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To know what we cannot know: To know what we cannot know: Global mapping of minimal Global mapping of minimal
detectable precipitation trendsdetectable precipitation trendsdetectable precipitation trendsdetectable precipitation trends
Efrat MorinEfrat MorinGeography DepartmentGeography Department
The Hebrew University of JerusalemThe Hebrew University of Jerusalem
To know what we cannot know:To know what we cannot know:To know what we cannot know: To know what we cannot know: Global mapping of minimal Global mapping of minimal
d t t bl i it ti t dd t t bl i it ti t dTalk outline
detectable precipitation trendsdetectable precipitation trendsTalk outline
The problemObjectivesjMethodologyMinimal trend detectionGlobal mappingControls of the minimal detectable trendHydrological implicationsHydrological implicationsSummary
The problemThe problem
Precipitation trendsPrecipitation trends are of special interest are of special interest i t t di t t din water resources management and in water resources management and
planningplanning
ButButButBut
The natural high variability of precipitationThe natural high variability of precipitationThe natural high variability of precipitation The natural high variability of precipitation data masks the possibility to detect existing data masks the possibility to detect existing
t dt dtrendstrends
The problemThe problem
T dT d i i ifi ti i ifi tTrends are Trends are insignificantinsignificantin most land areasin most land areas
Trenberth et al., 2007 (Ch. 3 in the 4th Assessment Report of the IPCC)
The problemThe problem
Insignificant trends
Trends in land Trends in land averaged data are averaged data are
l ftl ftalso often also often insignificantinsignificant
Trenberth et al., 2007 (Ch. 3 in the 4th Assessment Report of the IPCC)
--2121..11Israeli stations: Trends in mm per decadeThe problemThe problem
--1919..99
--1919..88
Israeli stations: Trends in mm per decadefor 1964-2003
99 44
--00..99
--99..44
--44..88--1919..3355..99Trends are Trends are insignificantinsignificant
in in allall examined stationsexamined stations
00..00
--55..99
--00..22
The problemThe problem
How to deal with insignificant trends?Option 1: assume there are no trendsOption 1: assume there are no trendsOption 2: understand that trends maybe exist but have a low chance to be detected (i.e., probability ofhave a low chance to be detected (i.e., probability of type II error is large)
T ki h 2 t l t ti t h t i thTaking approach 2 we can at least estimate what is the “minimal detectable trend” for specific conditions
ObjectiveObjective
The main objective is to quantify and to map The main objective is to quantify and to map j q y pj q y pover the globe lower limits of detectable over the globe lower limits of detectable annual precipitation linear trendsannual precipitation linear trendsp pp p
ObjectiveObjective
The minimal detectable trend:The minimal detectable trend:The minimal trend (absolute value) that th b bilit t id tif it i ifi tthe probability to identify it as significant (5% level) is higher than a pre-selected threshold (in this study: 20 and 50%).
In statistical terms: the minimal trend for which the power of the test (the complementary of the probability of type II error)the test (the complementary of the probability of type II error) is higher than a pre-selected threshold.
Assumptions:MethodologyMethodology
Assumptions:
• Record length: 50 years
A l i it ti d t
501 tt L• Annual precipitation data
501 PPL
• Linear trends
P β• Assumptions on residuals: depends on trend
iii tP εβα ++=Assumptions on residuals: depends on trend detection method
• 5% significance level
MethodologyMethodologyLinear trend detection
Simple linear regression method:Residuals are assumed to be independent, normally p , ydistributed with equal variance. Regression parameters are estimated from data using least square method, and their significance is examined with a t-test.
Mann-Kendall method:A good non-parametric, rank-based alternative for simple g p , plinear regression when the normality assumption cannot be met and an outlier effect should be reduced. Modifications
i t t t f i l l ti f id lexist to account for serial correlation of residuals.
MethodologyMethodology
⎞⎜⎛ − ij PP
diβ̂
Mann-Kendall method:
⎟⎠
⎜⎜⎝ −
=<
ij
ij
ji ttmedianβ
∑ ∑−
−=1
)(n n
ij PPsignS= +=1 1i ij
( )( )( )521181)( +−= nnnSVar
)(SsignS −
( )( )( )18
)(
)1,0(~)(
)( NSVar
SsignSz = for n>8
* Accounting for serial correlation* Accounting for ties
Minimal trend detectionMinimal trend detectionR li i f l i i i 50 d iRealizations of annual precipitation 50-year data series were generated for a given positive trend (β) and precipitation characteristics: mean ( ) and coefficient of variance ( )P )(PCV
2000:1951=itcharacteristics: mean ( ) and coefficient of variance ( ) P )(PCV
tP εβα ++=
2000:1951it
iii tP εβα ++=
)0( 2N ),0(~ 2σε Niindependent
tP ⋅−= βα tP βα
Minimal trend detectionMinimal trend detectionResidual variance is related to precipitation moments assuming zero covariance between the residuals and the time:
)(2 Var == εσ
( ))()(
2
2 tVarPVar =⋅−= β( ) )()( 22 tVarPPCV ⋅−⋅= β
d d/10β
Minimal trend detectionMinimal trend detectionExample:Example:
)(120)(
600=
=mmPStd
mmP decademm /10=β=> 8% in 50 years
Example:Example:
1000
20.0)( =PCV
700
800
900
mm
)
Non-significant trend(p-value=0.44)
500
600
700
cipi
tatio
n (m
300
400
Annu
al p
rec
0
100
200A
Realization 1Realization 2
Significant trend(p-value=0.03)
1951 1961 1971 1981 1991Year
Minimal trend detectionMinimal trend detectionExample:Example: 600= mmP decademm /10=βExample:Example:
200)(120)(
600
==
=
PCVmmPStd
mmP decademm /10β=> 8% in 50 years
Monte-Carlo simulations: 1000 realizations12% found significant 88% found insignificant at 5% level
20.0)(PCV
12% found significant, 88% found insignificant at 5% level(i.e., estimated probability of type II error is: 88%)Probability of this trend to be detected is estimated 12%Probability of this trend to be detected is estimated 12% which is lower from the pre-defined thresholds of 20 and 50%. The computations is done for different trend magnitudes until the thresholds are exceeded.For mean precipitation of 600 mm and CV of 0 2 theFor mean precipitation of 600 mm and CV of 0.2 the minimal detectable trends are:
12 5 /d d ith 20% b bilit th h ld12.5 mm/decade with 20% probability threshold20.0 mm/decade with 50% probability threshold
Minimal trend detectionMinimal trend detection1 200
20% threshold Response surfaces of the0.8
0.9
1
140
160
180
200 Response surfaces of the minimal detectable trend (in mm/decade) as a function of
CV
0.5
0.6
0.7
80
100
120
mm/decade) as a function of the precipitation mean and CV
0.2
0.3
0.4
20
40
60
Precipitation (mm)
100 200 300 400 500 600 700 800 900 1000
0.1 0
0.9
1
160
180
20050% threshold
CV
0.6
0.7
0.8
100
120
140
160
C
0.3
0.4
0.5
40
60
80
Precipitation (mm)
100 200 300 400 500 600 700 800 900 1000
0.1
0.2
0
20
Global mappingGlobal mappingThe GPCC VASClimO data set was used to compute meanThe GPCC VASClimO data set was used to compute mean annual precipitation and CV globally over land areas for years 1951-1999 at 0.5o x 0.5o resolution
Beck et al., 2004
Global mappingGlobal mappingMean annual Mean annual precipitationprecipitation
CVCV
Based on GPCC VASClimO data set
20% probability thresholdGlobal mappingGlobal mapping
Absolute minimal Absolute minimal detectable trenddetectable trend
Trend (mm/decade)50% probability threshold
Global mappingGlobal mappingTrend (mm/decade)Trend (mm/decade)
S th E dSouthern EcuadorMean = 850 850 mmmm CV = 00..99Min20 = 8585 mm/decademm/decade
Northern ChadMean = 155 155 mmmm CV = 00..11Min20 = 1717 55 mm/decademm/decadeMin20 = 85 85 mm/decademm/decade
Min50 = 148 148 mm/decademm/decadeChange 20 = 5050%%
Min20 = 1717..5 5 mm/decademm/decadeMin50 = 2727..5 5 mm/decademm/decadeChange 20 = 5656%%
50% probability threshold
The highest undetectable trends are mainly in the tropics due to
gChange 50 = 8787%%
Change 20 5656%%Change 50 = 8989%%
The highest undetectable trends are mainly in the tropics due to high precipitation mean and variability. In arid and semi-arid regions the minimal detectable trends are gconsiderably lower but are very substantial when translated into percent change in annual precipitation.
20% probability thresholdGlobal mappingGlobal mapping
Absolute relative Absolute relative minimal detectable minimal detectable t dt dtrendtrend
Change from the mean Israel (north)over 50 years
( )Mean = 535 535 mmmm CV = 00..2525Min20 = 15 15 mm/decademm/decade
Precipitation change (% in half century)
50% probability thresholdMin50 = 25 25 mm/decademm/decadeChange 20 = 1414%%Change 50 = 2323%%(% in half century)Change 50 = 2323%%
Global mappingGlobal mappingObserved trend(mm/decade)
P-value
Controls of the minimal detectable trendControls of the minimal detectable trend
• Mean and CV of annual precipitation• Record lengthRecord length• Temporal smoothing/Serial correlation• Spatial averaging/Spatial correlationSpatial averaging/Spatial correlation
Controls of the minimal detectable trendControls of the minimal detectable trendMean and CV of annual precipitationMean and CV of annual precipitationMean and CV of annual precipitationMean and CV of annual precipitation
50% threshold
0.9
1
180
20050% threshold
0.7
0.8
120
140
160C
V
0.5
0.6
80
100
120
0.3
0.4
40
60
80
100 200 300 400 500 600 700 800 900 1000
0.1
0.2
0
20
Precipitation (mm)100 200 300 400 500 600 700 800 900 1000
Controls of the minimal detectable trendControls of the minimal detectable trendRecord lengthRecord length
6
Record lengthRecord length
Example:Example:5
e)
20%50%120)(
600=
=mmPStd
mmPpp
4
/dec
ade %
20.0)()(=PCV
2
3
nd (m
m/
1
2
Tre
00 50 100 150 200
Record length (years)
Controls of the minimal detectable trendControls of the minimal detectable trendRecord lengthRecord lengthRecord lengthRecord length
1200
Jerusalem annual precipitation 1863-20071000
(mm
)
p p
600
800
cipi
tatio
n
400
nnua
l pre
c
Trends are often nonTrends are often non--linear especiallylinear especially
0
200An
Trends are often nonTrends are often non--linear, especially linear, especially for longer recordsfor longer records
0
1863
1868
1873
1878
1883
1888
1893
1898
1903
1908
1913
1918
1923
1928
1933
1938
1943
1948
1953
1958
1963
1968
1973
1978
1983
1988
1993
1998
2003
Year
Annual precipitation Filtered precipittion series Mean annual precipitation 95% range
Controls of the minimal detectable trendControls of the minimal detectable trendTemporal smoothing/Serial correlation (under study):Temporal smoothing/Serial correlation (under study):Temporal smoothing/Serial correlation (under study):Temporal smoothing/Serial correlation (under study):
Smoothing reduces variance but also increases the serialSmoothing reduces variance but also increases the serial correlation.
In general, if a time series is positively correlated then the trend identification test will find a significant trend more often than it will for an independent series (Kulkarni and von Storch, 1995).
Controls of the minimal detectable trendControls of the minimal detectable trendTemporal smoothing/Serial correlation (under study):Temporal smoothing/Serial correlation (under study):
1000
600P
Temporal smoothing/Serial correlation (under study):Temporal smoothing/Serial correlation (under study):Example:Example: CV=0.19
r =0 16
600
800
tion
(mm
)
200)(120)(
600=
=
PCVmmPStd
mmP r1=0.16
400
600
ual p
reci
pita
t
Original realization0
20.0)(=
=β
PCV
CV=0.08
0
200Ann
u Smoothed realization
Original data:Sl 15 9 /d d l 0 12
r1=0.98
01950 1960 1970 1980 1990 2000
Year
Slope=15.9 mm/decade p-value=0.12
Smoothed data without accounting for serial correlation:Smoothed data without accounting for serial correlation:Slope=23.4 mm/decade p-value<0.001
Smoothed data with accounting for serial correlation:Slope=23.4 mm/decade p-value<0.09
Controls of the minimal detectable trendControls of the minimal detectable trendSpatial averaging/Spatial correlation (under study):Spatial averaging/Spatial correlation (under study):Spatial averaging/Spatial correlation (under study):Spatial averaging/Spatial correlation (under study):Spatial averaging reduces variance and can improve trend detectability. However, averaging over large area (> 35o) is y , g g g ( )required to get significant trends and in this area may include different trend signs.
0.8
0.9
-1
0
p-val
Significant trends
0 5
0.6
0.7al
-3
-2
/dec
ade)
Slope
Example:Example:Averaging around
0.3
0.4
0.5
p-va
6
-5
-4
Slop
e (m
m/Averaging around
Israel pixel
0
0.1
0.2
-8
-7
-6 S
00 10 20 30 40 50 60
Averaged box size (degrees)
8
Hydrological implicationsHydrological implicationsIsrael (north)
Recharge to the Runoff volume in
Israel (north)Mean = 535 535 mmmm CV = 00..2525Min20 = 15 15 mm/decademm/decadeMin50 = 25 25 mm/decademm/decade
Yarqon-Taninim regional Aquifer
the Taninim catchment (51 km2)
Change 20 = 1414%%Change 50 = 2323%%
Taninim catchmentcatchment
Rain station
Hydrological implicationsHydrological implicationsContinuous hydrological model representing the main processesContinuous hydrological model representing the main processes of the water budget: rain, infiltration, runoff, evapotranspiration, and deep percolation.
ETRain (mm/day)
P ( / )
p p
β f ( /d ) s ( )
Runoff (mm/day)ET (mm/day)Pan evap. (mm/day)
z (mm),θs,θfc,θpwp
β f (mm/day), s (mm)
θParameters:Z: Effective depth (mm)
µ
p ( )θs: Saturation water contentθfc: Field capacityθpwp: Permanent wilting point
Deep percolation (mm/day)
p pf: Constant infiltration rate (mm/h) s: Surface water storage (mm)β: Pan evaporation coefficient
D l i ffi iµ: Deep percolation coefficientState variable:θ: Current water content Based on: Sheffer et al., 2009 with modifications
1.6
Hydrological implicationsHydrological implicationsEffect on aquifer recharge
1.2
1.4
rech
arge 10% increase in annual rainfall
19% increase in annual recharge
Effect on aquifer recharge Yarqon-Taninim aquifer
0.8
1
ange
in a
nnua
l
10% decrease in annual rainfall
0.4
0.6
0.7 0.8 0.9 1 1.1 1.2 1.3
Cha
10% decrease in annual rainfall20% decrease in annual recharge
1.61.8
2
off
10% increase in annual rainfall
0.7 0.8 0.9 1 1.1 1.2 1.3Change in annual rainfall
Effect on runoff volumeTaninim catchment
0 81
1.21.4
n an
nual
runo 10% increase in annual rainfall
31% increase in annual runoff
0.20.40.60.8
Cha
nge
i
10% decrease in annual rainfall28% decrease in annual runoff
00.7 0.8 0.9 1 1.1 1.2 1.3
Change in annual rainfall
SummarySummaryTrends in precipitation data are often masked by their highTrends in precipitation data are often masked by their high variance.Monte-Carlo simulations together with the Mann-Kendall method were applied to detect the probability of existing trends to be found significant.The minimal detectable trends were derived using probabilityThe minimal detectable trends were derived using probability thresholds of 20% and 50%.Minimal trends were mapped globally both in mm/decade and in
t f l Th GPCC VASCli O d t tpercent from mean annuals. The GPCC VASClimO data set was used for the mapping.The highest minimal detectable trends were found in the tropics andThe highest minimal detectable trends were found in the tropics and other wet regions, but in terms of percent from mean semi-arid and arid areas are emphasized.Th i t l f th i i l d t t bl t dThe main controls of the minimal detectable trends are: mean precipitation, CV, record length, temporal and spatial smoothing. It is demonstrated that the hydrological systems (aquifer recharge y g y ( q gand catchment runoff volume) magnifies trends in precipitation means.
In many cases the chance to detect a significant trend in
ConclusionsConclusionsIn many cases the chance to detect a significant trend in precipitation data series is low even if the trend exists.Onl trends abo e some threshold are detectableOnly trends above some threshold are detectable.The undetectable (in terms of statistical significance) t d i ti t li ibl d htrends are in practice not negligible and can have a crucial impact on water resources availability.Th k l d f th li it i i t t i tThe knowledge of these limits is important input especially for risk assessment that is related to adaption decision making.
Th k fTh k fThanks for Thanks for your your
attention!attention!
800
600
700m
m)
Taninim catchment
YRTN recharge(Jerusalem)
400
500
al v
alue
(m
(Jerusalem)
74% 68%
200
300
400
ean
annu
a
29%
0
100
200
Me
23%
3%3%0
Rain Surface runoff ET Recharge
1200
The problemThe problem
1000
mm
)Jerusalem annual precipitation 1863-2007
800
itatio
n (m
400
600
al p
reci
pi
200Ann
u
0
1863
1868
1873
1878
1883
1888
1893
1898
1903
1908
1913
1918
1923
1928
1933
1938
1943
1948
1953
1958
1963
1968
1973
1978
1983
1988
1993
1998
2003
Year* Reconstructed from two station records
Annual precipitation Filtered precipittion series Mean annual precipitation 95% range
1-lag autocorrelation = 0.09 (p-value = 0.29)Coefficient of variance = 169 / 568 = 0.30
Minimal trend detectionMinimal trend detection
0.8
0.9
1
200
250
0.8
0.9
1
200
250C
V
0.5
0.6
0.7
100
150
CV
0.5
0.6
0.7
100
150
0 1
0.2
0.3
0.4
0
50
0 1
0.2
0.3
0.4
0
50
Precipitation (mm)100 200 300 400 500 600 700 800 900 1000
0.1 0
Precipitation (mm)
100 200 300 400 500 600 700 800 900 1000
0.1 0
1 250
0 6
0.7
0.8
0.9
150
200
CV
0.3
0.4
0.5
0.6
50
100
Precipitation (mm)
100 200 300 400 500 600 700 800 900 1000
0.1
0.2
0
50