Precipitation

• Precipitation: water falling from the atmosphere to the earth.

– Rainfall

– Snowfall– Snowfall

– Hail (grandine), sleet (nevischio)

• Requires lifting of air mass so that it cools and condenses.

Mechanisms for air lifting

1. Frontal lifting

2. Orographic lifting

3. Convective lifting

Definitions

• Air mass : A large body of air with similar temperature

and moisture characteristics over its horizontal extent.

• Front: Boundary between contrasting air masses.

• Cold front: Leading edge of the cold air when it is

advancing towards warm air.

• Warm front: leading edge of the warm air when

advancing towards cold air.

Frontal Lifting

• Boundary between air masses with different properties is called a front

• Cold front occurs when cold air advances towards warm air

• Warm front occurs when warm air overrides cold air

Cold front (produces cumulus cloud) Warm front (produces stratus cloud)

Orographic liftingOrographic uplift occurs when air is forced to rise because of the physical

presence of elevated land.

Convective liftingConvective precipitation occurs when the air near the ground is heated by the Convective precipitation occurs when the air near the ground is heated by the

earth’s warm surface. This warm air rises, cools and creates precipitation. earth’s warm surface. This warm air rises, cools and creates precipitation.

Hot earth

surface

Precipitation formation

• Lifting cools air masses so moisture condenses

• Condensation nuclei

– Aerosols

– water molecules attachattach

• Rising & growing

– 0.5 cm/s sufficient to carry 10 µm droplet

– Critical size (~0.1 mm)

– Gravity overcomes and drop falls

Precipitation Variation

• Influenced by

– Atmospheric circulation and local factors

• Higher near coastlines

• Seasonal variation – annual oscillations in some • Seasonal variation – annual oscillations in some

places

• Variables in mountainous areas

• Increases in plains areas

Global precipitation pattern

Spatial interpolation of precipitations

Thiessen polygon method

P1

P2

P

A1

A2

• Any point in the watershed receives the same amount of rainfall as that at the nearest gage

• Rainfall recorded at a gage can be applied to any point at a distance halfway to the next station in any direction

• Steps in Thiessen polygon method

1. Draw lines joining adjacent gages P3

A3

2. Draw perpendicular bisectors to the lines

created in step 1

3. Extend the lines created in step 2 in both

directions to form representative areas for

gages

4. Compute representative area for each gage

5. Compute the areal average using the following

formula

∑=

=N

i

ii PAA

P1

1

P1 = 10 mm, A1 = 12 Km2

P2 = 20 mm, A2 = 15 Km2

P3 = 30 mm, A3 = 20 km2

mmP 7.2047

302020151012=

×+×+×=

Inverse distance weighting

P1=10

P2= 20

P3=30

• Prediction at a point is more influenced by nearby measurements than that by distant measurements

• The prediction at an ungaged point is inversely proportional to the distance to the measurement points

d1=25

d2=15 P3=30points

• Steps

– Compute distance (di) from ungaged point to all measurement points.

– Compute the precipitation at the ungaged point using the following formula ∑

∑

=

=

=N

i i

N

i i

i

d

d

P

P

12

12

1

ˆ

d3=10

mmP 24.25

10

1

15

1

25

110

30

15

20

25

10

ˆ

222

222

=

++

++

=

p

( ) ( )2

21

2

2112 yyxxd −+−=

Geostatistical (kriging)

interpolation

Example: breaks up topography into 3 elements: Drift (general trend), small deviations from the drift and random noise.

Geostatistical (kriging)

interpolation

To be stepped over

General structure of the model

All interpolation algorithms (inverse distance squared, splines,triangulation, etc.) estimate the value at a given location as a weightedsum of data values at surrounding locations. Almost all assign weightsaccording to functions that give a decreasing weight with increasingseparation distance.

Kriging assigns weights according to a (moderately) data-drivenweighting function, rather than an arbitrary function, but it is still aninterpolation algorithm and will give very similar results to others in

Kriging vs other methods

interpolation algorithm and will give very similar results to others inmany cases. In particular:

o If the data locations are fairly dense and uniformly distributedthroughout the study area, you will get fairly good estimatesregardless of interpolation algorithm.

o If the data locations fall in a few clusters with large gaps in between,you will get unreliable estimates regardless of interpolation algorithm.

o Almost all interpolation algorithms will underestimate the highs andoverestimate the lows; this is inherent to averaging.

Some advantages:

• Gives estimate of estimation error (kriging variance), along withestimate of the variable, Z, itself (but error map is basically a scaledversion of a map of distance to nearest data point, so not that

Kriging vs other methods

version of a map of distance to nearest data point, so not thatunique).

• Availability of estimation error provides basis for stochasticsimulation of possible realizations of Z(u).

IDW vs. Kriging

• Kriging appears to give

a more “smooth” look to

the data

• Kriging avoids the “bulls

eye” effect

The Kriging paradigm

Reproduced from: Goovaerts, P., Geostatistics for Natural Resources Evaluation

Experimental variogram

Stochastic simulation of possible realizations on the basis of estimation error

Stochastic simulation of possible realizations on the basis of estimation error

Snowmelting