A Statistical Method for Recovering the Depth
to Shallow Groundwater Table in China
袁 星
谢正辉，梁妙玲
中国科学院大气物理研究所
[email protected]
2006.08.10
 Background
 Data
 Methodology
 Validation
 Summary
& Application
Groundwater
在全球总水量中，海洋占97%以上，偏远而难以利用的两
极冰帽及冰川约占2%，其余不到1%才是人类可取用的水资
源，而其中地下水的储存总量居冠。
地下水的过量开采会造成地下水位的大幅下降，引起地面
沉降。地下水位过高会对农作物生长不利，造成渍害。因
此，研究地下水位的动态对国计民生具有重大意义。
气候条件、植被地形和人类活动的变化能引起地下水埋
深时空分布的变化；反之，大尺度地下水埋深的变化，
导致土壤含水量、地表径流和基流的改变，进一步影响
下垫面的蒸散发和低层大气感热和潜热的分配，从而对
气候产生影响。
估计浅层地下水埋深变化对水资源研究、陆面过程模拟、
陆地生态系统及陆气相互作用的研究具有重要意义。
Purpose
To recover monthly data for the depth to
shallow groundwater table since 1961 in
continental China.
Scheme
Transfer function-noise (TFN) models &
parameter transfer method.
Data
Meteorological Data (1961-2000)
Daily time-series of precipitation, maximum temperature,
and minimum temperature are obtained by interpolating
station values from 740 meteorological stations in China.
Soil Data
The soil texture information is derived from Food and
Agriculture Organization dataset (FAO).
Groundwater
Phreatic data from monitor wells.
Locations of the meteorological stations
Locations of wells interpolated into 60×60 km2 grids
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MethodologyⅠ:
Calibration
TFN model
Input:
precipitation surplus (precipitation minus potential
evapotranspiration).
The instantaneous evaporative demand (mm/s) is
calculated following Jarvis and McNaughton (1986):
 s

s 

pet 


 Rn

 s
2k 
s 

dpet 

 (uu  hn  vv  sin(hn))


TFN model
Gt  G  nt
*
t
G  G
*
t
*
t 1
  Pt
(nt  c)   (nt 1  c)  at
State-space representation
Gt*  1 0  Gt*1  0
 


0

1   nt 1 
0
 nt  
  Pt  0

a
c(1  1 )   1  1 t
0
A linear discrete stochastic system
State equation
X t  AX t 1  BUt  Dat
Measurement equation
Yt  Ct X t   t
Recursive application of the Kalman filter
X t  AXˆ t 1  BUt  BU
1 t 1
M t  APt 1 AT  D a2 DT
If there is an observation at time t
vt  yt  Ct X t
 v2,t  CMt CT   2
Kt  M t CT { v2,t }1
Xˆ t  X t  Kt vt
t  (  Kt C ) M t
If no observation taken at time t
Xˆ t  X t
t  M t
Running the Kalman filter for the calibration period
with a parameter set
α T  ( ,  ,  , c,  a2 )
resulting in the following objective function:
2
v
2
i ( )
J ( N ;  )  N ln(2 )   ln( v ,i ( ))   ( 2
)
i 1
i 1  v ,i ( )
N
N
Using SCE-UA (shuffled complex evolution method
developed at The University of Arizona) method to
minimize the objective function.
Identification of TFN model
Transform data to improve normality and stationarity,
Output calibrated parameters
and determine parameters which will be calibrated.
Yes
Representation of TFN model in
vector notations(state space form)
Generate sample
convergence criteria
satisfied?
No
Sample s points randomly in the feasible parameter space.
Running Kalman filter for Optimal prediction
Shuffle complexes
Combine the points into a single sample population.
(calculate the criterion values)
Rank points
Partition and evolve
Sort the s points in order of increasing criterion value.
Partition the s points into p complexes,evolve each complex
Flow chart: the calibration method of TFN model
Methodology Ⅱ:
Parameter Transfer
1 Tropical climate
2 Dry, cold climate
3 Rainy, midlatitude climate
4 Continental climate with hot summer
5 Continental climate with cool summer
6 Continental climate with short cool summer
聚类 (clustering)

基于平方误差的聚类


K均值(K-Mean)
基于概率密度估计的聚类


高斯混合模型：GMM
核密度估计：mean-shift
层次聚类
 基于图的聚类
 模糊聚类
 基于神经元网络的聚类

高斯混合模型(GMM)
(Mixture of Gaussians Model)
 基本思想：将聚类视为一个概率密度估
计问题

给定一堆多峰分布的数据，估计其概率密度
Expectation-Maximum likelihood
(EM) Algorithm
EM Algorithm
Validation of TFN models
Mean Absolute Error: 0.18m 0.15m 0.19m 0.15m
Estimation and 95% confidence intervals of the depth to
groundwater table for the four grids
Moving average parameter
of transfer model
Autoregressive parameter
of transfer model
Autoregressive parameter
of noise model
Variance of noise series
Cross Validation
Time series of errors for cross validation.
(a) ME(t); (b) RMSE(t); (c) MAE(t)
r=0.003
r=0.357
r=0.293
Reconstruction Scheme
Transfer function-noise (TFN) models are calibrated
by SCE-UA method coupled with Kalman filter in
each observed grids.
Parameters for gauged grids are transferred to
ungauged grids by GMM clustering method based
on soil property data and 40-years meteorologic data
such as precipitation and temperature.
The depth to groundwater table for continental
China are estimated by the TFN models with
parameters calibrated or transferred.
Conclusions
(1) Validated by phreatic data, TFN models not only
provide results with high accuracy, but also can
quantify the prediction uncertainty reasonably well.
(2) Cross validation shows that the parameter transfer
scheme is an effective way for the recovery.
(3) The seasonal variations of recharge and discharge
for groundwater in China are obtained by our scheme.
(4) The second EOF match the pattern of mean depth
of the groundwater reasonably well.
Future work
Further validation and modification of our data
by satellite data such as GRACE.
Assessing the improvement of Land surface
model or climate model, running with initial
conditions provided by the recovered data.
谢谢指导！
Thank You!