# Resarch Summary of Dr. Samuel Shen

I am an applied mathematician with research interests in the analysis of climate data and theory of nonlinear waves. Error estimation of the climate data and uncertainty quantification of the climate change assessment are the focus of my research. Our spectral optimal averaging (SOA) method for inhomogeneous fields and our findings of multiple solutions of forced nonlinear waves have influenced the international community's research in the relevant fields. The United Nations’ Inter-governmental Panel for Climate Change (IPCC) adopted our SOA theory to quantify the uncertainty in global warming assessment (IPCC Report 2001, Figures 2.7 and 2.8). The report cited six of our papers. I have published 3 books and over 80 papers including a paper in *Nature Geoscience*. 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’ honor of “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 20 agencies, including NSF, NSERC, and NOAA. This summary describes my past research achievements and their significance.

## 1. New methods and theories for analyzing the climate and agroclimate data

My group has been conducting climate data analysis research in collaboration with distinguished climatologists since 1988. We have published a series of high-quality papers on the analysis of both observed data and model output.

- SOA method:The spectral optimal averaging (SOA) method was developed in 1994 (Shen et al., 1994,
*J. Clim*.), and it is also known as the reduced space optimal averaging (RSOA). The SOA theory provides an approach that can use the climate model data and the recent high quality observations to build covariance functions and to use empirical orthogonal functions (EOFs) to represent climate patterns, such as the El Nino Southern Oscillation. The SOA can adequately treat spatial inhomogeneity and make obsolete the conventional objective analysis approach, which assumes a homogeneous covariance function. 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*.” Some further development and applications of the method were made in subsequent years (Shen et al., 1998; Folland et al., 2001). This method is now widely used for climate data error analysis, including that for the IPCC reports.

- Space-time optimal detection theory:In early 1990s, I worked with Jerry North of Texas A&M University and developed a space-time spectral filter to detect signals of the forced climate change (North et al., 1995,
*J. Clim.*). The method can overcome the problem of stationarity assumption in other detection methods, such as the fingerprint method of K. Hasselmann. Our method has successfully detected the carbon dioxide signal in the global average of surface air temperature. The filter is a solution of an integral equation. This rather general scheme establishes the mathematical foundation for constructing an iteration scheme for the future nonlinear detection strategy. IPCC considered our work as one of the three advances toward statistical climate change detection method in the early 1990s (IPCC 1995 report, p. 416; IPCC 2001 report, Appendix 12.2).

- Error estimate theory in the optimal reconstruction: This theory helps quantify uncertainties in the climate change assessment due to sampling errors and was developed in collaboration with Tom Smith, Chet Ropelewski, and Bob Livezey of the US Climate Prediction Center (CPC) (Shen et al., 1998, J. Clim.; Smith et al., 1998, J. Clim.). 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. Our work improved the CPC scheme for interpolating sparse observation data onto grid points. Our test shows that 4% of data can recover the original field with an error less than 10%.

- RVC method: Regression variance correction (RVC) method was developed in the late 1990s to overcome the space-time over smoothness of daily precipitation interpolation (Shen et al., 2001, 2007, J. Applied Meteo.). The RVC is a hybrid method that uses two steps to grid daily precipitation data. The first step is to interpolate the large scale monthly data, and the second step is to temporally downscale the monthly data into the daily gridded data.

- Randomization theory for point observations of grid box data: The point observation for a grid box data defined in climate models and physical nature is a fundamental problem in climate model validation. An example is to use surface rain-gauges to measure the rain rate over a grid box and to ask a question: what is the ground truth (North et al., 1994, J. Atmos. Ocean. Tech.) Our idea was to randomize the locations of the rain gauges. This general mathematical method helps objectively compare the point observational data with climate model output in order to avoid the apple-orange comparison.

- ECC method: To overcome the “spring barrier” in the US seasonal precipitation prediction, I worked with Bill Lau of the NASA Lab for Atmosphere and developed the ensemble canonical correlation (ECC) analysis method (Shen et al., 2001, NASA Tech. Memo.; Lau et al., 2002, Geophys. Rev. Lett.). Different from the traditional linear statistical forecasting methods, the ECC incorporates the global atmospheric circulation dynamics, and it is a quasi-nonlinear scheme. NASA Goddard Space Flight Center announced the results on January 15, 2002 in its press release as a TOP STORY. US Climate Prediction Center has adopted this method as one of its operational forecasts (Mo, 2002, J. Clim.)

- Nonlinear theory of critical precipitation for forest fires: With my student Robert Field, we developed a nonlinear theory and a new dataset for Indonesia forest fires back to 1960. The airport visibility data, satellite remote sensing data, and precipitation data were used to determine the parameters in the nonlinear model. A paper was published in Nature Geoscience (Field et al., 2009). The journal also published a backstory on the research. Nature made an official press release about the paper on February 22, 2009. The research has then been covered by many news media around the world, including New York Times, New Scientist, and Science Daily.

- Degrees of freedom of a climate field: With my student Xiaochun Wang, we developed a more accurate theory to answer an important climate assessment question: at least how many stations are needed to measure a climate field? Wang and Shen (1999, J. Clim.) found that the earlier estimates of degrees of freedom for annual mean surface air temperature anomalies by other authors were too low.

- Agroclimate database services: (a) Through eight years of collaboration with Alberta Agriculture, Food and Rural Development (AAFRD), we created the first comprehensive agroclimate database for Alberta province, called ABClim 1.0. It includes many variables, such as temperature, precipitation, and growing degree days. 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). I was one of the conceptual designers of this Oracle database system with GIS (Geographic Information System) interface. (b) 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. The information is important to AAFRD’s climate adaptation strategies. AAFRD made a press release on our research results, and several local newspapers reported the work (Shen et al., 2005, J. Appl. Meteo.). (c) 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 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. The atlas is a significant advancement of information technology for Alberta agricultural industry. 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 (fKdV) 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 non-conservation of momentum or other quantities. There exist some surprising phenomena, such as periodic upstream soliton radiation, and hydraulic fall in the fKdV 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 held a special session on nonlinear waves, with emphasis on forced evolution equations. Our results attracted the attention of many leading researchers in PDE and nonlinear waves, such as Ted Wu, David McLaughlin, Jerry Bona, Mark Ablowitz, and Peter Lax.

- Complete bifurcation diagram for fKdV: When a water flow over a bump in a channel has its upstream uniform velocity close to the characteristic speed of shallow water waves, the free-surface profiles of the flow can demonstrate some intriguing phenomena: trans-critical upstream soliton radiation waves, a sub-critical hydraulic fall, and super-critical multiple solitary waves. Using asymptotic analysis, we demonstrated that the forcing due to the bump could be modeled by a Dirac delta function. An analytic expression of the bifurcation diagram was found to classify the types of the surface waves (Shen et al., 1992, J. Fluid Mech., 1992; Gong and Shen, 1994, SIAM J. Appl. Math).

- Confirmation of the fKdV 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 calculation, experimental data, and mathematical theory, we demonstrated otherwise: the fKdV model can actually yield quantitatively accurate results with an error less than 10%. Such an unexpected high accuracy of the fKdV as a model equation encouraged further fKdV studies (Shen, 1995, Quarterly Appl. Math.).

- Spectral scheme, stability of solitary waves and collision of uniform soliton trains: A user-friendly C++ software was developed as a tool for both research and teaching. This package solves the forced KdV equations, forced nonlinear Schrodinger equations and forced sine-Gordon equations. This rather unique software can render numerical, graphic, and animated output. The tool is very convenient for checking the stability of the multiple stationary solutions and for simulating the collision process of the solitons of the same size (Danohue and Shen, 2010, J. Engr. Math.).

- 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. We 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 (Shen, 1993, A Course on Nonlinear Waves, Ch. 6; Shen, 1996, Wave Motion).

## 3. Press Release on the Research Results from Sam Shen’s Group

- 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 published in Geophysical Research Letters. I was responsible for developing the statistical theory of the forecasting. For the complete theory, see Shen et al. (2001), NASA Technical Memorandum. For the complete press release, go to: http://www.gsfc.nasa.gov/topstory/20020115forecast.html

- Fight Fire with Mathematics: Nature Press Release on February 22, 2009, and SDSU News Report on March 9, 2009

Nature made an official press release on Feb 22, 2009. For the complete press release, go to http://www.nature.com/ngeo/press_releases/ngeo0209.html. Many news media around the world, including New York Times, New Scientist, and Science Daily, followed the release and reported our research. In order to get their results, Shen and his colleagues used a statistical model to find how dry the weather had to be in order to trigger a forest fire. A mathematical model was used to study the pattern of air motion and wind direction common during fires. The paper also reconstructed a very important dataset of Indonesia forest fires since 1960. SDSU also made a news report on this work. “When thinking of the common tools used to fight a fire, mathematics does not make it on most people’s lists. But a recent paper published in Nature Geoscience has proven that thinking wrong.” For the complete SDSU press release, go to http://universe.sdsu.edu/sdsuniverse/news.aspx?s=70844. For the complete research paper, go to http://www.nature.com/ngeo/journal/v2/n3/abs/ngeo443.html. Nature also published a back story on the paper. http://www.nature.com/ngeo/journal/v2/n3/index.html#bkstory.

- Agri-News by Alberta Agriculture, Food and Rural Development Ministry, August 22, 2005

Alberta Agriculture, Food and Rural Development Ministry made a press release on August 22, 2005 about my group’s work on the agroclimate changes over Alberta (Shen et al., 2005, J. Appl. Meteo.) The details of the release can be found from http://www.math.ualberta.ca/~shen/Press_Release/Agric_Press.pdf. Newspapers from many cities and communities in Alberta reported this research release.

- Exclusive 12-minute radio interview by CBC Radio-Canada in 2004, and exclusive 15-minute CCTV-China interview in 2003. The interviews can be downloaded from http://www.ualberta.ca/~shen/media-eng.html.

- SDSU Mathematics Department's collaboration with History Channel in 2010 with a worldwide broadcast of the SDSU campus. The TV program can be downloaded from http://video.sdsu.edu/reference/user/dept_mathematics/SUPERHUMAN.mov.