LASSO: Perspective and Results

Graham Feingold1, Joseph Balsells2,3 , Franziska Glassmeier1,4, Takanobu Yamaguchi1,2, Jan Kazil1,2, Allison McComiskey1, Elisa Sena5

1 NOAA Earth System Research Laboratory, Boulder, Colorado

2 CIRES, University of Colorado, Boulder, Colorado3 Yale University, Connecticut4 NRC Fellow5 University of Sao Paolo, Brazil

Observations: 14 years of warm clouds at SGP Continental US

Correlation between rCRE and Aerosol Index Correlation between rCRE and L

Correlation between L and AICor

rela

tion

betw

een

rCR

Ean

d A

Id(lnL)

d(lnNd)=

d(lnk)

d(lnNd)=

rCRE = 1 � Fsw,all

Fsw,clr

rCRE ⇡ Ac ⇥ fc

A = Acfc +Aclr(1� fc)

Fsw = downwellingshortwave flux

Sena et al., 2016

ExpectedUnexpectedExpected

Importance of understanding co-variability between aerosol and meteorology/L

expectedunexpectedexpected

Cloud fraction fc

Scen

e A

lbed

o, A Engström et al.

(2015)

MODIS, CERESOcean 60°S – 60°N1°x1° monthly mean

Approaches to quantifying Aerosol-Cloud Radiative Effect

Drop N, size

Top-down

Ac = f(L, N)

Cloud field PropertiesCloud fraction, fcLiquid water path, LOptical depth, tCloud albedo, Ac

Cloud depth, H

Cloud fraction fc

Scen

e A

lbed

o, A Engström et al.

(2015)

MODIS, CERESOcean 60°S – 60°N1°x1° monthly mean

Cloud field PropertiesCloud fraction, fcLiquid water path, LCloud optical depth, tcCloud albedo, Ac

Ac

As

Approaches to quantifying Aerosol-Cloud Radiative Effect

Drop N, size

Top-down

Ac = f(L, N)

Cloud field PropertiesCloud fraction, fcLiquid water path, LOptical depth, tCloud albedo, Ac

Cloud depth, H

Scen

e A

lbed

o, A Engström et al.

(2015)

MODIS, CERESOcean 60°S – 60°N1°x1° monthly mean

Cloud field PropertiesCloud fraction, fcLiquid water path, LCloud optical depth, tcCloud albedo, Ac

How do aerosol cloud interactions manifest themselves in this kind of framework?

As

Ac

Cloud fraction fc

Tota

l sce

ne a

lbed

o

Aerosol influence is detectable but relatively weak

Set 1 (100 simulations)

Feingold et al., 2016, PNAS

Co-variability of inputs influences detectability of aerosol effects!

Aerosol effect is imperceptible

Cloud fractionCloud fraction

Tota

l sce

ne a

lbed

o

Engstrom et al. 2015Satellite data

Set 2 (120 simulations)Latin Hypercubesampling

Two sets of model simulations, differing only in the co-variability of aerosol and meteorology

Clo

ud A

lbed

oCloud fractionCloud fraction (tc > 2)

Engstrom et al. 2015Satellite data

Engstrom et al. 2015Satellite data

Closed-to-open cell Scu transition(modeling)

SGP observations

Scen

e A

lbed

o

SGP, 1999-2010 - shallow clouds (300-3000 m) - single layer, non-precipitating - Each point is a daily average

Relative Cloud Radiative Effect

SGP, 1999-2010 - shallow clouds (300-3000 m) - single layer, non-precipitating - Each point is a daily average

Clo

ud A

lbed

oCloud fractionCloud fraction

Engstrom et al. 2015Satellite data

Engstrom et al. 2015Satellite data

Closed-to-open cell Scu transition(modeling)

SGP observations

Scen

e A

lbed

o

tc > 5

Relative Cloud Radiative Effect

Cloud fraction requires careful definition!

tc > 2

Albedo,Cloudfraction

Albedo,Cloudfraction

Initial conditions Met: NWP/VA/RGCMAerosol: observed

CRECRE

Data

Process Model: LES (statistics)

L, t, Na,Radiation, Surface Fluxes..

L, t, Na,Radiation, Surface Fluxes..

Routine,long-term,regime-based.Naturalco-variabilityof met. and Na

See also Cabauw, HD(CP)2

LASSO (routine LES at SGP)

Gustafson and Vogelmann

Modeling

ImprovedPhysics

Albedo,Cloudfraction

Albedo,Cloudfraction

Albedo,Cloudfraction

Initial conditions Met: NWP/VA/RGCMAerosol: observed

CRECRE CRE

Data

Process Model: LES (statistics)

L, t, Na,Radiation, Surface Fluxes..

L, t, Na,Radiation, Surface Fluxes..

Routine,long-term,regime-based.Naturalco-variabilityof met. and Na

L, t, Na,Radiation, Surface Fluxes..

SCM or CAPT

CloudRadiativeForcing:Global (PD-PI)

GCM

See also Cabauw, HD(CP)2

LASSO

But what controls the shape of these A vs. fc curves?

Connect cloud processes to shape of A; fc curves

Simple models:

1. Stratocumulus 2. Cumulus

Feingold et al. (2017)

..Ac

As

fc

Scen

e A

lbed

o, A

Simple Stratocumulus Cloud Model (Considine 1997)

Vary Ac vs fc relationship

Ac=a fcb

b < 0

b > 0 (most plausiblefor SCu)

fc

A

b = 0

ipcc_model_documentation.php). We use an output pe-riod of the same length as the satellite data record, fromJanuary 1995 to December 1999. All model output is lin-early interpolated onto a common 2.58 3 2.58 grid to becomparable with the CERES–MODIS data.

For albedo calculation from model output, we usedeseasonalized monthly means of incoming and reflectedSW radiative fluxes at the TOA. For cloud fraction esti-mates from the model output, the model parameter‘‘total cloud fraction’’ is used in the analysis. This value isa result of different cloud overlapping schemes in dif-ferent models, but regardless of the cloud overlap algo-rithm the total cloud fraction parameter represents theradiatively effective cloud fraction, or the cloud fractionas seen from above, in each model.

d. Cloud fraction in models and observations

Defining and determining cloud fraction accuratelyis problematic and its value is to some degree methoddependent. Yet, f plays a significant role in the earth’sclimate system through its application in Eq. (1). Thisequation and the concept of f have been adapted to nu-merous surface and satellite observation schemes and toradiation and climate models (e.g., Hahn et al. 2001; Collins2001; Martins et al. 2002; Chepfer et al. 2008), and ac-cordingly it is also used as the basis for the present analysis.

The various satellite retrievals and models used havecloud detection schemes that are objective and internallyconsistent, but they do not have exactly the same defini-tion of cloud fraction, and the threshold values for cloud

formation and identification are not the same acrossmodels and observations. Satellite retrievals of globalcloud fraction are based on the occurrence of cloudyconditions in individual satellite pixels, or lidar profiles inthe case of CALIPSO, resulting in a gridded fractionalcloud cover. For GCMs, on the other hand, the fractionalcloud cover of each grid box is calculated based on theoverlapping of cloud at different levels in the atmosphericcolumn. Despite these differences, across the differentdatasets a near-linear relation between albedo and cloudfraction is generally present, which we see as addingcredibility to our approach.

4. Results

a. CERES and MODIS satellite observations

As seen in Fig. 1, monthly mean satellite observationsof albedo from CERES and cloud fraction from MODISon the Terra and Aqua satellites for the selected areas doindeed display closely linear relations for the sampledranges of f.

In other words, on a monthly mean time scale, Eqs.(1)–(3) seem to hold, and the slope and intercept of theregression line shown in the plot can be used to estimateacloud. The values of the correlation coefficient for therelations, as well as estimated cloud albedo values andrelated variability ranges, are summarized in Table 1.

The lack of data points for low values of f (below ap-proximately 0.3) prevents investigation of the linearity inthis range, and the physical meaning of the intercept of the

FIG. 1. Scatterplots showing the relations between monthly mean albedo from CERES and monthly mean cloudfraction from MODIS during 2002–07 in (left to right) three marine stratocumulus regions. Observations are from the(top) Terra and (bottom) Aqua satellites, respectively.

2142 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 50

Bender et al. 2011Peruvian SCu

Curvature indicates that cloud albedo increases with cloud fraction

Considine model 1997

Assumptions:- Cloud size distribution is a negative power law

- Power law relationship between cloud depth and area (cloud top varies)

- Cloud base is constant

where units are in meters. This empirical equation was derived from three stratocu-mulus airborne campaigns. A value of z

i

= 800 m is assumed. The results exhibitsome sensitivity to the choice of z

i

but are far more sensitive to the choice of Lmin

inEq. 7 (Fig. 2), a point that will be emphasized in the ensuing analysis. A value ofL

min

= 45 g m�2 matches the Engstrom et al. (2015) relationship quite well. Furthertuning of z

i

and/or modification to the parameters in Eq. 8 within their uncertaintiescan improve this fit even more. We note that we also explored relationships between�

cb

and z

i

based on large eddy simulation of closed to open cell transitions (Feingoldet al. 2015) but found they were not robust and exhibited hysteresis in the closedto open and open to closed transition.We surmise that this is a characteristic of aboundary layer in transition rather than a statement of the usefulness of Eq. 8.

2 Cumulus Clouds

A broken cumulus cloud field has characteristics that lend itself to a di↵erent ap-proach. The size distribution is known to be well described by a negative power lawdistribution in terms of cloud area a

P (a) = A a

�b

, (9)

where P (a) is the number of clouds per unit area and per unit cloud size a. Thenormalizing factor A is calculated based on satellite observations (Koren et al. 2008)and large eddy simulations (Jiang et al. 2009) of cumulus cloud fields (A ⇡ 2 ·10�7).Following from the discussion in Section 1, we assume that cloud depth is a functionof cloud area:

z = ↵ a

�

. (10)

We again rely onP (z)dz = P (a)da (11)

to deriveP (a) = A � ↵

1�b

a

�(1�b)�1

. (12)

Cloud fraction can be calculated as

f

c

=Z

a

max

a

min

P (a)da =A � ↵

(1�b)

�(1� b) + 1

ha

�(1�b)+1

max

� a

�(1�b)+1

min

i. (13)

We now assume that entrainment-mixing dilutes the cloud water below the adiabaticvalue:

L = �p z

3

3+

q z

2

2, (14)

5

z = cloud depth

a = cloud area

a

P(a)

L =

c

w

z

2

2

(8)

f

c

=

ZL

max

L

min

P (L) dL (9)

�

cb

= 0.051 z

i

� 8.4, (10)

P (z) = A z

�b

, (11)

P (a) = A a

�b

, (12)

z = ↵ a

�

. (13)

P (z)dz = P (a)da (14)

P (a) = A � ↵

1�b

a

�(1�b)�1. (15)

f

c

=

Ra

max

a

min

aP (a)da

L

x

L

y

=

A

(2� b)

ha

(2�b)max

� a

(2�b)min

i

L

x

L

y

. (16)

f

c

=

Ra

max

a

min

aP (a)da

L

x

L

y

=

A � ↵

(1�b)

�(1� b) + 1

ha

�(1�b)+1max

� a

�(1�b)+1min

i

L

x

L

y

. (17)

L = �p z

3

3

+

c

w

z

2

2

, (18)

¯

L =

RLP (L)dL

RP (L)dL

=

(b

0+ 1)

(b

0+ 3)

[L

(b0+3)/2max

� L

(b0+3)/2min

]

[L

(b0+1)/2max

� L

(b0+1)/2min

]

; b

0= (1� b)/� (19)

2

b ~ 0.3

Simple Cumulus Cloud Model (Feingold et al., 2017)

integrate numerically

cloud optical depthcloud albedo

Adiabatic:

Subadiabatic:

L =

c

w

z

2

2

(8)

f

c

=

ZL

max

L

min

P (L) dL (9)

�

cb

= 0.051 z

i

� 8.4, (10)

P (z) = A z

�b

, (11)

z = ↵ a

�

. (12)

P (z)dz = P (a)da (13)

P (a) = A � ↵

1�b

a

�(1�b)�1. (14)

f

c

=

Ra

max

a

min

aP (a)da

L

x

L

y

=

A � ↵

(1�b)

�(1� b) + 1

ha

�(1�b)+1max

� a

�(1�b)+1min

i

L

x

L

y

. (15)

L = �p z

3

3

+

c

w

z

2

2

, (16)

¯

L =

RLP (L)dL

RP (L)dL

=

(1� b)

(3� b)

[L

(3�b)/2max

� L

(3�b)/2min

]

[L

(1�b)/2max

� L

(1�b)/2min

]

(17)

⌧

c

= 1.5

¯

L

r

e

, (18)

⌧

c

= 1.5

L

r

e

, (19)

2

L =

c

w

z

2

2

(8)

f

c

=

ZL

max

L

min

P (L) dL (9)

�

cb

= 0.051 z

i

� 8.4, (10)

P (z) = A z

�b

, (11)

z = ↵ a

�

. (12)

P (z)dz = P (a)da (13)

P (a) = A � ↵

1�b

a

�(1�b)�1. (14)

f

c

=

Ra

max

a

min

aP (a)da

L

x

L

y

=

A � ↵

(1�b)

�(1� b) + 1

ha

�(1�b)+1max

� a

�(1�b)+1min

i

L

x

L

y

. (15)

L = �p z

3

3

+

c

w

z

2

2

, (16)

¯

L =

RLP (L)dL

RP (L)dL

=

(1� b)

(3� b)

[L

(3�b)/2max

� L

(3�b)/2min

]

[L

(1�b)/2max

� L

(1�b)/2min

]

(17)

⌧

c

= 1.5

¯

L

r

e

, (18)

⌧

c

= 1.5

L

r

e

, (19)

2

L =

c

w

z

2

2

(8)

f

c

=

ZL

max

L

min

P (L) dL (9)

�

cb

= 0.051 z

i

� 8.4, (10)

P (z) = A z

�b

, (11)

z = ↵ a

�

. (12)

P (z)dz = P (a)da (13)

P (a) = A � ↵

1�b

a

�(1�b)�1. (14)

f

c

=

Ra

max

a

min

aP (a)da

L

x

L

y

=

A

(2� b)

ha

(2�b)max

� a

(2�b)min

i

L

x

L

y

. (15)

f

c

=

Ra

max

a

min

aP (a)da

L

x

L

y

=

A � ↵

(1�b)

�(1� b) + 1

ha

�(1�b)+1max

� a

�(1�b)+1min

i

L

x

L

y

. (16)

L = �p z

3

3

+

c

w

z

2

2

, (17)

¯

L =

RLP (L)dL

RP (L)dL

=

(1� b)

(3� b)

[L

(3�b)/2max

� L

(3�b)/2min

]

[L

(1�b)/2max

� L

(1�b)/2min

]

(18)

⌧

c

= 1.5

¯

L

r

e

, (19)

2

L = �p z

3

3

+

c

w

z

2

2

, (20)

¯

L =

RLP (L)dL

RP (L)dL

=

(b

0+ 1)

(b

0+ 3)

[L

(b0+3)/2max

� L

(b0+3)/2min

]

[L

(b0+1)/2max

� L

(b0+1)/2min

]

; b

0= (1� b)/� (21)

�

0= (1� b)/� (22)

¯

L =

RLP (L)dL

RP (L)dL

=

(b

0/2)

(b

0/2 + 1)

[L

b

0/2+1

max

� L

b

0/2+1

min

]

[L

b

0/2

max

� L

b

0/2

min

]

; b

0= (1� b)/� (23)

⌧

c

= 1.5

¯

L

r

e

, (24)

⌧

c

= 1.5

L

r

e

, (25)

A

c

=

(1� g)⌧

c

2 + (1� g)⌧

c

(26)

⌧

c

= CN

1/3Z(�pz

2+qz)

2/3dz = CN

1/3

"0.6z(qz(1� pz/q))

2/32F1(�2/3, 5/3; 8/3; pz/q)

(1� pz/q)

2/3

#

,

(27)

3

L = �p z

3

3

+

c

w

z

2

2

, (20)

¯

L =

RLP (L)dL

RP (L)dL

=

(b

0+ 1)

(b

0+ 3)

[L

(b0+3)/2max

� L

(b0+3)/2min

]

[L

(b0+1)/2max

� L

(b0+1)/2min

]

; b

0= (1� b)/� (21)

�

0= (1� b)/� (22)

¯

L =

RLP (L)dL

RP (L)dL

=

(b

0/2)

(b

0/2 + 1)

[L

b

0/2+1

max

� L

b

0/2+1

min

]

[L

b

0/2

max

� L

b

0/2

min

]

; b

0= (1� b)/� (23)

⌧

c

= 1.5

¯

L

r

e

, (24)

⌧

c

= 1.5

L

r

e

, (25)

A

c

=

(1� g)⌧

c

2 + (1� g)⌧

c

(26)

⌧

c

= CN

1/3Z(�pz

2+qz)

2/3dz = CN

1/3

"0.6z(qz(1� pz/q))

2/32F1(�2/3, 5/3; 8/3; pz/q)

(1� pz/q)

2/3

#

,

(27)

3

cloud fraction

scene albedo

Simple Cumulus Cloud Model (Feingold et al., 2017)

L = �p z

3

3

+

c

w

z

2

2

, (20)

¯

L =

RLP (L)dL

RP (L)dL

=

(b

0+ 1)

(b

0+ 3)

[L

(b0+3)/2max

� L

(b0+3)/2min

]

[L

(b0+1)/2max

� L

(b0+1)/2min

]

; b

0= (1� b)/� (21)

�

0= (1� b)/� (22)

¯

L =

RLP (L)dL

RP (L)dL

=

(b

0/2)

(b

0/2 + 1)

[L

b

0/2+1

max

� L

b

0/2+1

min

]

[L

b

0/2

max

� L

b

0/2

min

]

; b

0= (1� b)/� (23)

⌧

c

= 1.5

¯

L

r

e

, (24)

⌧

c

= 1.5

L

r

e

, (25)

Ac

=

(1� g)⌧

c

2 + (1� g)⌧

c

(26)

⌧

c

= CN

1/3Z(�pz

2+qz)

2/3dz = CN

1/3

"0.6z(qz(1� pz/q))

2/32F1(�2/3, 5/3; 8/3; pz/q)

(1� pz/q)

2/3

#

,

(27)

3

Analysis of Albedo vs. Cloud Fraction

Relationships

Graham Feingold

March 2, 2017

1 Background

A = f

c

Ac

+ (1 � f

c

)As

(1)

↵ = � @ ln r

e

@ ln AI

�����L

(2)

↵ =

1

3

· d ln N

d

d ln AI

(3)

A = f

c

(Ac

� A

s

) + As

, (4)

A

c

= a1fb1c

(5)

P (z) =

1p2⇡�

cb

exp

h� (z � z)

2

2�

2cb

i; (6)

P (z)dz = P (L)dL; (7)

1

Simple Cumulus Model: Sub-adiabatic Clouds

fc

A

Remote sensing line Engström (2015)

Adiabatic

Subadiabatic

a

P(a)

Change intercept(slope is constant)

Linear A vs. fcwhen intercept varies and slope is constant

Feingold et al. (2017)

Simple Cumulus Model: Sub-adiabatic Clouds

Remote sensing line Engström (2015)

Adiabatic

Subadiabatic

a

P(a)

Steep slope

a

P(a)

a

P(a)A

Curvature derives from co-variability of slope and intercept

fc

Feingold et al. (2017)

Summary Points• (Scene) Albedo vs. cloud fraction framework for

understanding cloud field responses to perturbations

• Co-variability in meteorology and aerosol influences detectability of aerosol-cloud interactions; LASSO

• Link shape of rCRE, Albedo, fc plots to:– Micro/macrophysical processes, scale, regime

• Simple models to understand shape of A - fc curves and its relationship to convective processes