GIST 24/10/06

Jeff Settle, ESSC

Reconciling Point and Area Data for RADAGAST

GIST 24 October 2006

GIST 24/10/06

The problem

RADAGAST compares fluxes at the surface and at the TOA. The former are measured at a point, the latter are averaged over an area ~40kmX40km.Is that OK?

Many previous attempts to estimate RBs under the ARM programme have taken the time-average of the point ARM measurements to be a sound estimate of the same time-average over a much wider area.

1 hours average at a point ~ 1hours average over a 50km square (According to ARM Wiscombe, pers.comm).

GIST 24/10/06

GIST 24/10/06

€

Sampling error = E ξ obs −ξ A( )2

{ }

A denotes an average of over an area A(and a time period T).

obs is our observation, an average over a different area (a point in this case).

€

A = x r, t( )drdtA

∫T

∫ ξ obs = x robs, t( )dtT

∫

Definition

GIST 24/10/06

Thus, let p denote a model prediction of A.

€

E ξ obs −ξ p( )2

{ }

The comparison we make1 2 4 4 3 4 4

= E ξ A −ξ p( )2

{ }

The comparison we are really interested in

1 2 4 4 3 4 4 + E ξ obs −ξ A( )

2{ }

Sampling Error Variance1 2 4 4 3 4 4

Similar considerations apply when: validating satellite-derived geophysical variables

testing large-scale models with observational data

GIST 24/10/06

Sampling error arises from heterogeneity in the underlying variable being averaged, caused by heterogeeity in some other underlying variable (e.g. cloud cover). The variations are characterised by:

Their point varianceThe size of the area AThe period of time averagingCorrelation functions in space and time

spatial scale lengthtemporal scale time

The spatial scale length defines the size of a “lump” in the underlying variable. The time scale, compared to the integration period, defines a number of semi-independent observation periods.

GIST 24/10/06

To keep it simple,I make the assumption: A is very large compared to the scale length(2) of the heterogeneity(GERB and clouds: 40km2 vs 4-10km2

The sampling error is then dominated by the statistical variance in the observation.

€

E ξ obs −ξ A( )2

{ } ~ Var ξ obs{ }

GIST 24/10/06

Example: How is the SW downward flux at the surface for a GERB pixel related to that at ARM.

The variation in time t of this flux at a point r be written:

x(r,t) = D(r,t) + d(r,t).[1-(r,t)](D = diffuse irradiance, d = direct irradiance)

takes the value 0 or 1 (according to whether a cloud is intercepting the sun).

GIST 24/10/06

Cloud statistical model

To proceed we need to consider (r,t) as random variable, with mean (say).

Since 2= , the expected value of 2 is also , and so the variance of is:

Var() = - 2. The instantaneous error on

is: d√(µ-µ2)

€

SW ↓

GIST 24/10/06

€

A = D + d 1− μ( )

ξ obs = D + d 1− μobs( )

μobs = 1

Tχ robs, t( )dt

0

T

∫(effectively using the AMF to estimate cloud cover)

sampling error variance = d 2 μ − μobs( )2

GIST 24/10/06

The error after averaging over time depends on the persistence of cloud obscuration, captured in the autocorrelation function of

€

E χ t1( ) − μ( ) χ t2( ) − μ( ){ } = μ − μ 2( )γ

t2 − t1CT

⎛

⎝ ⎜

⎞

⎠ ⎟

where CT is the correlation time. We find, for the Shortwave downward flux:

€

E ξ obs −ξ A( )2

{ } = d2 μ − μ 2( )

Point variance1 2 4 3 4

2

T 2T − t( )γ

t

CT

⎛

⎝ ⎜

⎞

⎠ ⎟dt

0

T

∫Averaging factor

1 2 4 4 4 3 4 4 4

= d2 μ − μ 2( )κ

2 (say)

GIST 24/10/06

is shown below for two cases.

Black line: simple exponential

Blue line: Gaussian

GIST 24/10/06

GIST 24/10/06

For long integration times (T>>CT),

€

~ 2CT

T

e.g. with µ = 0.5 (worst case) d ~ 300Wm-2

CT = 20 minutes

We have SW ~ 120/√T Wm-2 (T in hours) = 30 Wm-2 T = 16hr

Most of the required numbers can be found directly fromthe time series of flux at the observation point.

GIST 24/10/06

Our approach for SW irradiance is:

(1) to use the SEVIRI cloud mask to determine the cloud cover across the area of comparison.

(2) to use data from the two ARM sites to generate time series of clear and cloudy SW irradiances.

(3) combine these to provide the required area irradiance.

€

ˆ ξ A = D + d AMF

1 2 3 − d AMF{ × μ

SEVIRI{

GIST 24/10/06

The theoretical first-order error is now driven by the accuracy of the cloud mask. If the mask is binary, determined by a simple threshold test, the error depends on the true distribution of cloud cover.

Assume: the cloud mask predicts cloud whenever the true cloudiness is>50%, predicts clear when the true cloudiness is <50%, and that true cloudiness is uniformly distributed for partial cover. The relative error on a day’s irradiance (SHI pixel) is then:

€

(0.14 − 0.18)fmixed

27 < 0.035 fmixed

fmixed is the proportion of SEVIRI pixels not completely cloudy or completely clear. d=300Wm-2 as before, error <10Wm-2

GIST 24/10/06

SW Upward/NetThis is really just the shortwave-downward times (1-the albedo). The latter is quite heterogeneous, and variable in time.

Left: SPOT image of Niamey, June. Right: August

GIST 24/10/06

Error on SW net

€

SW ↑ = a r, t( )SW ↓

δ SWnet( ) ≈ 1− a ( ).δ SW ↓( ) − δa.SW ↓

a may be systematically in error if we use the fluxes at the AMF to define the albedo. We minimise that term by using the mean albedo in calculations.

GIST 24/10/06

Current statusWe are presently working on the cloud mask. This is needed for the conditional sampling of the various meteorological conditions at the site, as well as for the calculation of the SRB.

GIST 24/10/06

Conclusion

Time-averaged SW fluxes at a point are a poor estimate of the time-area average over a footprint much larger than typical cloud length scales.

The sampling error decreases like 1/√T (effectively, 1/√N, where N is the number of times the sun is switched on and off by clouds).

SEVIRI to the rescue, provided the cloud mask is up to scratch.

(not shown) LW downward fluxes less of a problem, in theory, because some spatial averaging is effectively done.

GIST 24/10/06