Geophysical Research Letters
Supporting Information for
A framework for combining multiple soil moisture retrievals based on maximizing
temporal correlation
Seokhyeon Kim1, Robert M. Parinussa1, Yi. Y. Liu2, Fiona M. Johnson1, Ashish Sharma1
1School
of Civil and Environmental Engineering, University of New South Wales, Sydney, Australia, 2ARC
Centre of Excellence for Climate Systems Science & Climate Change Research Centre, University of New
South Wales, Sydney, Australia
Contents of this file
Derivation of equation for optimal weighting factor
Tables S1 to S3
Figures S1 to S15
Introduction
The following supplementary data contains a derivation of the equation for calculating
the optimal weighting factor, Table S1 showing details of datasets used in this study,
Figure S1 presents the locations of ground stations used for evaluating combination
performances from the soil moisture retrievals from the night-time observations
(descending satellite path). Figure S2 shows the locations of ground stations for the daytime observations (ascending satellite path).
The results from the day-time observations are provided for comparisons through Figure
S3, S4 and S5 which correspond with Figure 1, 2 and 3 respectively in the main
manuscript.
In addition, combination results with another reference, MERRA-Land top soil layer soil
moisture content (SFMC), are presented through Figure S6 to S8 for descending data
and Figure S9 to S11 for ascending data. The results for the descending and ascending
data with the two references, ERA-Interim and MERRA-Land, are summarized in Table
S2.
Finally, cross-validation results are presented and summarized in Figures S12 to S15
and Table S3. The cross-validation results are based on combinations using the
ascending and descending data with ERA-Interim and MERRA-Land which are
interchangeably used for combination and validation. Unlike the cases in which one
reference (ERA-Interim or MERRA-Land) is used for both combination and validation,
negative values are observed over a number of regions but there is still an improvement
at the global scale.
1
Derivation of equation for optimal weighting factor
Two sets of unbiased soil moisture retrievals 𝜽𝟏 and 𝜽𝟐 (n×1) are linearly combined into
𝜽𝒄 by applying a weighting factor w, 0 to 1.
𝜽𝒄 = 𝑤𝜽𝟏 + (1 − 𝑤)𝜽𝟐
(1)
The Pearson correlation coefficient (R) between 𝜽𝒄 and a reference ( 𝜽𝑹 ) can be
expressed as a function of w according to the definition of R and equation (3), and this is
an optimization problem of the following function.
Maximize 𝑅 = 𝑓(𝑤) =
𝐸[(𝜽𝒄 −𝜇𝐶 )(𝜽𝑹 −𝜇𝑅 )]
𝜎𝑐 𝜎𝑅
(2)
Subject to 0≤ 𝑤 ≤ 1
Where 𝜇𝐶 and 𝜇𝑅 are the mean values, and 𝜎𝑐 and 𝜎𝑅 are the standard deviations
of 𝜽𝒄 and 𝜽𝑹 respectively. From equation (1),
𝜇𝐶 = E[𝜽𝑪 ]
= E[𝑤𝜽𝟏 + (1 − 𝑤)𝜽𝟐 ]
= 𝑤𝜇1 + (1 − 𝑤)𝜇2
(3)
𝜎𝑐2 = Var(𝜽𝑪 )
= E[(𝜽𝒄 − 𝜇𝐶 )2 ]
= E[𝜽2𝑪 ] − 𝜇𝐶 𝟐
= E[(𝑤𝜽𝟏 + (1 − 𝑤)𝜽𝟐 )2 ] − (𝑤𝜇1 + (1 − 𝑤)𝜇2 )2
= [Var(𝜽𝟏 ) + Var(𝜽𝟐 ) − 2𝐶𝑜𝑣(𝜽𝟏 , 𝜽𝟐 )] ∙ 𝑤 2 − 2 ∙ [𝑉𝑎𝑟(𝜽𝟏 ) − 𝐶𝑜𝑣(𝜽𝟏 , 𝜽𝟐 )] ∙ 𝑤 + Var(𝜽𝟐 )
(4)
Therefore, from equation (3) and (4),
𝑓(𝑤) =
=
𝐸[(𝜽𝒄 − 𝜇𝐶 )(𝜽𝑹 − 𝜇𝑅 )]
𝜎𝑐 𝜎𝑅
[𝐶𝑜𝑣(𝜽𝟏 , 𝜽𝑹 ) − 𝐶𝑜𝑣(𝜽𝟐 , 𝜽𝑹 )] ∙ 𝑤 + 𝐶𝑜𝑣(𝜽𝟐 , 𝜽𝑹 )
Var(𝜽𝑹 ) ∙ [Var(𝜽𝟏 ) + Var(𝜽𝟐 ) − 2𝐶𝑜𝑣(𝜽𝟏 , 𝜽𝟐 )] ∙ 𝑤 2 …
√
−2 ∙ Var(𝜽𝑹 ) ∙ [𝑉𝑎𝑟(𝜽𝟏 ) − 𝐶𝑜𝑣(𝜽𝟏 , 𝜽𝟐 )] ∙ 𝑤 + Var(𝜽𝑹 ) ∙ Var(𝜽𝟐 )
=
𝐴∙𝑤+𝐵
√𝐶 ∙ 𝑤 2 + 𝐷 ∙ 𝑤 + 𝐸
(5)
Where,
A = 𝐶𝑜𝑣(𝜽𝟏 , 𝜽𝑹 ) − 𝐶𝑜𝑣(𝜽𝟐 , 𝜽𝑹 )
B = 𝐶𝑜𝑣(𝜽𝟐 , 𝜽𝑹 )
C = Var(𝜽𝑹 ) ∙ [Var(𝜽𝟏 ) + Var(𝜽𝟐 ) − 2𝐶𝑜𝑣(𝜽𝟏 , 𝜽𝟐 )]
D = −2 ∙ Var(𝜽𝑹 ) ∙ [𝑉𝑎𝑟(𝜽𝟏 ) − 𝐶𝑜𝑣(𝜽𝟏 , 𝜽𝟐 )]
2
E = Var(𝜽𝑹 ) ∙ Var(𝜽𝟐 )
Differentiating equation (5) with respect to w,
𝑓 ′ (𝑤) =
𝐴
√𝐶 ∙ 𝑤 2 + 𝐷 ∙ 𝑤 + 𝐸
−
(𝐵 + 𝐴 ∙ 𝑤)(𝐷 + 2 ∙ 𝐶 ∙ 𝑤)
2 ∙ (𝐶 ∙ 𝑤 2 + 𝐷 ∙ 𝑤 + 𝐸)2/3
(6)
Therefore the optimal weighting factor is calculated by letting equation (6) 0 and
simplified as
𝑤=
=
2∙𝐴∙𝐸−𝐵∙𝐷
2∙𝐵∙𝐶−𝐴∙𝐷
𝜎2 (𝜌1𝑅 − 𝜌12 ∙ 𝜌2𝑅 )
𝜎1 (𝜌2𝑅 − 𝜌12 ∙ 𝜌1𝑅 ) + 𝜎2 (𝜌1𝑅 − 𝜌12 ∙ 𝜌2𝑅 )
(7)
3
Table S1. Details of data used for this study
Data source
AMSR2-JAXA
AMSR2-LPRM
AMSR2-LPRM
ERA-Interim
ERA-Interim
MERRA-Land
ISMN
ESA CCI
Temporal
resolution
Variable
Level 3 geophysical parameter
SMC
Level 3 Surface Soil Moisture
(X-band)
Vegetation optical depth (Cband)
Soil water contents level 1 (00.07m depth)
Soil temperature level 1 (00.07m depth)
Top soil layer soil moisture
consent (SFMC)
In-situ measured soil moisture
from 8 networks
topographic complexity, wetland
fraction
Spatial
resolution
Units
Daily
0.25º
m3/m3
Daily
0.25º
m3/m3
Daily
0.25º
-
6 hourly
0.25º
m3/m3
6 hourly
0.25º
K
Hourly
0.25º
(Resampled)
m3/m3
Hourly
Point
m3/m3
-
0.25º
%
Table S2. Summary of combination results from the ascending and descending data
with ERA-Interim and MERRA-Land as references. Generally, the combination
performance is more prominent for the JAXA and the ascending data than the LPRM
and the descending data
Descending
Scale
Statistics
Product
ERAInterim
0.52
MERRALand
0.51
ERAInterim
0.50
MERRALand
0.43
JAXA
0.35
0.31
0.33
0.26
LPRM
0.45
0.44
0.36
0.30
0.31
0.46
0.43
0.57
Combined
Mean R
Global
Mean w
Mean R
In-situ
stations
Mean w
Ascending
JAXA
0.37
LPRM
0.63
0.69
0.54
Combined
0.56
0.45
0.53
0.52
JAXA
0.35
0.34
0.34
0.34
LPRM
0.56
0.45
0.49
0.49
JAXA
0.24
0.22
0.26
0.29
LPRM
0.76
0.78
0.74
0.71
4
Table S3. Summary of cross-validation results (i.e. correlation coefficients) from
combination using the ascending and descending data with ERA-Interim and MERRALand which are interchangeably used for combination and validation.
Satellite path
Combination
Reference
Validation
Reference
Combined
Descending
Ascending
ERA-Interim
MERRA-Land
ERA-Interim
MERRA-Land
MERRA-Land
ERA-Interim
MERRA-Land
ERA-Interim
0.47
0.49
0.39
0.46
JAXA
0.31
0.35
0.26
0.33
LPRM
0.44
0.45
0.30
0.36
Remark
Figure S12
Figure S13
Figure S14
Figure S15
5
Figure S1. Locations of 159 in-situ stations from 8 networks, selected from the ISMN,
that were used for evaluating performances of combination using the soil moisture
retrievals from the night-time observations (descending satellite path).
6
Figure S2. Locations of 164 in-situ stations from 8 networks, selected from the ISMN,
that were used for evaluating performances of combination using the soil moisture
retrievals from the day-time observations (ascending satellite path). The reason for the
different number of stations compared to Figure S1 is that number of observations at
day-time is generally larger than ones at night-time due to moderately higher
temperature reducing data mask.
7
Figure S3. The spatial distribution of the optimal weights for the JAXA and LPRM soil
moisture products at the day-time (ascending satellite path) using ERA-Interim soil water
contents level 1 as the reference.
8
Figure S4. Results of combination using datasets at the ascending satellite path with
ERA-Interim soil water contents level 1 as the reference: Spatial distribution of Pearson’s
correlation coefficients between the reference and a) the combined product (RCOM), b)
the JAXA product (RJAXA) and c) the LPRM product (RLPRM). Where, the global mean of
RCOM is 0.50, RJAXA, 0.33 and RLPRM, 0.36 respectively. Panel d) shows the differences in
between correlation coefficients of the combined and JAXA products (RCOM minus RJAXA),
and e), the combined and LPRM products (RCOM minus RLPRM).
9
Figure S5. Results for evaluating improvements in correlation coefficients through
combinations of data at the ascending satellite path using ERA-Interim soil water
contents level 1 as the reference. a) Scatter plot showing correlation coefficients of the
JAXA and LPRM products (RJAXA and RLPRM on y-axis respectively) against correlation
coefficients of the combined product (RCOM on x-axis). b) Boxplots for three sets of
correlation coefficients for the JAXA, LPRM and combined products against the
reference. Where, the mean of correlation coefficients for the JAXA product is 0.34, the
LPRM product, 0.49 and the combined product, 0.53 respectively. c) Boxplot for
weighting factors (w) from all in situ stations. Where, the mean of weighting factors is
0.26 for the JAXA product and 0.74 for the LPRM product.
10
Figure S6. The spatial distribution of the optimal weights for the JAXA and LPRM soil
moisture products at the night-time (descending satellite path) using MERRA-Land top
soil layer soil moisture consent as the reference.
11
Figure S7. Results of combination using datasets at the descending satellite path with
MERRA-Land top soil layer soil moisture content as the reference: Spatial distribution of
Pearson’s correlation coefficients between the reference and a) the combined product
(RCOM), b) the JAXA product (RJAXA) and c) the LPRM product (RLPRM). Where, the global
mean of RCOM is 0.51, RJAXA, 0.31 and RLPRM, 0.44 respectively. Panel d) shows the
differences in between correlation coefficients of the combined and JAXA products
(RCOM minus RJAXA), and e), the combined and LPRM products (RCOM minus RLPRM).
12
Figure S8. Results for evaluating improvements in correlation coefficients through
combinations of data at descending satellite path using MERRA-Land top soil layer soil
moisture content as the reference. a) Scatter plot showing correlation coefficients of the
JAXA and LPRM products (RJAXA and RLPRM on y-axis respectively) against correlation
coefficients of the combined product (RCOM on x-axis). b) Boxplots for three sets of
correlation coefficients for the JAXA, LPRM and combined products against the
reference. Where, the mean of correlation coefficients for the JAXA product is 0.34, the
LPRM product, 0.45 and the combined product, 0.45 respectively. c) Boxplot for
weighting factors (w) from all in situ stations. Where, the mean of weighting factors is
0.22 for the JAXA product and 0.78 for the LPRM product.
13
Figure S9. The spatial distribution of the optimal weights for the JAXA and LPRM soil
moisture products at the day-time (ascending satellite path) using MERRA-Land top soil
layer soil moisture consent as the reference.
14
Figure S10. Results of combination using datasets at the ascending satellite path with
MERRA-Land top soil layer soil moisture content as the reference: Spatial distribution of
Pearson’s correlation coefficients between the reference and a) the combined product
(RCOM), b) the JAXA product (RJAXA) and c) the LPRM product (RLPRM). Where, the global
mean of RCOM is 0.43, RJAXA, 0.26 and RLPRM, 0.30 respectively. Panel d) shows the
differences in between correlation coefficients of the combined and JAXA products
(RCOM minus RJAXA), and e), the combined and LPRM products (RCOM minus RLPRM).
15
Figure S11. Results for evaluating improvements in correlation coefficients through
combinations of data at descending satellite path using MERRA-Land top soil layer soil
moisture content as the reference. a) Scatter plot showing correlation coefficients of the
JAXA and LPRM products (RJAXA and RLPRM on y-axis respectively) against correlation
coefficients of the combined product (RCOM on x-axis). b) Boxplots for three sets of
correlation coefficients for the JAXA, LPRM and combined products against the
reference. Where, the mean of correlation coefficients for the JAXA product is 0.34, the
LPRM product, 0.49 and the combined product, 0.52 respectively. c) Boxplot for
weighting factors (w) from all in situ stations. Where, the mean of weighting factors is
0.29 for the JAXA product and 0.71 for the LPRM product.
16
Figure S12. Cross-validation results of combination using data at the descending
satellite path with ERA-Interim soil water contents level 1 as the reference: Spatial
distribution of Pearson’s correlation coefficients between MERRA-Land top soil layer soil
moisture content and a) the combined product (RCOM), b) the JAXA product (RJAXA) and
c) the LPRM product (RLPRM). Where, the global mean of RCOM is 0.47, RJAXA, 0.31 and
RLPRM, 0.44 respectively. Panel d) shows the differences in between correlation
coefficients of the combined and JAXA products (RCOM minus RJAXA), and e), the
combined and LPRM products (RCOM minus RLPRM).
17
Figure S13. Cross-validation results of combination using data at the descending
satellite path with MERRA-Land top soil layer soil moisture content as the reference:
Spatial distribution of Pearson’s correlation coefficients between ERA-Interim soil water
contents level 1 and a) the combined product (RCOM), b) the JAXA product (RJAXA) and c)
the LPRM product (RLPRM). Where, the global mean of RCOM is 0.49, RJAXA, 0.35 and
RLPRM, 0.45 respectively. Panel d) shows the differences in between correlation
coefficients of the combined and JAXA products (RCOM minus RJAXA), and e), the
combined and LPRM products (RCOM minus RLPRM).
18
Figure S14. Cross-validation results of combination using data at the ascending satellite
path with ERA-Interim soil water contents level 1 as the reference: Spatial distribution of
Pearson’s correlation coefficients between MERRA-Land top soil layer soil moisture
content and a) the combined product (RCOM), b) the JAXA product (RJAXA) and c) the
LPRM product (RLPRM). Where, the global mean of RCOM is 0.39, RJAXA, 0.26 and RLPRM,
0.30 respectively. Panel d) shows the differences in between correlation coefficients of
the combined and JAXA products (RCOM minus RJAXA), and e), the combined and LPRM
products (RCOM minus RLPRM).
19
Figure S15. Cross-validation results of combination using data at the ascending satellite
path with MERRA-Land top soil layer soil moisture content as the reference: Spatial
distribution of Pearson’s correlation coefficients between ERA-Interim soil water contents
level 1 and a) the combined product (RCOM), b) the JAXA product (RJAXA) and c) the
LPRM product (RLPRM). Where, the global mean of RCOM is 0.46, RJAXA, 0.33 and RLPRM,
0.36 respectively. Panel d) shows the differences in between correlation coefficients of
the combined and JAXA products (RCOM minus RJAXA), and e), the combined and LPRM
products (RCOM minus RLPRM).
20