Forecasting occurrences of wildfires & earthquakes using point processes with directional covariates

Frederic Paik Schoenberg, UCLA StatisticsCollaborators: Haiyong Xu, Ka Wong. Also thanks to: Yan Kagan, James Woods, USGS, SCEC, NCEC, & Harvard catalogs.

1) Background

2) Existing point process models for wildfires & earthquakes

3) Problems, esp. wind & moment tensors

4) Directional kernel direction & wind

5) Using focal mechanisms in ETAS

Los Angeles County wildfire centroids, 1960-2000

Background Brief History.

• 1907: LA County Fire Dept.• 1953: Serious wildfire suppression.• 1972/1978: National Fire Danger Rating System.

(Deeming et al. 1972, Rothermel 1972, Bradshaw et al. 1983)

Damages.• 2003: 738,000 acres; 3600 homes; 26 lives.(Oct 24 - Nov 2: 700,000 acres; 3300 homes; 20 lives)• Bel Air 1961: 6,000 acres; $30 million.• Clampitt 1970: 107,000 acres; $7.4 million.

Global Earthquake Data:

Local e.q. catalogs tend to have problems, esp. missing data.1977: Harvard (global) catalog created. Considered the most complete. Errors best understood.

• Harvard Catalog, 1/1/84 to 4/1/07• Shallow events only (depth < 70km)• Mw 3.0+• Only focal mechanism estimates of high or medium quality

2. Existing models for forecasting eqs & fires

NFDRS’s Burning Index (BI): Uses daily weather variables, drought index, and

vegetation info. Human interactions excluded.

Some BI equations: (From Pyne et al., 1996:)

Rate of spread: R = IR (1 + w+ s) / (b Qig). Oven-dry bulk density: b = w0/.

Reaction Intensity: IR = ’ wn h Ms. Effective heating number: = exp(-138/).

Optimum reaction velocity: ’ = ’max ( / op)A exp[A(1- / op)].

Maximum reaction velocity: ’max = 1.5 (495 + 0.0594 1.5) -1.

Optimum packing ratios: op = 3.348 -0.8189. A = 133 -0.7913.

Moisture damping coef.: M = 1 - 259 Mf /Mx + 5.11 (Mf /Mx)2 - 3.52 (Mf /Mx)3.

Mineral damping coef.: s = 0.174 Se-0.19

(max = 1.0).

Propagating flux ratio: = (192 + 0.2595 )-1 exp[(0.792 + 0.681 0.5)( + 0.1)].

Wind factors: w = CUB (/op)-E. C = 7.47 exp(-0.133 0.55). B = 0.02526 0.54. E = 0.715 exp(-3.59 x 10-4 ).

Net fuel loading: wn = w0 (1 - ST). Heat of preignition: Qig = 250 + 1116 Mf.

Slope factor: s = 5.275 -0.3 (tan 2. Packing ratio: = b / p.

1

Point Process Models Conditional rate (t, x1, …, xk; ): [e.g. x1=location, x2 = area.]a non-neg. predictable process such that ∫ (dN -d) is a martingale.

• Wildfire incidence seems roughly multiplicative.(only marginally significant in separability test)

• Roughly exponential in relative humidity (RH), windspeed (W), precipitation (P), avg precip over prior 60 days (A), temperature (T), and date (D).

• Tapered Pareto size distribution f, smooth spatial background .

(t,x,a) = f(a) (x) 1exp{2R(t) + 3W(t) + 4P(t) + 5A(t) + 6T(t) + 7[8 - D(t)]2}

Or, split by season:(t,x,a) = f(a) (x) 1,i exp{2,i R(t) + 3,i W(t) + 4,i P(t) + 5,i A(t) + 6,i T(t)}

Aftershock activity described by modified Omori Law: K/(t+c)p

3. Some problems with existing models

• BI has low correlation with wildfire. Corr(BI, area burned) = 0.09 Corr(date, area burned) = 0.06 Corr(windspeed, area burned) = 0.159

• Too high in Winter (esp Dec and Jan) Too low in Fall (esp Sept and Oct)

3. Some problems with existing models, continued

• Wildfires: no use of wind direction. Santa Ana winds (from NE) typically hot & dry.

• ETAS: no use of focal mechanisms.Summary of principal direction of motion in an earthquake, as well as resulting stress changes and tension/pressure axes.

4. Directional kernel regression and wind:∑i yi g( - i) / ∑ g( - i),

using a circular kernel g, such as the von-Mises densityg() = exp{ cos()}/2 I0().

4. Directional kernel regression and wind:f() estimated via ∑i yi g( - i) / ∑ g( - i),

using a circular kernel g, such as the von-Mises densityg() = exp{ cos()}/2 I0().

RH< 15% 15% < RH < 30%

Improvement in forecasting

5. Using focal mechanisms in ETAS

Distance to next event, in relation to nodal plane of prior event

In ETAS (Ogata 1998), (t,x,m) = f(m)[(x) + ∑i g(t-ti, x-xi, mi)],where f(m) is exponential, (x) is estimated by kernel smoothing,

i.e. the spatial triggering component, in polar coordinates, has the form:

g(r, ) = (r2 + d)q .

Looking at inter-event distances in Southern California, as a

function of the direction of the principal axis of the prior event,

suggests: g(r, ; ) = g1(r) g2( - | r),

where g1 is the tapered Pareto distribution,

and g2 is the wrapped exponential.

and

tapered Pareto / wrapped exp. biv. normal (Ogata 1998) Cauchy/ ellipsoidal (Kagan 1996)

Thinned residuals:

Data tapered Pareto / wrapped exp. Cauchy/ ellipsoidal (Kagan 1996) biv. normal (Ogata 1998)

Tapered pareto / wrapped exp. Cauchy / ellipsoidal

Cauchy/ ellipsoidal(Kagan 1996)

biv. normal(Ogata 1998)

tapered pareto /wrapped exp.

Conclusions:

• The impact of directional variables on a scalar response can readily be summarized using directional kernel regression.

• The resulting function can then be incorporated into point process models, to improve forecasting of the response variable.

• Wildfires: wind direction is very significant, and models incorporating wind direction and other weather variables forecast about twice as well as the BI (which uses these same variables).

• Earthquakes: focal mechanism estimates should be used to improve triggering functions in ETAS models.

Greenness (UCLA IoE)

(IoE)

On the Predictive Value of Fire Danger Indices:

From Day 1 of Toronto workshop (05/24/05):• Robert McAlpine: “[It] works very well.”• David Martell: “To me, they work like a charm.”• Mike Wotton: “The Indices are well-correlated with fuel moisture and fire

activity over a wide variety of fuel types.”• Larry Bradshaw: “[BI is a] good characterization of fire season.”

Evidence?

• FPI: Haines et al. 1983 Simard 1987 Preisler 2005Mandallaz and Ye 1997 (Eur/Can), Viegas et al. 1999 (Eur/Can), Garcia Diez et al. 1999 (DFR), Cruz et al. 2003 (Can).

• Spread: Rothermel (1991), Turner and Romme (1994), and others.