RESEARCH SUMMARY
Samuel
S.P. Shen, McCalla Professor at the University of Alberta
E-mail: shen@ualberta.ca
I am an applied
mathematician and statistician with research interests in statistical
climatology and nonlinear waves. Scientific computing is my main research tool.
Optimal estimation is my currently main research direction. This summary
describes my past research achievements and their significance. Briefly, my group’s spectral method for
inhomogeneous spatial statistics and our method for finding multiple solutions
of forced nonlinear waves have influenced the international community's
research methods and directions. We established a theory to estimate the
sampling errors of an inhomogeneous field. The United Nations’
Inter-governmental Panel for Climate Change (IPCC) adopted our theory to
estimate the uncertainty in global warming (IPCC Report 2001, Figures 2.7 and
8). The report cited six of our papers. I have been personally recognized with
international honors and awards, such as the prestigious US National Academy of
Science’s NRC Associateship award in 1999 and the Chinese Academy of Sciences’
“Well-known Overseas Chinese Scholar” in 2001. I was elected as President-elect
of the Canadian Applied and Industrial Mathematics Society in 1999, and
Vice-President of the Canadian Mathematical Society in 2003. My research team
has been supported by more than ten agencies, including NOAA and NASA.
1.
Statistical
Climatology and Agroclimatology
My group has been conducting statistical
climatology research in collaboration with distinguished climatologists since
1988. We have published a series of high-quality papers on climate data analysis
and on optimal detection and prediction of climate changes.
(i)
Spectral approach to estimate the global
average temperature and higher order spherical harmonics:
An optimal averaging scheme was developed to estimate the various orders of the
spherical harmonic components of the surface air temperature from the surface
station data (J. Climate, 1994,
1996). By using empirical orthogonal
functions (EOFs), our method can adequately treat spatially inhomogeneous
fields and make obsolete the conventional approach, which assumes a homogeneous
covariance function. The algorithm of the spectral method clearly reflects the
physical meaning of climate patterns such as the El Nino Southern Oscillation.
The editor of J. Climate highly
praised our contribution to climate data analysis: “This work makes an important advance over previous studies involving
optimal averaging in its consideration of inhomogeneous, anisotropic covariance
structure. The continued use of obviously inappropriate assumptions about this
structure will do nothing more than delay the optimum exploitation of existing
historical data networks or the optimum design of the future ones. Your point
about the relative insensitivity of error reductions to the exact shapes of the
eigenvectors is extremely important.”
(ii)
Optimal linear filter design:
A linear optimal filter was constructed to detect forced climate signals such
as the change in the global average of surface air temperature caused by the
increase of carbon dioxide or the variation of solar radiation (J. Climate, 1995). The filter is
formulated as a solution of a linear integral equation resulting from
minimizing the mean square error of detection.
This rather general scheme establishes the mathematical foundation for
constructing an iteration scheme for the future nonlinear detection strategy.
This work has been cited by more than 42 papers, and, more importantly, it was
cited by the IPCC Reports (Climate Change
1995 and 2001). The IPCC Report is the world's most authoritative and
comprehensive publication on climate change. Our method is considered as a
significant step forward in climate change detection.
(iii)
Regional optimal averaging and gridding
for a nonhomogeneous climate variable: This research was
done in collaboration with Bob Livezey, Chet Ropelewski and Tom Smith of the
National Center for Environmental Modeling (NCEP), NOAA (J. Climate, 1998a, b). The optimal averaging method uses
extrapolated eigenvalues, the area factor in computing EOFs, and cross
validation. This method is considered the most accurate regional averaging
scheme currently available and has been applied to the Tropical Pacific SST
(sea surface temperature) field. The gridding scheme of NCEP was improved to
optimally interpolate sparse observation data onto grid points. Our test shows
that 4% of data can recover the original field with an error less than 10%.
(iv)
Degrees of freedom of climate fields:
A statistical theory was developed to answer an important climate assessment
question: at least how many stations are needed to measure a climate field?
This paper examined four statistical models for estimating degrees of freedom
and showed that the earlier estimates by other authors were too low (J. Climate, 1999).
(v)
Rainfall measurement and ground
validation problem: Both surface rain-gauges and
multiple-satellites were used to measure the rain rate in the tropics and
sub-tropics (J. Appl. Meteo., 1993). Several data sets, including Global
Atlantic Tropical Experiment data of precipitation experiments were
studied. The satellite microwave
sensing of rainfall was calibrated with ground station validation. Numerical
simulations were based upon Tropical Rainfall Measurement Mission satellite
paths. Also see J. Atmos. Ocean. Tech.,
1994.
(vi)
Development of the benchmark
observational dataset for model validation: The
improvement of climate models needs validation from accurate historical
observational data. The most robust signal is the monthly surface temperature
and has been accurately reconstructed from 1856-1998. Our idea of using the
spectral method for the reconstruction has proved to be successful to produce
the benchmark data for model validation. The resulting datasets are more
reliable than that of NCEP's Reanalysis. See Environmetrics, 2004.
(vii)
An accurate seasonal forecasting method:
In collaboration with Bill Lau’s group of the NASA Goddard Space Flight Center,
we developed a new seasonal forecasting scheme for regional precipitation. It
is called the CEC (canonical ensemble correlation) prediction and is a
quasi-nonlinear scheme. The forecasting skill is better than that of NCEP. NASA
GSFC announced the results on January 15, 2002 in its press release as a TOP
STORY.
(viii)
Creation of agroclimate database:
We created the first comprehensive agroclimate database, called ABClim 1.0.
Developed originally for Alberta Agriculture, Food and Rural Development
(AAFRD), this database includes daily precipitation, maximum temperature,
minimum temperature, incoming solar radiation, relative humidity, wind speed,
and wind direction on every township, ecodistrict polygon, and soil landscape
polygon in Alberta from January 1, 1901 to present. It also includes many
derivative products useful to farmers and ranchers, such as growing degree
days, corn heat units, and evaportranspiration. Precipitation frequency and
extreme events are required for the interpolated daily precipitation data
between stations to retain both spatial and temporal variances. Our hybrid
interpolation method can meet these requirements (J. Appl. Meteo.,
2001). Nationally, ABClim 1.0 serves as a prototype model for the
climate component of Canada’s digital agriculture system called NLWIS (National
Land and Water Information Service), currently under development. I was one of
the conceptual designer of this Oracle database system with GIS (Geographic
Information System) interface. The design was completed in December 2004.
(ix)
Assessment of Alberta’s agrcolcimatic
changes from 1901 to 2002: My group, in collaboration with AAFRD,
developed the first documentation on the details of the Alberta agroclimatic
changes since 1901. We analyzed the long-term (1901-2002) temporal trends in
the agroclimate of Alberta, and explored the spatial variations and the
potential crop-growing area in Alberta, and made several important conclusions
based on analyzing the data. The information will help Albertans optimally
manage the land usage for crops and livestock, and is important to AAFRD’s
climate adaptation strategies. See J. Appl. Meteo., 2005.
(x)
Generation of the Agroclimatic Atlas of
Alberta: The Agroclimatic Atlas of Alberta is a 97-page
document that presents climatic information of importance to the agriculture
community in Alberta. The latest edition was published in 2003 with major
additions, reflecting the new issues faced by the farming community. The current
atlas has been well distributed to Alberta farm communities, schools, and
relevant governmental ministries, as well as relevant research centers
worldwide. It is also displayed on the AAFRD website (http://www1.agric.gov.ab.ca/$department/deptdocs.nsf/all/sag6278?opendocument).
The website version includes maps for previous 30-year periods (1901 to 1930,
1911 to 1940 and so on to 1961 to 1990) as well as additional maps as they
become available. The atlas is a significant advancement of information
technology for Alberta agricultural industry. My group was responsible for
providing the data used to generate the maps, and ABClim 1.0 was the backbone
of the atlas’s map data. See S. Chetner and the Agroclimatic Atlas Working
Group, Agroclimatic Atlas of Alberta, 2003.
2.
Fluid
Dynamics and Forced Nonlinear Waves
Before 1994, my group
was mainly investigating the waves modeled by forced evolution equations, such as
the forced Korteweg-de Vries equations, forced nonlinear Schrodinger equations
and forced sine-Gordon equations. The mathematical difficulty of this research
results from the lack of the group symmetries associated with unforced problems
due to the inconservation of momentum or other quantities. There exist some
surprising phenomena, such as periodic upstream soliton radiation and hydraulic
falls in the forced Korteweg-de Vries equations, which do not occur in the
unforced cases. The mathematical community
has recognized the importance of our research in this direction. In 1997, the
American Mathematical Society arranged a special session on nonlinear waves,
with emphasis on forced evolution equations. Our results attracted the
attention of some leading researchers in PDE and nonlinear waves, such as Ted
Wu, David McLaughlin, Jerry Bona, Mark Ablowitz, and Peter Lax.
(i)
The finding of the complete bifurcation
diagram: A channel flow over a bump was considered. The
upstream-velocity was uniform and close to its critical speed. The free-surface
profiles of the flow were determined by the upstream velocity, when the bump
was given. Using asymptotic analysis, we demonstrated that the forcing due to
the bump could be modeled by a Dirac delta function. Finally, the surface waves were analytically classified according
to the upstream velocity (J. Fluid Mech.,
1992, SIAM J. Appl. Math, 1994).
(ii)
Confirmation of the fKdV equation as an
accurate model equation: Since the fKdV equations were derived
based upon the assumption of small amplitude and long wave, many researchers
believed that these models could provide only
qualitative results. Using numerical, experimental data, and
mathematical theory, we demonstrated that the fKdV model can actually yield
quantitatively accurate results with an error less than 10%. Such a high degree
of accuracy of the fKdV equation as a model equation, although somewhat
unexpected, encouraged further studies on the fKdV equations (Quarterly Appl. Math., 1995).
(iii)
Spectral scheme, stability of solitary
waves and collision of uniform soliton trains: A C++
software was developed by using the spectral method. This package solves the
forced KdV equations, forced nonlinear Schrodinger equations and forced
sine-Gordon equations. This rather unique software has a user-friendly
interface. It can render numerical, graphic, and animated output. It is an
important, effective and convenient tool for studying the evolution of an
initial profile governed by one of the above three types of equations. In particular,
it is very useful for checking the stability of the multiple stationary
solutions and for simulating the collision process of the solitons of the same
size.
(iv)
Mechanical energy for transcritical flows:
In the transcritical regime, solitons are periodically generated and radiated
upstream, and a depression zone and a modulated wake zone are simultaneously
generated downstream. It was found that the depth of the depression can be
determined by the solvability condition of a boundary value problem for an ordinary
differential equation. Upon obtaining the depression depth, the flow
characteristics, such as the period of the soliton generation, can be
analytically determined (A Course on
Nonlinear Waves, Ch. 6, 1993, and Wave
Motion, 1996).
3.
Important
Contributions to Other Research
The diverse expertise
of the people in my group enables us to conduct research in other important
areas of applied mathematics. With Bill Perry at Texas A&M University, we
completed a project on reactive toxic mass spreading. An interesting result is
the computation of the “waiting time” for nonlinear parabolic equations. Our
1990 paper appeared to be the first to give concrete values for the “waiting
time.” With Yarlong Wang, now a Senior Engineer in the oil industry based at
Calgary, we worked on the hydraulic fracturing of rocks for studying the
stability of oil-production wells. The research involved multi-phase flows and
rock mechanics. Wang's company has developed commercial software based on our
finite element models.
4.
NASA Goddard Space
Fight Center’s Top Story Press Release on January 15, 2002: -New Method Greatly Improves
U.S. Seasonal Forecasts
A new technique could raise the bar for predicting
seasonal precipitation by 10 to 20 percent for all seasons in the United
States, a NASA-funded study finds. The new method looks at changes in sea
surface temperatures in various ocean basins, and then weighs their individual
impacts on regional climate to greatly increase predictability of precipitation
during all seasons. Changes in sea surface temperatures strongly influence
atmospheric winds, climate and weather. “The paper presents results
applied to the U.S. continent, where we show that the potential predictability
can be raised 10 to 20 percent above traditional methods,” said William Lau, a senior
researcher at Goddard and lead author of the paper. “The scheme can be applied
to other regions as well. It raises the bar for seasonal and inter-annual
climate forecasts." The paper was
presented on January 15 at the American Meteorology Society meeting in Orlando,
Fla. The study will also be published in an upcoming issue of the journal Geophysical
Research Letters. (W.K.M. Lau, K.M. Kim and S.S.P. Shen, Potential
predictability of seasonal precipitation over the United States from canonical
ensemble correlation predictions,
Geophys. Res. Lett. , accepted for publication (2002).) I was
responsible for developing the statistical theory of the forecasting. See
S.S.P. Shen, K.M. Lau, K.M. Kim and G. Li, Error estimation of an ensemble
statistical seasonal precipitation prediction model, NASA Technical
Memorandum, NASA-TM-2001-209989, 2001. For the complete article on the use
of sea surface temperatures to forecast seasonal precipitation, go to: http://www.gsfc.nasa.gov/topstory/20020115forecast.html