Hydrologic trend analysis Dennis P. Lettenmaier Department of Civil and Environmental Engineering University of Washington GKSS School on Statistical Analysis in Climate Research Lecce, Italy October 15, 2009

Outline of this talk 1.Water cycle observations 2.Long-term trend analysis of hydrologic variables 1.Nonparametric approach (seasonal Man n Kendall) 2.Examples 3.Some pitfalls in trend analysis 3.Analysis and trends in hydrologic extremes

1. Water cycle observations Land surface water balance Atmospheric water balance Land surface and atmospheric water and energy balances both contain evapotranspiration (with a multiplier)

Precipitation measurement (dominant hydrological forcing) In situ methods use gauges (essentially points) –Issues with representativeness, changes of instrumentation in time, biases (see Phil Jones lecture) –Wind catch deficiency is critical problem for solid precipitation, of somewhat lesser (but not necessarily negligible) magnitude –Small scale spatial variability (tends to average out with time) Surface radars –Provides near spatially continuous coverage –Terrain blockage issues, also tilt angle, other issues –Complications in producing climate quality records Remote sensing –Indirect (aside from TRMM radar) –Sampling issues aside from geostationary –Solid precipitation issues

Stream discharge (streamflow) measurement Stage (water height), not discharge is measured. Discharge is derived via a (usually power law) rating curve derived from discrete stage and discharge measurements, applied to time- continuous stage measurements Visuals courtesy USGS

Snow Water Equivalent (SWE) measurement – manual snow courses Visuals courtesy NRCS

SWE measurement – automated snow pillows Typical SnoTEL installation Enhanced SnoTEL installation Visuals courtesy NRCS

Other hydrological variables Soil moisture –few climate quality observations with consistent observation methods –In situ methods are complicated by short scale (as small as 1 m) spatial variability Evapotranspiration –Most common long-term measurement is pan evaporation, which can be considered a rough index to potential (not actual) evapotranspiration –Flux towers (AmeriFlux, EuroFlux, FluxNet) provide estimates of latent heat flux (essentially actual evapotranspiration) via either eddy correlation or Bowen ratio methods. However, record lengths are short from a climatological perspective, and generally not of trend quality Groundwater –Relatively few well observations of long length that are not affected by management (withdrawals) –Satellite (GRACE) data provide an alternative over large areas, but record is short (less than a decade), and measurement is effectively all changes in moisture content (atmospheric, soil moisture, lakes, etc) Lakes and wetlands –Very few records suitable for trend analysis –Some work on high latitude lakes (surface area) from remote sensing Glaciers –Relatively small number of glaciers have detailed mass balance records, but many changes are not subtle –Changes in area are generally easier to detect than storage (and are amenable to visible satellite imagery with record lengths exceeding 30 years –Satellite altimetry (e.g. ICESAT) provides basis for storage estimates, but record lengths are short

Hydrologic data characteristics Precipitation: usually measured as accumulations over time, statistics are characterized by intermittency and high Cv and skewness for accumulation intervals < multiple days. Correlation lengths increase with accumulation intervals, generally greater for winter (synoptic scales events) than summer (convective); skewness and Cv decrease with accumulation intervals (and intemittency vanishes) Streamflow: Most variable of land surface fluxes; can be intermittent for short accumulation intervals in arid areas and in some cases small drainage areas. Controls include precipitation and evaporative demand, but also land surface characteristics and drainage areas. It is an areal integrator. Spatial correlation lengths usually longer than for precipitation. Evapotranspiration: Least variable of three major land surface hydrologic fluxes. Near-direct measurement difficult, and mostly applicable to points, large area estimates from remote sensing (essentially indirect/model), or by difference Soil moisture: Few long-term observations available, most viable approach (not without shortcomings) is model reconstruction Groundwater: Few long-term observations available that are not dominated by management effects; large area estimates (which include other storage terms) now possible via satellite microgravity (GRACE) Snow water equivalent: Point measurements from snow courses (now increasingly replaced with automated snow pillows)

2.1 Hydrologic trend analysis – nonparametric approaches

Testing for Trends Ho: Distribution (F) of R.V. X t is same for all t H1: F changes systematically with time We may also want to describe the amount or rate of change, in property (e.g. central tendency) of the distribution

Parametric vs Nonparametric statistics Parametric: Assume the distribution of X t (often Gaussian) Nonparametric: Form of distribution not assumed (but often are some assumptions, e.g. common distribution aside from change in central tendency) Nonparametric tests are usually more robust to violation of assumptions that must be made for parametric tests, however when parametric tests are appropriate, the range of quantitative inferences that can be made is usually greater

Monotonic Trend: Continuing (and not reversing) with time Parametric test example: linear regression (with time) Nonparametric test examples: Kendall’s tau; spearman’s rho (essentially rank correlation with time) Step Trend: One-time change, of fixed amount Parametric test example: t-test Non-parametric test example: Mann Whitney

Kendall’s Tau (  ) Tau (  ) measures the strength of the monotonic relationship between X and Y. Tau is a rank-based procedure and is therefore resistant to the effect of a small number of unusual values. Because  depends only on the ranks of the data and not the values themselves, there are adjustments for missing or censored data (essentially treated as ties) – tests work with a “limited amount of” such data In general, for linear associations,  0.9 corresponds to  > 0.7. For trend test, Y can be time

The test statistic S measures the monotonic dependence of X on t: – S = P - M –where : P = # of (+), the # of times the X’s increase with t, or the # of X i < X j for all t i < t j (“concordant pairs”). –M = # of (-), the # of times the X’s decrease with t, or the number of X i > X j for all t i <t j (“discordant pairs”). –i = 1, 2, … (n-1); and j = (i+1), …, n. There are n(n-1)/2 possible comparisons to be made among the n data pairs. If all y values increased along the x values, S = n(n- 1)/2. In this situation,  = +1, and vice versa. Therefore dividing S by n(n-1)/2 will give a -1 <  < +1. Adjustment can be made for ties (missing or censored data)

 is defined as : Critical value of S can be determined by enumerating the discrete distribution of S, when the data are randomly ranked with time For n > about 10, there is a large sample approximation to the test statistic; for smaller values, tables of the exact distribution are available

Key assumptions for Kendall’s tau (or Mann-Kendall test) Common distribution of X t (most importantly homoscedastic) Independence (no temporal correlation)

Large sample approximation The large sample approximation Z s is given by: And, Z s = 0, if S = 0, and where: The null hypothesis is rejected at significance level  if Z s > Z crit where Z crit is the critical value of the standard normal distribution with probability of exceedance of  /2 (i.e., S is approximately normally distributed with mean 0).

Kendall slope estimator Med {(X j -X i )/(t j -t i )} for all j>I

Seasonality effects Usually result in violation of key assumption, as distributions of most hydrologic (and climatic) variables change with season One approach is to “homogenize” time series e.g. by seasonal transformation (can be left with issues as to seasonally varying correlation)

where t i is number of ties in season i From Hirsch et al (1982) Seasonal Kendall Test (per Hirsch et al, 1982)

Absent missing data (note that g is season index, p is number of seasons): Where r gh is Spearman’s rho (rank correlation) between seasons g and h

From Hirsch et al (1982)

2.2 Examples

Minimum flow Increase No change Decrease Mann Kendall analysis -- annual minimum flow from to Visual courtesy Bob Hirsch, figure from McCabe & Wolock, GRL, 2002

About 50% of the 400 sites show an increase in annual median flow from to Median flow Increase No change Decrease Visual courtesy Bob Hirsch, figure from McCabe & Wolock, GRL, 2002

About 10% of the 400 sites show an increase in annual maximum flow from to Maximum flow Increase No change Decrease Visual courtesy Bob Hirsch, figure from McCabe & Wolock, GRL, 2002

USGS streamgage annual flood peak records used in study (all >=100 years) Visual courtesy Bob Hirsch

Number of statistically significant increasing and decreasing trends in U.S. streamflow (of 395 stations) by quantile (from Lins and Slack, 1999)

Annual hydroclimatic trends over the continental U.S., from Lettenmaier et al, 1994

Monthly streamflow trends over the continental U.S., from Lettenmaier et al, 1994

Estimated spatial correlation functions (anisotropic)

Field significance levels (from Lettenmaier et al, 1994)

Model Runoff Annual Trends period selected to account for model initialization effects Positive trends dominate (~28% of model domain vs ~1% negative trends) Positive + Negative Drought trends in the continental U.S. – from Andreadis and Lettenmaier (GRL, 2006)

HCN Streamflow Trends Trend direction and significance in streamflow data from HCN have general agreement with model-based trends Subset of stations was used (period ) Positive (Negative) trend at 109 (19) stations

Soil Moisture Annual Trends Positive trends for ~45% of CONUS (1482 grid cells) Negative trends for ~3% of model domain (99 grid cells) Positive + Negative

2.3 Pitfalls in trend analysis 1)Spurious trends (e.g., changes in instruments; site-specific effects). Solution: understand the data and adjust as necessary; evaluate spatial consistency of trends (site specific effects should not have a spatial signature) 2)Multiple comparison problem (“fishing expeditions”). Solution: test field significance; pre-specify the tests, time periods, etc to be tested. 3)Strong conclusions from short record lengths (e.g. satellite data)

References Fowler, H.J., and C.G. Kilsby, A regional frequency analysis of United Kingdom extreme rainfall from 1961 to 2000, International Journal of Climatology 23, Hirsch, R.M., J.R. Slack, and R.A. Smith, Techniques of trend analysis for monthly water quality data, Water Resources Research 18, Hirsch, R.M., and J.R. Slack, A nonparametric trend test for seasonal data with serial dependence, Water Resources Research 20, Lettenmaier, D.P., E.F. Wood, and J.R. Wallis, Hydro- climatological trends in the continental U.S., , Journal of Climate 7, Livezey, R.E., and W.E. Chen, Statistical field significance and its determination by Monte Carlo techniques, Monthly Weather Review 111,

3. Analysis and trends in hydrological extremes

Probability weighted moments and L-moments

Clearwater River flood frequency distribution (from Linsley et al 1975)

Fitted flood frequency distribution, Potomac River at Pt of Rocks, MD Visual courtesy Tim Cohn, USGS

Problems with traditional fitting methods –mixed distributions

Pecos River flood frequency distribution (from Kochel et al, 1988)

Inferred elasticity (“sensitivity”) of extreme floods with respect to MAP as a function of return period (from regional flood frequency equations) Q T = K A b1 * P b2 dQ/Q)/dP/P = dln[Qp]/dln[P] = b2

JANUARY FLOODS JANUARY 12, 2009 When disaster becomes routine Crisis repeats as nature’s buffers disappear Disaster Declarations Federal Emergency Management Agency disaster declarations in King County in connection with flooding: January 1990 November 1990 December 1990 November 1995 February 1996 December 1996 March 1997 November 2003 December 2006 December 2007 Mapes 2009

Urban Stormwater Infrastructure Urbonas and Roesner 1993 Minor Infrastructure Roadside swales, gutters, and sewers typically designed to convey runoff events of 2- or 5-year return periods. Major Infrastructure Larger flood control structures designed to manage 50- or 100-year events.

Objectives 1.What are the historical trends in precipitation extremes across Washington State? 2. What are the projected trends in precipitation extremes over the next 50 years in the state’s urban areas? 3.What are the likely consequences of future changes in precipitation extremes on urban stormwater infrastructure?

Literature Review

Karl and Knight % increase in total precip (nationally) since 1910 Mostly due to trends in highest 10% of daily events Kunkel et al. 1999, % increase in frequency of 7-day extremes (nationally) from Some frequencies nearly as high at beginning of 20th century as at end of 20th century No significant trend found for Pacific Northwest

Literature Review Madsen and Figdor 2007 Statistically significant increase of 30% in frequency of extreme precipitation in Washington from Statistically significant increase of 45% in Seattle Statistically significant decrease of 14% in Oregon Non-significant increase of 1% in Idaho

Literature Review Two main drawbacks with prior research: Not focused on sub-daily extremes most critical to urban stormwater infrastructure Not focused on changes in event intensity most critical to urban stormwater infrastructure

Literature Review Fowler and Kilsby 2003 Used “regional frequency analysis” to determine changes in design storm magnitudes from 1960 to 2000 in the United Kingdom Employed framework that we adapted for our study

Historical Precipitation Analysis

Study Locations

Visual Inspection Divided precipitation records into two 25-year time periods ( and ). Compared annual maxima between two periods at storm durations ranging from 1 hour to 10 days. Time series of 1-hour and 24-hour annual maxima on following six slides for SeaTac, Spokane, and Portland Airports (shown in color, with other stations in each region shown in gray).

1-Hour Annual Maxima at SeaTac Avg at airport = 0.34” Avg at airport = 0.36”

1-Hour Annual Maxima at Spokane Avg at airport = 0.36”

1-Hour Annual Maxima at Portland Avg at airport = 0.39” Avg at airport = 0.40”

24-Hour Annual Maxima at SeaTac Avg at airport = 2.00” Avg at airport = 2.48”

24-Hour Annual Maxima at Spokane Avg at airport = 1.04” Avg at airport = 1.12”

24-Hour Annual Maxima at Portland Avg at airport = 1.95” Avg at airport = 1.97”

Regional Frequency Analysis Principle: Annual precipitation maxima from all sites in a region can be described by common probability distribution after site data are divided by their at- site means. Larger pool of data results in more robust estimates of design storm magnitudes, particularly for longer return periods.

Regional Frequency Analysis Methods: Annual maxima divided by at-site means. Regional growth curves fit to standardized data using method of L-moments. Site-specific GEV distributions obtained by multiplying growth curves by at-site means. Design storm changes calculated for various return periods. Sample procedure shown on following slides.

1. Annual maxima calculated for each station in region. Average = 2.00” Average = 2.48”

2. Each station’s time series divided by at site mean. Average = 1

3.Standardized annual maxima pooled and plotted using Weibull plotting position.

4.Regional growth curves fitted using method of L-moments.

5.Site-specific GEV distributions obtained by multiplying regional growth curves by at-site means.

6.Probability distributions checked against original at-site annual maxima

7.Changes in design storms calculated for various return periods. +37% +30% Change in Average Annual Maximum = +25%

Statistical Significance General indication of how likely a sample statistic is to have occurred by chance. We tested for: →differences in means (Wilcoxon rank-sum) →differences in distributions (Kolmogorov-Smirnov) →non-zero temporal trends (Mann-Kendall) Tests performed at a 5% significance level.

Results of Historical Analysis Changes in average annual maxima between 1956–1980 and 1981–2005: SeaTacSpokanePortland 1-hour+7%-1%+4% 3-hour+14%+1%-7% 6-hour+13%+1%-8% 24-hour+25%+7%+2% 5-day+13%-10%-5% 10-day+7%-4%-10% * * Statistically significant for difference in means

Decadal changes in regional growth curves, UK from Fowler and Kilsby, 2003

Future Precipitation Projections

Global Climate Models ECHAM5 Developed at Max Planck Institute for Meteorology (Hamburg, Germany) Used to simulate the A1B scenario in our study CCSM3 Developed at National Center for Atmospheric Research (NCAR; Boulder, Colorado)Used to simulate the A2 scenario in our study

Global Climate Models Mote et al 2005 ECHAM5 CCSM3

Dynamical Downscaling Courtesy Eric Salathé Global Model Regional Model

Results of Future Analysis SeaTacSpokanePortland 1-hour+16%+10%+11% 24-hour+19%+4%+5% 1-hour-5%-7%+2% 24-hour+15%+22%+2% * Statistically significant for difference in means and distributions, and non-zero temporal trends ECHAM5 CCSM3 * * * * Changes in average annual maximum precipitation between 1970–2000 and 2020–2050:

Future Runoff Simulations

Overview: Bias Correction Bias Correction and Statistical Downscaling Performed at the grid point from each simulation that was closest to SeaTacBias corrected data used to drive hydrologic model Area (ac) Imp Area Thornton714029% Juanita435234%

Overview: Bias Correction Bias Correction and Statistical Downscaling of hourly precipitation Raw RCM output differs from observed record in both frequency of events and amounts of precipitation.For example, from 1970 to 2000 for SeaTac Airport: -CCSM3/A2 simulation resulted in 11,734 hours of nonzero precipitation for a total of 225 inches during the month of January, -Observations recorded 4144 hours of nonzero precipitation for a total of 162 inches during the months of January.

Overview: Bias Correction Bias Correction and Statistical Downscaling Despite biases in modeled data, projections may still prove useful if interpreted relative to the modeled climatology rather than the observed climatology.Performed separately for each calendar month.

Overview: Bias Correction Bias Correction and Statistical Downscaling Procedure based on probability mapping as described by Wilkes (2006) and Wood et al. (2002): 1.Simulated 1970–2000 data truncated so that each month had the same number of nonzero hourly values as the corresponding observed record. 2.Simulated 2020–2050 data truncated with same thresholds. 3.Monthly totals recalculated, and Weibull plotting position used to map those totals from the modeled empirical cumulative distribution function (eCDF) to those from the observed eCDF. 4.Modeled hourly values rescaled to add up to new monthly totals. 5.New hourly values mapped from their eCDF to the hourly values from the observed eCDF, and once again rescaled to add up to the monthly totals derived in the first mapping step.

Overview: Bias Correction Bias Correction and Statistical Downscaling Raw RCM output differs from observed record in both frequency of events and amounts of precipitation.Despite these biases, projections may still prove useful if interpreted relative to the simulated climatology rather than the observed climatology.

Overview: Bias Correction Bias Correction and Statistical Downscaling Procedure based on probability mapping as described by Wilks (2006) and Wood et al. (2002)Performed at the grid point from each simulation that is closest to SeaTacBias corrected data used to drive hydrologic model in Thornton and Juanita Creek watersheds.

7590 (11, ) hours w/ smallest amounts of nonzero precip eliminated from 1970–2000 simulated record, coinciding w/ a truncation threshold of 0.012”.Any hour during the 2020–2050 simulated record w/ a nonzero precip of less than 0.012” also eliminated (6824 out of 10,322, for a remainder of 3498 hours). Overview: Bias Correction Step 1: Truncate simulated data so that each month has the same number of nonzero hourly values from 1970 to 2000 as the observed data. Observed ,734 Hours of Nonzero Precipitation in January 10,322 CCSM CCSM Observed CCSM CCSM

Corresponding precipitation total reduced from 5724 mm to 5272 mm from 1970 to 2000, and from 4960 mm to 4573 mm from 2020 to Overview: Bias Correction Step 1: Continued Observed mm 5724 mm Total Precipitation in January 4960 mm CCSM CCSM Observed mm 4573 mm CCSM CCSM mm

Step 2: Recalculate simulated monthly totals, and map those totals from the simulated eCDF of to the observed eCDF of Simulated monthly totals replaced with values having the same nonexceedance probabilities, with respect to the observed climatology, that they have with respect to the simulated climatology. Simulated hourly values rescaled to add up to new monthly totals.

Step 3: Map new hourly values from simulated eCDF of to the observed eCDF of Simulated hourly values replaced with values having the same nonexceedence probabilities, with respect to the observed climatology, that they have with respect to the simulated climatology. Hourly values again rescaled to add up to monthly totals derived in the first mapping step.

Step 4: Recalculate annual maxima at durations ranging from 1-hr to 10-days New simulated annual maxima roughly match observed annual maxima from 1970 to 2000

Results of Bias Correction Raw BiasCorrected Bias 1-hour-19%-7% 24-hour+11%-2% 1-hour-33%-13% 24-hour-22%+3% ECHAM5 CCSM3 Improvements to bias of average annual maximum:

Results of Bias Correction Raw ChangeCorrected Change 1-hour+16%+14% 24-hour+19%+28% 1-hour-5%-6% 24-hour+15%+14% ECHAM5 CCSM3 Comparison of changes in average annual maximum between 1970–2000 and 2020–2050: * * Statistically significant for difference in means and distributions, and non-zero temporal trends * **

Results of Bias-Correction (CCSM3/A2 )

Results of Bias-Correction (ECHAM5/A1B )

Hydrologic Model Used HSPF (Hydrologic Simulation Program – Fortran), a continuous rainfall-runoff model that has been regionally validated and endorsed by EPA, USGS, FEMA, and WA-DOE for several decades. Primary inputs are hourly precipitation, daily potential evapotranspiration. Typically accurate given calibration with good contemporaneous precipitation and flow data.

Thornton Creek

Changes in Average Streamflow Annual Maxima ( to )

Results of Hydrologic Modeling Changes in average streamflow annual maxima at outlet of watershed between and : Juanita CreekThornton Creek CCSM3+25%+55% ECHAM5+11%+28% * Statistically significant for difference in means **

The November Surprise JANFEBMARAPR MAYJUNJULAUG SEPOCTNOVDEC Courtesy Eric Salathé NOV

INCREASE? 25-yr 24-hr Design Storms at SeaTac VARIABILITY? ECHAM5? CCSM3?

Concluding thoughts on hydrologic extremes Much of the work in the climate literature on “extremes” doesn’t really deal with events that are extreme enough to be relevant to risk analysis (typically estimated from the annual maximum series) Regional frequency analysis methods help to filter the natural variability in station data Decadal scale differences in flood risk are detectable in the historical record, to what extent are these manifestations of decadal (vs long-term) climate variability? RCMs help to make extremes information more regionally specific, but nonetheless contain information that may be “smoother” than observations Extent to which RCM-derived changes in projections of extremes are controlled by GCM-level extremes is unclear Use of ensemble approaches is badly needed, however RCM computational requirements presently precludes this