,𝑗𝑗
(𝛤𝛤11
)𝑖𝑖,𝑗𝑗
2ℎ
(𝑛𝑛+1)
(𝑛𝑛+1)
+
12ℎ2
(𝑛𝑛)
(𝑛𝑛)
(𝑛𝑛) 𝑓𝑓𝑖𝑖+1,𝑗𝑗 −𝑓𝑓𝑖𝑖−1,𝑗𝑗
2ℎ
(𝑛𝑛+1)
(𝑛𝑛+1)
(𝑛𝑛)
(𝑛𝑛)
−𝑓𝑓
𝑓𝑓
2 (𝑛𝑛) 𝑖𝑖,𝑗𝑗+1 𝑖𝑖,𝑗𝑗−1
(𝛤𝛤11
)𝑖𝑖,𝑗𝑗
2ℎ
(𝑛𝑛+1)
−𝑓𝑓𝑖𝑖,𝑗𝑗+2 + 16𝑓𝑓𝑖𝑖,𝑗𝑗+1 − 30𝑓𝑓𝑖𝑖,𝑗𝑗
1
= (𝛤𝛤22
)𝑖𝑖,𝑗𝑗
(𝑛𝑛+1)
12ℎ2
(𝑛𝑛+1)
+
(𝑛𝑛)
𝐿𝐿𝑖𝑖𝑖𝑖
(𝑛𝑛)
(S1-1)
(𝑛𝑛)
�𝐸𝐸𝑖𝑖,𝑗𝑗 +𝐺𝐺𝑖𝑖,𝑗𝑗 −1
(𝑛𝑛+1)
(𝑛𝑛+1)
+ 16𝑓𝑓𝑖𝑖,𝑗𝑗−1 − 𝑓𝑓𝑖𝑖,𝑗𝑗−2
(𝑛𝑛)
(𝑛𝑛)
(𝑛𝑛) 𝑓𝑓𝑖𝑖,𝑗𝑗+1 −𝑓𝑓𝑖𝑖,𝑗𝑗−1
2ℎ
2
+ (𝛤𝛤22
)𝑖𝑖,𝑗𝑗
(𝑛𝑛+1)
+ 16𝑓𝑓𝑖𝑖−1,𝑗𝑗 − 𝑓𝑓𝑖𝑖−2,𝑗𝑗
(𝑛𝑛+1)
𝑓𝑓𝑖𝑖+1,𝑗𝑗+1 − 𝑓𝑓𝑖𝑖+1,𝑗𝑗 − 𝑓𝑓𝑖𝑖,𝑗𝑗+1 + 2𝑓𝑓𝑖𝑖,𝑗𝑗
2ℎ2
+
(𝑛𝑛)
𝑁𝑁𝑖𝑖𝑖𝑖
(𝑛𝑛)
(𝑛𝑛+1)
(S1-2)
(𝑛𝑛)
�𝐸𝐸𝑖𝑖,𝑗𝑗 +𝐺𝐺𝑖𝑖,𝑗𝑗 −1
(𝑛𝑛+1)
(𝑛𝑛+1)
+ 𝑓𝑓𝑖𝑖−1,𝑗𝑗 − 𝑓𝑓𝑖𝑖,𝑗𝑗−1 + 𝑓𝑓𝑖𝑖−1,𝑗𝑗−1
1
=
(𝑛𝑛)
(𝑛𝑛)
−𝑓𝑓
𝑓𝑓
1 (𝑛𝑛) 𝑖𝑖+1,𝑗𝑗 𝑖𝑖−1,𝑗𝑗
(𝛤𝛤12
)𝑖𝑖,𝑗𝑗
2ℎ
+
(𝑛𝑛)
(𝑛𝑛)
−𝑓𝑓
𝑓𝑓
2 (𝑛𝑛) 𝑖𝑖,𝑗𝑗+1 𝑖𝑖,𝑗𝑗−1
(𝛤𝛤12
)𝑖𝑖,𝑗𝑗
2ℎ
+
(𝑛𝑛)
𝑀𝑀𝑖𝑖𝑖𝑖
(𝑛𝑛)
(S1-3)
(𝑛𝑛)
�𝐸𝐸𝑖𝑖,𝑗𝑗 +𝐺𝐺𝑖𝑖,𝑗𝑗 −1
where Ei,j( n ) , Fi,j( n ) and Gi,j( n ) are the iterants of the first fundamental coefficients at the
nth iterative step; L(i,jn ) ,
and N i,j( n ) represent the iterants of the second
2 ( n)
fundamental coefficients at the nth iterative step; (Γ111 )(i ,nj) , (Γ11
)i , j , (Γ122 )(i ,nj) and
(Γ 222 )(i ,nj) the iterants of the Christoffel symbols of the second kind at the nth iterative
step, which depend only upon the first fundamental coefficients and their derivatives.
The matrix formulation of HASM master equations can be respectively
expressed as,
(S1-4)
(S1-5)
(S1-6)
where A , B and
represent coefficient matrices of the first equation, the second
( n)
equation and the third equation; and d , q( n ) and
the
three
equations
z ( n +1)
f ( ) ,..., f ( ) ,......, f ( ) ,..., f ( ) ) ( z (
(=
fi ,( nj )
is the value of the nth iteration of
n +1
1,1
n +1
1, J
are right-hand side vectors of
n +1
I ,1
T
n +1
I ,J
1
n +1)
respectively;
)
,
(xi , yi )
;
,..., z J( n +1) ,..., z((In−+11))⋅ J +1 ,..., z I( n⋅ J+1)
f (x, y ) at grid cell
T
z((in−+1)1)⋅ J + j = fi ,( nj +1) for 1 ≤ i ≤ I , 1 ≤ j ≤ J .
If f i , j is BCS at the pth sample plot (xi , y j ) , s p ,(i −1)× J + j = 1 , and k p = f i , j .
There is only one non-zero element, 1, in every row of the coefficient matrix, S ,
making it a sparse matrix. The solution procedure of HASM, taking the BCS at the
sampled plots as its optimum control constraints and results from SOA as its driving
2
field, can be transformed into solving the following linear equation set in terms of least
squares principles,
(S1-7)
The parameter λ is the weight of the sample plots and determines the
contribution of the sample plots to the simulated surface. λ could be a real number,
which means all sample plots have the same weight, or a sector, which means every
sample plot has its own weight. An area affected by a sample plot in a heterogeneous
region is smaller than in a homogeneous region. Therefore, a smaller value of λ is
selected in a heterogeneous region and a bigger value of λ is selected in a
homogeneous region.
3