( 15 ). Finally, by combining Eq. 1 and Eq. 2
and approximating the integral in Eq. 1 as a
product, we obtain a characteristic avulsion
time scale,
tc≈4 h^2 c
kS^20ð 3 ÞA full semi-analytical solution to Eqs. 1 and 2
with discrete slip events ( 15 ) (fig. S1) cor-
roborates the approximate framework and
suggests a response independent of whether
slip is continuous or sequenced in individ-
ual earthquakes. Provided a long-term fault
slip rate (vx), a channel becomes unstable
and susceptible to avulsion when the chan-
nel offset approaches a critical offsetdc=vxtc.
In this way, through a known slip rate, the chan-
nel offset is an independent chronometer of the
channel response. We can thus relate the advec-
tive and fluvial history of offset channels at the
time of avulsion to determine a critical offset,dc≈4 h^2 cvx
kS^20ð 4 ÞEquation 4 is the first key result of the model.
It defines the offset that resets fault-crossing
channels and depends entirely on measurable
quantities.
We tested this relationship on a collection
of active and abandoned fault-crossing chan-nels in the Carrizo Plain. Although it is rare to
estimatedcdirectly, Eq. 4 predicts observed
offsets (dobs) that are less thandcon active
channels and more thandcon abandoned
channels. We measured values of total offset
(dobs), avulsion threshold height (hc), and ini-
tial slope (S 0 ) interpreted from LiDAR data
(e.g., Fig. 2, A to C) for a set of 55 active and
abandoned channels in the Carrizo Plain ( 15 )
(table S1 and figs. S6 to S64). We avoided
channels where misalignment across the fault
appears to result from the deflection of flow
rather than progressive fault offset (e.g., down-
stream from an alluvial fan). We estimated the
diffusivity coefficient usingk≈0.1Lr, where
Lis the corresponding catchment length
(Fig. 2D) andris the mean annual regional
rainfall ( 13 ). We expect that the main source
of error in our approach is this approxima-
tion ofkand its constancy through time
( 15 ). We recorded partial overspilling on an
active channel or minimal offset on the active
tributary of an abandoned channel, as these
respectively indicate an incipient or a recent
avulsion. In these cases, we assumed that the
measured offset approximates the critical
offset,dc.
We compared measured offsets to corre-
sponding predictions ofdc(Fig. 2E). A logistic
regression through the log-transformed data
finds that the expressiondobs=dc¼ 0 : 6 þ ^00 ::^53
(99% confidence interval on 10,000 bootstrap
samples) best separates active and abandoned
channels, indicating that the model prediction
corresponds to observed critical offsets within
a factor of 2 (Fig. 3). If the timing of avulsions
were independent of the diffusive channel re-
sponse, the ratiodobs/dcwould not separate
active and abandoned channels (P= 0.0007,
following a Wald test) and has no reason to
approach unity. Instead, the scaling anal-
ysis predicts the life span of fault-crossing
channels well within model uncertainty,
particularly in the parameterization of the
diffusivity coefficient ( 15 ).
Under the assumptions of the model, any
one parameter in Eq. 4 is constrained from
the others. Posing Eq. 4 as an equality and
isolating for the slip rate, the channel diffu-
sivity coefficient ( 15 ) and channel geometry
(table S1) on active and abandoned channels
bracket a slip rate, respectively under- and
overestimating it. A logistic regression then
yields a 2: 1 þ ^11 ::^70 cm/year slip rate in the Carrizo
Plain (99% confidence interval on 10,000 boot-
strap samples), compatible with geologic and
geodetic estimates of 3.5 cm/year ( 4 , 16 ). With
a known fault slip rate, a similar approach cali-
brates the diffusivity (and hence the sediment
transport capacity) of channels. The critical
offset encodes channel response best in arid
environments where transport capacity is low
relative to slip rate, which perhaps is a distin-
guishing feature of the Carrizo Plain ( 15 ). Where206 9JULY2021•VOL 373 ISSUE 6551 sciencemag.org SCIENCE
Fig. 2. Channel geometry along the
San Andreas Fault in the Carrizo
Plain.(AtoE) Measurements
of (A) offsetdobs, (B) initial avulsion
threshold heighthc, (C) initial slope
S 0 , (D) channel reachL, and (E)
normalized offsetdobs/dc(table S1).
Solid markers indicate active
channels; open markers indicate
abandoned channels. Yellow markers
indicate channels that have evidence
for incipient or recent avulsions.
No individual set of measurements
[(A) to (D)] appears sufficient to
separate active and abandoned
channels. Offsets in (A) and a slip rate
of 3.5 cm/year are consistent with
a channel response over millennial
time scales. The uncharacteristically
large offsets in (A) collapse near
unity in (E) when normalized by the
critical offset.
Fig. 3. Channel classification.
The dashed line is a logistic regression
through active and abandoned channels
(collapsing Fig. 2E along thexaxis)
with respect to the normalized offset.
Separation of the data, with a class
boundary near unity, indicates consistency
with the model framework. The gray box
indicates the 99% confidence interval
on the class boundary using 10,000 bootstrap samples. Near-critical channels (incipient or recent avulsion in
yellow) are summarized in the histogram as an additional qualitative test.
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