Signature Redacted

Hillslope Evolution in Response to Lateral Base Level Migration

by Jennifer L. Hamon

submitted to the Department of Earth, Atmospheric, and Planetary Science in partial fulfillment of the requirements for the degree of

Bachelor of Science

The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created.

Signature of Author: Signature redacted

Department of Earth, Atmospheric and Planetary Science 7 May 2010 Signature redacted Certified by:

Professor Taylor Perron Thesis Advisor Signature redacted Accepted by:

Professor Samuel Bowring Chair, Committee on Undergraduate Program

1 Contents

1 Introduction 6 1.1 Background on Hillslope Evolution ...... 7

2 Analytical Model of Lateral Hillslope Migration 7 2.1 Developing the 1-D Model ...... 7 2.2 A ssum ptions ...... 10

3 Case Study 11 3.1 Choosing a field site ...... 11 3.1.1 Previous work on seepage networks ...... 12 3.1.2 Previous work on the Apalachicola Bluffs ...... 14 3.2 Goals of the Case Study ...... 15 3.3 Measuring Elevation Profiles ...... 15 3.4 Numerical simulation: exploring the 2-D Effect ...... 18

4 Data 19 4.1 Measured profiles and computed A values ...... 19 4.2 Quantifying the 2-D effect of channel width ...... 22

5 Analysis and Discussion 23 5.1 Assessing the 2-D Effect ...... 23 5.2 Distribution of A values and observed spatial relationships ...... 25 5.3 Comparison of v with Abrams et al. [2009a] ...... 29

6 Concluding Remarks 30

7 Appendix 1: MATLAB functions 34 7.1 pthandler.m ...... 34 7.2 strtprof.m ...... 35 7.3 extractor.m ...... 40 7.4 qualitycontrol.m ...... 42

8 Appendix 2: Data Tables 43

9 Appendix 3: Elevation Plots 44

List of Figures

1 1-D linear transport coordinate setup ...... 8 2 Elevation data for sapping network near Bristol, FL ...... 12 3 Truncating measured profiles ...... 17 4 Schematic topography described by the 1-D linear transport model ...... 18 5 Representative surfaces generated by numerical hillslope evolution model ...... 20 6 2-D effect vs. velocity ...... 20 7 Histogram of measured A values ...... 22

2 8 Relative standard error of measured A values ...... 23 9 Divergence of measured A from true A increases with v ...... 24 10 Map of measured heads, colored by calculated A ...... 26 11 Measured A vs geometric drainage area ...... 27 12 Total curvature map, Florida seepage network ...... 54 13 Gradient map, Florida seepage network ...... 55

List of Tables

1 Longitudinal profile locations and extents ...... 45 2 Measured A and error values ...... 46

3 Abstract

Hillslopes evolve in response to base level change, sediment production, and sediment transport. Many previous studies have focused on hillslopes undergoing vertical base level migration due to tectonic forcing and bedrock incision. Many geomorphic features, however, are characterized by lateral hillslope retreat and have not been adequately studied. Here I adapt a theory of linear diffusive hillslope evolution to relate the velocity of lateral hillslope retreat to the steady-state hillslope form. A case study in a Florida sapping network, in which headward migration of seepage faces in a sandy soil sets the base level for the surrounding hillslopes, provides numerous opportunities to test the analytical model by direct measurement. Measurements of hillslopes in the Florida sapping network found quantitative agreement between the predicted and observed hillslope morphology. An expected relationship between geometric drainage area and channel growth velocity was not borne out in the data, but the distribution of measured v/K ratios is consistent with what I expect based on my preferential sampling of slow-moving gently-sloped heads. Several explanations are given to explain why the expected relationship with drainage area is not observed, and suggestions for future work based on these findings is offered.

4 Acknowledgments

This project would not have been possible without the help of several people. First I would like to thank Professor Taylor Perron for his help at every stage of this project. Taylor's positive attitude, guidance, and support helped keep me on track even when other things conspired to distract me. His advice- about geomorphology and about life-has been a tremendous resource. Thank you, Taylor, I could not have done this without you. Thanks to Professor Dan Rothman and members of his group, Alexander Petroff and Olivier Devauchelle, for furnishing me with the original LIDAR elevation data from the Florida Panhandle channel network. Testing the analytical model on their data proved to be a central part of this project. And, finally, I would like to thank my girlfriend Renata Cummins for her tireless support and encouragement during my writing and coding endeavors of the past several months. She endured my gripes and weird hours without complaint, and kept me fed in the chaotic final days. I love you. :)

5 1 Introduction

Hillslopes evolve in response to base level change and the production and transport of sediments

and soils. Base level change creates a topographic gradient while the transport of sediment leads

to erosion over time [Dietrich et al., 2003]. Many 'models and field studies at the scale of individual

hillslopes have focused on the response of hillslopes to vertical base level change under conditions of

either tectonic uplift and subsidence [Dietrich et al., 2003, Roering et al., 1999, Burbank et al., 1996]

or bedrock incision by rivers [Whipple, 2002]. However, a much less studied class of geomorphic

features undergo hillslope evolution by lateral base level migration. Examples of lateral base level

migration include lateral movement of meandering rivers [Nanson and Hickin, 1986, Smith, 1976], retreating escarpments and cliffs [Rosenbloom and Anderson, 1994], and headward advance of

drainage networks [Dunne, 1980, Howard, 1988b].

In this thesis I will build on existing hillslope theory to address the evolution of hillslopes

undergoing lateral base level change. Combining a linear diffusive transport law with a steady-

state assumption and boundary conditions corresponding to lateral migration, I will formulate a

1-D analytical model that relates the measurable topography to the rate of hillslope advection.

Once I have formulated the model, I will test its predictions with measurements of seepage channel

heads in a Florida sapping network.

Seepage channel networks are just one type of geomorphic feature that evolve in response

to lateral base level migration. However, they are particularly well-suited to testing my analytical

model because they occur in undissected low-relief areas in response to the approximately horizontal

erosion of groundwater springs. Additionally, a single seepage channel network provides many

opportunities to measure and compare lateral migration rates.

The primary goal of the case study is to show that some of the channel heads have a hillslope form consistent with that predicted by the analytical model. A secondary goal will be to compute rates of headward growth at each channel tip and see whether the distribution of measured velocities is consistent with existing theories of sapping network growth.

6 1.1 Background on Hillslope Evolution

In developing a mathematical description of hillslopes in seepage channels I rely fundamentally on

previous work establishing the relationship between hillslope form and slope-dependent geomorphic

transport laws. The occurrence of convex hillforms was first remarked upon by G.K. Gilbert 1877

[Gilbert, 1877, Dietrich et al., 2003]. Years later W.M. Davis proposed that a combination of

repeated dilation and contraction would result in downhill creep of material over time and give

rise to the observed convex hillform [Davis, 1892, Dietrich et al., 2003]. Gilbert eventually made

the observation that Davis's dilation and contraction theory implied a direct relationship between

material transport and slope [Gilbert, 1909, Dietrich et al., 2003].

The slope-dependent transport hypothesis was mathematically formalized and tested by W.E.H.

Culling, whose work described how mass conservation and slope-dependent transport of individual

soil particles will result in macroscopic flow analogous to Fick's Law of Diffusion [Dietrich et al.,

2003, Culling, 1963, 1960, Fernandes and Dietrich, 1997]. M.J. Kirby's recognition that soil mantled

hillslopes-said to be "transport limited" -evolving under certain fixed boundary conditions will

come to a characteristic steady-state form independent of the initial topographic conditions provided

a final key pillar in modern day hillslope theory and modeling [Dietrich et al., 2003, Kirkby, 1971].

2 Analytical Model of Lateral Hillslope Migration

2.1 Developing the 1-D Model

Consider an idealized one-dimensional geometry with an infinite flat surface being eroded at the base by a spring, as shown in Figure 1. This hypothetical geometry simplifies the setup of the problem by neglecting the effects of two-dimensional contour curvature, drainage area, and other factors. Seepage erosion at the base will incise the toe of the hillslope until the surface reaches a characteristic form. Once the steady-state form has been achieved, the shape of the hillslope will not change as it advects horizontally. It is the form of this steady-state profile that I are seeking in my model because relating the observable form of the hillslope to the parameters governing its development would allow me to measure the current channel tip growth rate from the topography.

7 Z

V -

Figure 1: I define the reference frame to move at velocity v to the right such that the spring always remains at the origin. This plot shows my chosen coordinate system, with q(x) flux of material downhill toward the spring.

First I will choose to work in a moving reference frame. The coordinate system will move in time

such that the spring is always fixed at the origin, even as the spring cuts back into the hillslope and

advances spatially with time. The origin, then, is assumed to move into the hillside with velocity

v as shown in Figure 1.

The linear transport assumption is stated mathematically as

q(x) = -K dz(x)(1) dx

where q(x) is the volumetric flux per unit width at which material is transported downhill toward

the spring [L3/LT], K is a material transport coefficient analogous to diffusivity [L2/T], and z is

elevation, and x is horizontal distances from the base of the hillslope [Dietrich et al., 2003, Fernandes

and Dietrich, 1997, Kirkby, 1971, Culling, 1960, 1963].

Continuity requires that the volumetric flux per unit width at each point x is equal to the total

volume per unit width eroded upslope of x per unit time as the steady-state form of the curve

propagates horizontally. Using the assumption that the elevation difference between the spring and

the upper plane remains fixed in time, the continuity equation can be written

q(x) = -v(zo - z(x)) (2)

where v is the hillslope advection velocity, z, is the height of the far-field uphill plain, and z(x) is

8 the elevation as a function of horizontal distance from the spring located at the origin.

Now I have two expressions for the material flux. If I set equations 1 and 2 equal to each other

at x and take v = jvj I have dz v(z"" - z) = K . dx (3)

Rearranging and defining A = I get

dz - = A(z,, - z). (4) dx

This differential equation is separable, so I rearrange and integrate

dz = Adx (5) zoo - z

0 ~Iz dz =JAdx (6) zCOc - z

- ln(zoo - z) + C1 = Ax + C2 . (7)

Applying the boundary condition z(x = 0) 0, 1 find C1 + C2 = - In zoc. Thus,

- [ln(zoo - z) + ln(zoc)] = Ax (8)

And by raising e to both sides to eliminate the natural log terms I arrive at a final expression for the elevation above the spring z:

z = zoo (I - e-AX (9)

Now I have ostensibly met my goal of relating z to v, but to compare this prediction to a measured topographic profile it is easiest to work with a linear expression; there are many good ways to assess the goodness of fit when data follows a linear relationship. I already have the linear form in Equation 8:

- [ln(zo,, - z) + ln(zo,)] = A x + 0 (10) dependent variable slope intercept

This linear form has an expression on the left that is a function only of A and horizontal distance.

9 However, to compute this expression I must know z,, the elevation of the far-field plane. This

value may be difficult to guess for some measured topographic profiles if the far-field upland plain

does not asymptotically approach z, as the model assumes, so a form without z, in the dependent variable would be preferable.

The z, term was introduced to the model as an integration constant. I can avoid z, by taking

the derivative of Equation 9, dz = zoOe-AX(1 dz and then rearranging terms to arrive at a linear form of the relationship between a measurable quantity, z', and the unknown quantity A:

ZI In = -Ax (12) A

-nz' - In (Az,)+ A x (13)

dependent variable intercept slope

This form no longer has z, contained in the dependent variable, so I can fit a line to measured

- In z' data to get the slope A without being affected by an unknown parameter.

2.2 Assumptions

This model makes several major assumptions: 1. A steady-state form has been reached. The model assumes that the landscape will respond to changes in channel tip velocity on very short timescales, or that the tip velocity has been constant long enough to allow the hillslope to reach steady-state. This assumption is codified in Equation 2. A steady-state assumption is important for placing confidence in the channel tip propagation velocities my model produces. If the landscape does not respond quickly to changes in the channel tip velocity, velocity estimates made from the measured profiles may be too high or too low if channel propagation is accelerating or decelerating.

2. Linear slope-dependent transport processes are dominant on some hillslopes. Specifically, the model assumes that the flux of material at any point in the landscape is directly proportional to the slope of the landscape as stated in Equation 1. When this assumption is violated the elevation profiles will not have the expected exponential form; I can verify the assumption is being met, then, by comparison of the measured elevation profiles with the profiles predicted by the theory. 3. There is no uplift of the far-field plain relative to the hillslope baseline. A horizontally migrating hillslope form is implicit in my choice of a moving coordinate system.

10 4. A 1-D description is sufficient to describe the form of the laterally migrating hillslope. The model neglects the potential effects of 2-D features such as contour curvature and drainage area. The magnitude of the potential 2-D effect will be assessed with a numerical model.

5. Material parameters are spatially and temporally constant. The model treats the diffusion coefficient K as a constant value, neglecting any material heterogeneities or temporal changes in vegetation, biogenic activity, precipitation rate, or other factors that would be accounted for in a more comprehensive description of hillslope evolution.

6. There are no far-field effects on hillslope form. This assumption is used in the case study to disregard far-field elevation data that deviates from the predicted asymptotic approach toward z,, as x -+ oo.

3 Case Study

3.1 Choosing a field site

With a theoretical prediction for hillslope form in hand, I needed to choose a field site to test the theoretical prediction of that an exponential hillslope profile will result from lateral base level change and linear diffusive material transport. An ideal test site would need to be consistent with the model assumptions described in Section 2.2. A good test site would need to be flat with very little topographic relief over its area to approximate the geometry shown in Figure 1. Since I have assumed diffusive transport is occurring and that K is spatially constant, the test site should be characterised by a mechanically homogeneous material such as sand. A mechanism for horizontal base level migration must be present. The test area should cover a large enough geographic extent to allow for analysis of many hillslopes advecting at different velocities. In addition to all of these requirements, high-resolution elevation data is needed.

The hillslope profile of channel heads in sapping networks-stream networks that grow primarily by horizontal propagation of channel tips-are one type of geomorphic feature that my model should be able to describe. Their branching geometry results in many distinct channel heads for analysis, and these networks often form in unconsolidated sediments or soft rocks where many of my model assumptions should be true.

A sapping network near Bristol, Florida, meets all of the site selection criteria described here.

This area provides an ideal test because conditions in the network are consistent with many of

11 70

I60 3000 50 2500

40 2000

30 1500

20 1000

500 10

0 1000 2000 3000 4000 5000 6000 7000 8000

Figure 2: Elevation data for a sapping network near Bristol, FL. The LIDAR measurements provide me with 1-m resolution elevation data over the entire 32 km2 site.

the analytical model's assumptions. A homogenous, sandy soil allows me to assume the diffusion

coefficient K is spatially constant [Abrams et al., 2009b, Schumm et al., 1995]. Infiltration is high

and there is little topographic relief, which suggests subsurface flow will be a dominant mechanism

for transporting precipitation to channels. This subsurface flow emerges in springs, which erode

the base of each channel head and drive lateral base level migration as I have assumed in my model

setup. Additionally, the many separate channel heads each provide an opportunity to measure a

rate of lateral migration.

High resolution LIDAR data available from a previous study of this area provides an opportunity

to measure elevation profiles at the 1-m scale, giving me a very detailed picture of the topography.

This data is shown in Figure 2.

3.1.1 Previous work on seepage networks

Stream networks on Earth are predominantly formed by the capture and concentration of runoff into valleys. However, processes other than runoff concentration and overland flow are active in shaping some landscapes. Seepage erosion is the process by which groundwater emerging at the

Earth's surface shapes topography [Dunne, 1980, Howard, 1988b]. Runoff and seepage processes

12 are probably active to some extent in all stream networks, and networks showing a combination

of these effects are said to be of "composite" origin [Howard, 1988a]. Subsurface flow, including

deep groundwater flow and seepage, is an especially important mechanism in conveying runoff into

channels in humid regions [Dunne, 1980, Kirkby and Chorley, 1967]. Improving our understanding

of the way seepage erosion processes are expressed in the topography, then, is necessary for a comprehensive understanding of the way valleys and channel networks may evolve over the full range of possible physiographic and hydrologic settings.

Along with overland flow, seepage erosion is believed to play an important role in the initiation and growth of sapping channel networks. Networks where seepage erosion is a dominant control on topography have many unusual morphological characteristics and should be thought of as end- member cases in stream evolution [Dunne, 1980, Howard, 1988a, Kirkby and Chorley, 1967]. Most notable among these features is the occurrence of steep amphitheater-headed channels [Howard, 1988a, Abrams et al., 2009a, Laity and Malin, 1985, Schumm et al., 1995].

Because of its relatively minor apparent role in the evolution of most terrestrial stream systems, seepage erosion processes received little attention from researchers until they emerged as a possible explanation for the dry channel networks visible on Mars in the images returned by the Viking and Mariner missions of the 1970s[Higgins, 1982, Howard, 1988a, Malin and Carr, 1999, Laity and Malin, 1985]. Mars surface images show valleys hundreds of kilometers long being fed by angular-dendritic tributaries[Higgins, 1982, Malin and Carr, 1999]. The surrounding upland areas, however, show little evidence of dissection as would be expected with an atmospheric source, so groundwater seepage has been conjectured as a likely explanation for the observed features [Howard, 1988a, Malin and Carr, 1999, Laity and Malin, 1985]. Understanding the groundwater discharge volume and rate involved in forming these features could have important consequences for our understanding of past climate on Mars. In the absence of flowing water to observe and measure on

Mars, terrestrial analogues have been sought to understand the mechanics of seepage erosion and the evolution of sapping networks.

Most of the theoretical and experimental work done on sapping processes has dealt with unconsolidated cohesionless or near-cohesionless sediments [Howard, 1988a, Dunne, 1980]. Major terrestrial analogs have been networks in unconsolidated sedimenents or relatively porous sedimen-

13 tary rock, while the Martian networks are cut through bedrock. Recent work by Lamb et al. [2006]

has questioned whether similarities in morphology between terrestrial and Martian stream net-

works are sufficient to conclude sapping processes dominated their formation given this difference

in lithology. In an examination of bedrock canyons with morphological similarity to those formed

in seepage networks, Lamb et al. [2006] found no mechanism to explain how resistant rock could

be eroded by seepage processes and concluded that the presence of amphitheatre-headed channels

does not automatically imply formation by sapping processes.

3.1.2 Previous work on the Apalachicola Bluffs

In addition to being a good test location for my lateral hillslope migration model, the Apalachicola

Bluffs have been studied as a terrestrial analogue for Martian sapping. Schumm et al. [1995] doc-

umented that the Apalachicola tributary network shared most of the same characteristics as the

Colorado Plateau and Hawaiian sapping networks that had previously been studied as Martian analogues, establishing the Florida steephead channels as a good candidate for further study. Notably, Schumm et al. [1995] found no evidence for a uniform lithologic confining layer in the sandy soils of the Citronelle Formation-a feature seen on all other large-scale terrestrial analogues [Schumm et al., 1995, Laity and Malin, 1985, Kochel and Piper, 1986]. This observation, along with experimental results by Howard [1988b], verified that sapping networks could develop in low-cohesion soils without impermeable zones or other structural controls on groundwater flow [Howard, 1988b].

More recent work on the Florida site by Abrams et al. [2009a] established a correlation between channel tip drainage area and the rate of headward growth experienced by channel tips in the

Apalachicola Bluffs sapping network. Ground penetrating radar measurements of the water table elevation level were correlated with distance to seepage channels to establish that the groundwater table adjusts rapidly on the timescale of channel tip growth [Abrams et al., 2009a]. This observation, coupled with Dunne's (1980) earlier hypothesis that headward growth of the channel network should be proportional to the groundwater flux to the channel tip, enabled them to reconstruct the growth of the network and estimate the rate of headward advance. From their reconstruction, Abrams et al. [2009a] estimated the time-averaged rate of channel tip growth to be approximately 5.3 mm yr- 1 for all the tips in the network, with a current mean rate of -0.3 mm yr- 1 . Because each of

14 the channel tips in the network is competing for a finite amount of groundwater flux, this slowdown

is the average tip velocity is expected as the number of channel tips increases.

3.2 Goals of the Case Study

The Florida test site allows us to work toward several important goals.

1. Demonstrate that some of the longitudinal hillslope profiles in the Florida sapping

network are of the exponential form predicted by Equation 9. By measuring each of

the network tips I am able to look at a large number of laterally migrating hillslopes. I expect

that some, but not all, of these hillslopes will display the exponential form predicted by the

model. Those hillslopes that do not have this form may not meet all the criteria assumed in

my 1-D linear diffusion model.

2. Obtain measurements of A for each channel tip. Without knowing the diffusivity for

a particular area it is impossible to get an absolute velocity measurement for each growing

channel tip, but since I assume K is spatially constant the A values can be interpreted as

direct proxies for growth velocity. By assuming a K used by Abrams et al. [2009a] I can check

whether these migration rates are reasonable.

3. Check whether the distribution of measured A values is consistent with previous

work and our understanding of seepage network geometry. Howard [1988b], Dunne

[1980] have suggested seepage channel growth rates should be proportional to groundwater

flux to the eroding spring. Others have assumed this flux proportional to geometric area

[Abrams et al., 2009a]. If this hypothesis and the drainage area assumption are correct, I

should expect to find an asymmetric distribution of growth velocities and see a relationship

between geometric drainage area and measured A values.

3.3 Measuring Elevation Profiles

Elevation data from the Florida Panhandle sapping network was obtained from the authors of

Abrams et al. [2009a]. The raw elevation data was collected by the National Center for Airborne

Laser Mapping, with the resulting elevation grid at 1-meter spacing [Abrams et al., 2009a]. Before

15 any other elevations measurements were made, the LIDAR data was processed by Taylor Perron's gradcurv.m to derive maps of total curvature and downhill gradient. gradcurv.m estimates the

Laplacian and gradient at each point in the grid by fitting a least-squares polynomial surface to points within a given radius.

In order to apply my mathematical model to this real landscape data I wrote several MATLAB routines to sample elevation profiles from the elevation grid and carry out the data processing

(see Appendix). The interpolation, mathematical transforms, and regression operations are all handled by my strtprof routine that takes as input an estimated spring location and upland profile endpoint and returns profile elevations, horizontal distance, gradient, log transform, fit parameters, and other relevant metadata. A wrapper function pthandler takes a large list of spring locations and profile endpoints and passes them one at a time to strtprof for analysis, then stores the output in an aggregate data matrix. A complementary function, extractor, was written to easily access the desired information from all profiles stored in the data matrix. The methods implemented in these functions are described in the paragraphs that follow, and the full text of these routines is available for review in the Appendix.

Approximate starting and ending points were manually from the elevation and gradient data

(computed using gradcurv) using the built-in MATLAB function getpts. These lists were sorted into a vector of starting points and a vector of ending points by the pthandler routine before being passed to the elevation sampling routine strtprof .m. Once passed a starting and ending point, strtprof uses the built-in MATLAB command linspace to create x and y vectors with 1-m spacing between the sampled coordinates. Next the built-in function interp2 is used to interpolate elevations at coordinate locations other than those specified by the original LIDAR elevation grid.

This linear profile, however, captures more of the landscape than I am interested in for my analysis. I must truncate the profile so that only the parts with negative curvature and elevation increasing with x are included; areas with positive curvature near the origin suggest my initial guess for the location of the spring was slightly off and I have measured part of the stream longitudinal profile, while decreases in elevation in the upland portion signal that that part of the landscape is not part of the steady-state hillslope profile. Decreases in elevation in the upper part of the profile are often indicative of logging-related human activities or non-planar topographic features

16 - >0 d 0 dx2 dx 0 0 0 0 0 0 Sdz * d< 0 dz> 0 -< Sd d dx manually chosen * topographic start anomalies approximate spring location

Figure 3: This schematic shows how this hypothetical profile data from a manually chosen start and end point would be handled by the strtprof routine. The lighter shaded areas on the ends of this profile would be discarded by strtprof before any transformations or linear regressions are performed. Usable profile data must have negative curvature and positive slope everywhere. that predate the development of the sapping network.

Truncating the profile is done by first interpolating values of total curvature along the profile from the curvature grid computed using Taylor Perron's gradcurv. Starting at the origin, strtprof discards all points until it finds one with negative curvature; this inflection in curvature marks the beginning of the exponential form predicted by the analytical model and is a probably a better approximation of the spring location than the one selected by visual inspection of the elevation data. A loop then checks that elevation is always increasing away from the spring, and truncates the profile if any decrease in elevation away from the spring is detected. An example of how this processing would be carried out is shown in Figure 3.

Beginning my profiles at the first encounter of negative curvature seems intuitive given the predicted form of the hillslope profile I am interested in measuring. Why I am justified in disregarding points on the upper plain may seem less obvious, but subtle hills and topographic depressions on these plains appear to be remnant features of the surface topography created before the development of the sapping network. In addition, logging companies have been active bulldozing roads and planting grids of trees in the relatively flat upland areas. These human modifications to topography are clearly visible in the LIDAR elevation data but the exact nature and amount of anthropogenic disruption to the naturally occurring plain topography is difficult to discern. Consequently, data from these areas has been disregarded in my analysis.

Once elevation profiles have been sampled from the LIDAR data, I need to compute z'. The

17 Z

stream surface water

groundwater

Figure 4: Schematic diagram showing the idealized topography described by the analytical model. Ground- water levels draw down toward a spring (shown not to scale), which is eroding an infinite flat plane. Effects of 2-D features such as channel width and plan-form curvature of the channel head are neglected in this model. strtprof routine does this using a centered difference method where the gradient at point j along a measured elevation profile is taken to be

Z zj+1 - Zj1 (14) i j+1 -- Xj-1

Now, to measure A, strtprof uses the built-in function robustf it to get a fit of - In z' vs x. The line that fits this log transformation of the gradient data should have slope A according to the analytical expression developed in Equation 13. I use robustf it instead of a least-squares fit because I don't want the resulting slope (A) to be significantly biased by outliers. robustf it iteratively downweights outliers, resulting in a more accurate fit of the data. This A and all associated data is returned by strtprof and stored in the data matrix by pthandler.

3.4 Numerical simulation: exploring the 2-D Effect

The analytical model that I have developed only describes one dimension of the laterally migrating hillslope profile. Figure 4 is a sketch of that 1-D model problem setup projected into space; this

18 is the idealized topography that my model was designed to analyze. For some situations in which laterally migrating hillslopes occur the linear geometry may be a reasonable approximation, but in the case of the advancing seepage channel heads there is significant contour curvature. In light of this, it is important to quantify the magnitude of the effect the contour curvature could have on steady-state hillslope form so that I have a context in which to interpret my measured A values.

I used numerical simulations to try to understand the magnitude of this 2-dimensional affect.

Taylor Perron adapted a numerical model to find the steady-state form of a hillslope experiencing linear diffusive transport of material under conditions of fixed spring elevation, fixed height z, of the upland plain, and a base level migrating horizontally at a constant velocity. By forcing the simulated stream width to be very wide-hundreds of meters-I was able to approximate the idealized geometry assumed in the 1-D mathematical model. Running the model again with stream width set instead to a single line of grid points (infinitesimally narrow) allows me to compare the computed A values and gauge the magnitude of the 2-D effect. The ratio of the A measured from an infinitesimally narrow channel to the A measured from a very wide channel approximating the

1-D assumption gives an estimate of the deviation from the true A I can expect when measuring narrow seepage channels using the 1-D model.

All numerical experiments were carried out using these typical values: a diffusivity K of 0.01, z, of 20 m, ten meter grid spacing, a timestep of 1000 years. Each trial was allowed to run to steady- state. I wrote a loop that passed different advection velocities and "streamwidths." Once each run reached steady-state I extracted hillslope profiles from the numerical solution, and computed a value of A by fitting to Equation 13 which is based on the 1-D assumption.

Representative output from the model is shown in Figure 5, and the results of all runs are summarized in Figure 6.

4 Data

4.1 Measured profiles and computed A values

In total, 171 longitudinal profiles were measured from channel tips in the Florida LIDAR data.

These profiles measured all of the major channel terminations visible on the elevation map in

19 0

75./40a m2530

Figure 5: Vertically exaggerated surface plots show the output two runs of the numerical model. The plot on the left shows the steady-state topography resulting from an infinitesimally narrow channel. The plot on the right shows the steady state resulting from the same parameters only using an extremely wide channel geometry. Note that the grid spacing in this model is 10m, so this plot represents a forced width of 250 meters. When widths are forced to large values the longitudinal profile converges to the idealized form predicted by the analytical model.

1.25

1.20 K

1.15 - Anarrow A wide 0 1.10 V 0 00 00

1.05

0.0002 0.0004 0.0006 0.0008 velocity, v (m/yr)

Figure 6: This plot shows the ratio of A computed on an infinitesimally narrow channel to A computed on a very wide channel approximating the idealized geometry of Figure 4; this ratio gives an estimate of the amount by which A measured with the 1-D linear diffusion model will deviate from the true value in seepage channel heads with significant contour curvature. The magnitude of the effect is a function of velocity. The effect here seems counterintuitive, with the percent effect being larger for small values of v.

20 Figure 2, as well as a large number of smaller channel "buds" that are clearly visible from the curvature and gradient maps but that have not yet achieved significant headward growth.

Of the measured elevation profiles, eight were discarded because the automated sampling procedure did not return enough points to perform meaningful regressions; this usually means lumps and other topographic anomalies were significant enough that the sampling routine was forced to truncate the section very close to the spring.

Visual inspection of hillslope profiles was used to remove those that do not reflect the steady- state and linear creep assumptions underlying the mathematical model described in Section 2.

Profiles with significant "lumps" or other topographic anomalies suggest the measured profile is not at steady-state, while those with abrupt changes in slope or steep non-exponential profiles are probably the result of significant nonlinear transport effects. Secondary screening was carried out to remove profiles whose log transformed gradient data (after Equation 13) did not follow an approximate linear trend; either these hillslopes are dominated by processes other than linear creep or else they are not at steady state.

Many of the discarded profiles follow a steep linear trend near the spring before tapering off exponentially toward the upland plain. Convexo-straight profiles such as these are indicative of significant nonlinear transport effects [Dietrich et al., 2003, Roering et al., 1999] and are therefore not properly described by the linear transport law I assumed in the model. These profiles may be addressed separately in a future study. Other discarded elevation profiles had near constant slope, and may represent angle-of-repose slopes.

After these quality control measures, 58 hillslope profiles remained for further analysis. The average profile encompassed 7.16 meters of measured relief from spring to upland termination. A few profiles encompassed up to 14 m of relief. Profiles had a mean length of 36.0 meters, with a range from 9 to 70 meters. Many profiles were truncated prior to completely leveling out on the upland plain so the wide range of measured profile extents should not be regarded as a true reflection of the total relief on each channel tip.

For each profile, robustf it approximates A by finding the slope of the linear best fit line associated with the transform described by Equation 13. Each calculated A is associated with a standard error giving an indication of how well the log transformed gradient values are represented

21 16

14-

o 12 -

2 10 -

4 - -

2 -- - - 0 L------0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Figure 7: Plot shows a histogram of 58 measured longitudinal hillslope profiles. The peak shown on this histogram corresponds to the A range 0.0328-0.470 m- 1 . Mean: 0.0523 m 1 ; Median: 0.0410 m- 1 . by a line. Figure 8 shows a plot of A versus the relative standard error.

The 58 profiles measured have a mean A of 0.0523 m- 1 , median of 0.0410 m-1, and mode at approximately 0.04. The distribution has a positive skewness of 0.9775, as shown in Figure 7.

Plotting A vs relative standard error reveals seven outliers in the range A=0-0.06, where values with relative standard error in excess of 30% were considered to be outliers. When these outliers are neglected, the mean A decreases to 0.0618 and skewness falls to 0.441.

After discarding those hillslope profiles with very large standard errors, 51 profiles remain.

Data on the location and extent of each of these profiles is presented in Table 1 in the Appendix.

Measured A values and associated statistics are compiled in Table 2. Elevations, log transformed gradient data, and linear fit for each of these 51 hillslope profiles is also available in the Appendix.

4.2 Quantifying the 2-D effect of channel width

Numerical simulations were performed to simulate the effect of 2-D geometry on the longitudinal profile and computed values of A = v/K over headward growth velocities from 0.1 mm/yr to 0.7 mm/yr. The results, summarized in Figure 6, show that a 10% overestimation bias should be

22 35%

30% -

25% -

20% -

15% -- *

10% * *

5% * 4* ***

S I I I I I i I I I 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Figure 8: Plot shows relative standard error, the standard error associated with each fit line as a percentage of the corresponding A measurement. The relative standard error is fairly consistent across the entire range of observed A values. Seven outliers with relative standard error well in excess of 30% not shown; all belonged to short profiles with relatively few points constraining the fit. expected in profiles moving with moderate velocity. Errors up to 25% are possible for profiles advecting at velocities on the lower end of the expected range.

5 Analysis and Discussion

5.1 Assessing the 2-D Effect

One question to consider before proceeding further into the discussion of these measured A values is whether the 2-D shape of the channel tip is adding a significant systematic bias to the measured values.

The findings shown in Figures 6 are good news for the linear diffusion model because they tell me I can compute A values and be confident that I am getting an estimate that does not significantly diverge from the true value due to the 1-D assumption. Overestimations of 10-15% will not prevent me from drawing general conclusions about the relative rates of channel tip growth.

Though Figure 6 shows an inverse relationship between percentage error and velocity, Figure 9 shows that this trend is reversed when considering the absolute magnitude of the A overestimation.

23 0.0008

0.0006

o 0.0004

-o *

o 0.0002

0 0 0.0002 0.0004 0.0006 0.0008 true velocity, (m/yr)

Figure 9: This plot shows how, in spite of the unusual trend in Figure 6 where there is an inverse relationship between the magnitude of the effect and velocity, the absolute deviation of a measured A from the true A due to 2-D effects increases with v. The red line provides a 1:1 reference. This effect leads to a small amount of dilation in the relative A values measured over a range of A values. These points follow a linear trend with slope 1.06179, zero intercept; R2=0.903828.

24 When measured, the hillslope profiles advecting at high speeds will give a measured A value that

exceeds the true value by a larger amount than hillslopes advecting at slower speeds. And, as

Figure 9 shows, there is a divergent linear trend between the true and measured values as a result

of the 2-D effect.

The linear trend in Figure 9 is somewhat strong, with an R2 value of about 0.9. Future work

might add to this plot with more runs at greater v resolution. Other numeric simulations also need

to be done to assess the importance of diffusivity K, which in all these numeric experiments has

been held constant at 0.01. If further numerical experiments show weak dependence on K and

the lineartrend of 9 remains strong with greater v resolution, it would be possible to calibrate the

measured A values to correct for this divergent trend.

5.2 Distribution of A values and observed spatial relationships

As shown in Figure 7, the measured A values follow a skewed distribution. Similar amounts of relative error are present over the entire range of measured A values, as shown in Figure 8 extreme values is comparable to that observed for values near the mean, so they should not be regarded as outliers.

On the contrary, this skewed distribution of measured A values actually seems to be in line with what I expect considering the observable geometry of the network and what I know about my sampling biases. The network geometry, shown in the elevation map of Figure 2, shows that there are many more small channels impinging on each other-growing into drainage areas shared by many other heads-than there are channels growing out into open space. If the groundwater flux hypothesis put forward by Dunne [1980] is correct and the velocity of headward growth is proportional to groundwater flux to each tip, I would expect to have many channels growing with relatively slow velocity and only a handful moving at relatively fast velocities.

This predicted trend is generally consistent with what I see when I look at the spatial distribution of computed velocities. A map view of tip velocities, as in the excerpt shown in Figure 10, shows that slow-moving channel heads occur exactly where I expect based on Dunne's theory: in close competition with many other seepage channel heads. The faster moving heads, shown in hot colors in Figure 10, occur only on the outside of arcing branches where there is relatively more room to

25 Figure 10: This excerpt from the full map shows a closeup view of a few network branches. A perfect drainage area to velocity relationship doesn't appear to be in effect, but in general the tips with higher A (colored in yellows, oranges, and reds) appear on the outside of the curved branches while slower heads (colored in blue, aqua, and green) are confined to the interior regions. grow.

My sampling procedure is also likely to have reinforced the skewed distribution I expect for the entire network. Many of the discarded profiles show signs of nonlinear transport, which have steepened to the point that their slopes have approached a critical slope. In order for slopes to become that steep while creep is occurring I assume the nonlinear profiles are generally moving faster than the linear creep hillslope profile included in my analysis. It seems, then, that I have systematically discarded many of the high A heads while retaining all of the slow-moving linear creep dominated profiles. My model is only applicable to linear creep dominated hillslopes, and the resulting sampling bias produces additional skew in resulting distribution of A values.

To see if the relationship between drainage area and A is rigorously borne out in my measurements I wrote an additional MATLAB routine that computed drainage area using distance to nearest neighbor. The drainage area at each head was taken to be the geometric locus of points closer to that head than any other point in the stream network. The resulting drainage areas are included in Table 2. Plotting this drainage area against A, as shown in Figure 11, reveals no direct relationship between drainage area and A as I would expect if Dunne's theory is correct and geometric drainage area is a good proxy for groundwater flux[Dunne, 1980, Howard, 1988b,

Abrams et al., 2009a]. Exploring possible power-law relationships between drainage area and A by

26 0.18

0.16

0.14 T

0.12 IL -

~I0. 1

0.08 - IIi

0.06 -

0.04 --

0.02 - X 0 0.002 O.00 0.00 0.008 0.01 0.01 2 0.014 0.016 0.018 0.02 drainage area, km2

Figure 11: Plot shows relationship of calculated A values vs geometric drainage area. Drainage area for each head was taken to be the locus of points closer to that head than any other part of the stream network. Error bars are i2SE,\, where SEA is the standard error associated with the A for each longitudinal profile analyzed using robustf it. log-scaling the axes did not reveal any significant trends.

While the apparent independence of A and drainage area is surprising, it could have several possible interpretations. The first of these is simply that my geometric definition of the drainage area based on proximity to the channel network may differ significantly from the area over which groundwater is actually draining into a particular head. Though the proximity rule sounds reasonable, some of the polygons it produces have very unusual elongate shapes that seem somewhat divorced from my intuitive sense of what drainage the drainage area should be. Estimates of drainage area could be refined in future work by considering drainage direction in addition to euclidean distances.

A second possibility is that drainage area at the surface is not a good proxy for three-dimensional groundwater flux to the head, as I have assumed. However, Abrams et al. [2009a] addresses this assumption at length with a combination of analytical, numerical, and observational methods and concludes that the assumption is sound [Abrams et al., 2009b]. Other factors such as elevation of the spring, or elevation of the spring relative to adjacent neighbors, may factor significantly into the volume of groundwater flux.

A third possibility is that there is a linear relationship between drainage area and velocity

27 but there is so much noise in the trend that it is not observable with only these 50 points from slow-moving linear diffusive hillslope profiles I have measured. Perhaps if I were able to obtain measurement of A from steeply-sloped fast-moving profiles that are likely undergoing nonlinear diffusive transport I would see a weak trend.

A fourth possible interpretation is that one of the assumptions going into the 1-D linear diffusion model has been significantly violated in its application to this area. The existence of a linear trend in the - In z' data provides a check of the steady-state and linear diffusion assumptions. Though, admittedly, the strength of the linear trend varies significantly from one profile to the next, a more stringent criterion in which only the very best profiles (those with R2 > 0.9) were plotted against drainage area did not turn up any hidden trends. Numerical experiments show that I can expect the 2-D effect to be small; though the divergent effect shown in Figure 9 could be partly to blame for the lack of an observable trend, I would expect this to skew the relationship between drainage and A rather than totally confound it. Measurements on synthetic profiles in which the upland portion was discarded showed that good estimates of A can still be reached even when the far-field uphill plain is not factored in to the measurement.

That leaves my assumption of spatially constant K as a suspect. Abrams et al. [2009a] and

Schumm et al. [1995] describe the soil in the area of the Florida Panhandle sapping network as homogenous and sandy with very little clay. Variations in vegetation from one head another could contribute to small differences in K, but satellite imagery show dense vegetation in all the heads.

Consequently I do not think this is a large source of error in my measurement.

Given that the standard errors associated with the measured A values are so small and the assumptions of the model are so consistent with the conditions present in the seepage network, I do not think the 1-D linear diffusion model is at fault. I cannot evaluate the third explanation-that a trend might emerge if I had the missing data from heads with large A-until a way for measuring those advecting profiles undergoing nonlinear diffusive transport is developed. Of these possibilities,

I favor the second; the relationship between spring elevation, elevation relative to nearest neighbors, and other potential controls on groundwater flux measured A will definitely be an area of continued inquiry.

28 5.3 Comparison of v with Abrams et al. [2009a]

Field and experimental studies undertaken to constrain the diffusion coefficient have produced

results varying over orders of magnitude [Fernandes and Dietrich, 1997, Dietrich et al., 2003. Any

absolute velocity v determined from A will only be as certain as the diffusivity term used in the

conversion. If I take the diffusion constant to be K ~0.02 m2yr- 1 [Abrams et al., 2009a, Fernandes

and Dietrich, 1997]-the only published estimate made in an area with climate conditions similar

to the study area-our observed A values indicate current rates of headward growth from 0.1 mm

yr- 1 to 2.9 mm yr- 1 with a mean rate of 1.1 mm yr-1. Our estimate of average headward growth velocity is therefore a factor of three larger than

that obtained by Abrams et al. [2009a] using the same diffusivity. Part of this difference in the

average growth rates could be attributed to a sampling bias; the longitudinal profiles I found

suitable for analysis with the linear diffusion model represent a small non-random sample of the

total population of channel heads, though intuition suggests the relatively gentle-sloped profiles

dominated by linear diffusion will be advecting slower-not faster-than the steeper profiles I

discarded. Other sources of uncertainty have already been mentioned in previous sections. I expect

the 2-D effect to contribute to a 10-15% overestimation of the velocity at each head, accounting

for only about one eighth of the difference in our estimates.

A reassessment of the steady-state assumption may provide insight. I have assumed that each

longitudinal elevation profile analyzed represents a steady-state characteristic form. Abrams et al.

[2009a] used ground-penetrating radar to show that the water table in an area surrounding the

Florida sapping network should respond very rapidly-on the order of a few days-to changes in groundwater sources and sinks [Abrams et al., 2009a,b]. If the response of the topography to a change in headward velocity is much slower than the response of the water table, continuous changes in the groundwater table due to the growing network geometry, increased competition for drainage, and the decrease in flux to each spring may result in channel heads that are never able to reach a true steady-state form. If the average head is slowing down as the sapping network grows and more sinks are competing for finite groundwater discharge, I expect the lagging intermediate form to express a headward growth velocity that is much greater than the true current velocity.

29 Future numerical experiments into the intermediate forms produced by the 1-D linear transport

model will investigate the effects of step changes and continuous systematic changes in velocity

on the topography. In particular, I am interested to know whether the intermediate forms will be detectable based on their - In z' plots.

Even if further investigation of the channel head topographic response time reveals no significant

lag in the way velocity is expressed in the profile, it is important to remember that I have included

a fair number of head measurements from small "stubs" in my analysis. These recently initiated

branches or split tips may not have had time to reach a characteristic form even if other older heads

are able to achieve steady state. These younger channel heads could themselves be a significant

source error in the population of measured profiles. However, the bias introduced by including these

stubby heads is not likely to increase the estimate of average channel tip velocity; they often occur

in close proximity to other heads competing for groundwater and move slower than the average.

6 Concluding Remarks

The absence of a clear relationship between the geometric drainage area and headward growth rate raises questions about our understanding of the way in which sapping networks develop. In particular, these results question whether drainage area at the surface is the only important factor to consider when estimating groundwater discharge to a spring in soils with high infiltration.

These initial findings should not discourage further development of models relating the kinemat- ics of lateral hillslope migration to topography. Good agreement between the predicted exponential form and the observed longitudinal profiles suggests that the linear diffusion model produces useful estimates of the headward growth rate. Future work should prioritize the use of numerical models to understand and correct for potential sources of systematic error, including a better understanding of how two-dimensional effects and a slow response of the landscape to continuous gradual changes in groundwater discharge affect the headward growth velocities measured with the 1D analytic model.

Once the linear transport case is better understood, further development of the model to account for nonlinear transport effects will enable researchers to gather information about lateral migration

30 of hillslopes across a much broader range of conditions than is possible with the 1-D linear diffusion transport model alone.

31 References

Daniel M. Abrams, Alexander E. Lobkovsky, Alexander P. Petroff, Kyle M. Straub, Brandon McEl- roy, David C. Mohrig, Arshad Kudrolli, and Daniel H Rothman. Growth laws for channel networks incised by groundwater flow. Nature Geoscience, 2(3):193-196, 2009a.

Daniel M. Abrams, Alexander E. Lobkovsky, Alexander P. Petroff, Kyle M. Straub, Brandon McEl- roy, David C. Mohrig, Arshad Kudrolli, and Daniel H Rothman. Supplementary information for "Growth laws for channel networks incised by groundwater flow". Nature Geoscience, 2:1-14, 2009b.

D.W. Burbank, J. Leland, E. Fielding, R.S. Anderson, N. Brozovic, M.R. Reid, and C. Duncan. Nature, 379(6565):505-510, 1996.

W.E.H. Culling. Analytical Theory of Erosion. The Journal of Geology, 68(3):336-344, May 1960.

W.E.H. Culling. Soil creep and the development of hillside slopes. The Journal of Geology, 71(2): 127-161, 1963.

W.M. Davis. The convex profile of badland divides. Science, 20:245, 1892.

William E. Dietrich, Dino G. Bellugi, Leonard S. Sklar, Jonathan D. Stock, and Joshua J. Roer- ing. Geomorphic Transport Laws for Predicting Landscape Form and Dynamics. Geophysical Monograph, pages 1-30, 2003.

Thomas Dunne. Formation and controls of channel networks. Progress in Physical Geography, 4 (2):211-239, 1980.

W W Emmett and R A Freeze. Mathematical models of hillslope hydrology. Overland, pages 452-460, 1972.

F. Fernandes and E. Dietrich. Hillslope evolution by diffusive processes: The timescale for equilib- rium adjustments. Water Resources, 33(6):1307-1318, 1997.

G.K. Gilbert. Report on the Geology of the Henry Mountains (Utah). U.S. Geological Survey, Washington, D.C., 1877.

G.K. Gilbert. The convexity of hillslopes. Journal of Geology, 344-350(17), 1909.

C.G. Higgins. Drainage systems developed by sapping on Earth and Mars. Geology, 10(3):147, 1982.

A. D. Howard. Groundwater sapping on Mars and Earth, chapter 1, pages 1-5. NASA, 1988a.

A.D. Howard. Groundwater sapping experiments and modeling, chapter 5, pages 71-83. NASA, 1988b.

Alan D. Howard, R. Craig Kochel, and Henry E. Holt. Sapping Features of the Colorado Plateau: A Comparative Planetary Geology Field Guide. NASA, 1988.

M J Kirkby. Hillslope process-response models based on the continuity equation, Inst. Brit. Geogr, Spec. Publ., 3:15-30, 1971.

32 M.J. Kirkby and R.J. Chorley. Throughflow, overland flow and erosion. Bulletin of the International Association of Scientific Hydrology, 12(3):5-21, 1967.

R. C. Kochel and J. F. Piper. Morphology of Large Valleys on Hawaii: Evidence for Groundwater Sapping and Comparisons with Martian Valleys. Journal of Geophysical Research, 91(B13): E175-E192, 1986.

Julie E. Laity and Michael C. Malin. Sapping processes and the development of theater-headed valley networks on the Colorado Plateau. Geological Society Of America Bulletin, 96:203-217, 1985.

Michael P. Lamb, Alan D. Howard, Joel Johnson, Kelin X. Whipple, William E. Dietrich, and J. Taylor Perron. Can springs cut canyons into rock? Journal of Geophysical Research, 111(E7): 1-18, 2006.

M. C. Malin and M. H. Carr. Groundwater formation of Martian valleys. Nature, 397:589-591, 1999.

G.C. Nanson and E.J. Hickin. A statistical analysis of bank erosion and channel migration in western Canada. Bulletin of the Geological Society of America, 97(4):497, 1986.

Joshua J. Roering, James W. Kirchner, and William E. Dietrich. Evidence for Nonlinear, Diffusive Sediment Transport on Hillslopes and Implications For Landscape Morphology. Water Resources Research, 35(3):853-870, 1999.

Nan A Rosenbloom and Robert S Anderson. Hillslope and channel evolution in a marine terraced landscape. Journal of Geophysical Research, 99(94):013-14, 1994.

S. Schumm, K. F. Boyd, C. G. Wolff, and W.J. Spitz. A ground-water sapping landscape in the Florida Panhandle. Geomorphology, 12(4):281-297, July 1995.

D.G. Smith. Effect of vegetation on lateral migration of anastomosed channels of a glacier meltwater river. Bulletin of the Geological Society of America, 87(6):857, 1976.

Kelin X. Whipple. Implications of sediment-flux-dependent river incision models for landscape evolution. Journal of Geophysical Research, 107(B2), 2002.

33 7 Appendix 1: MATLAB functions

7.1 pthandler.m function [profiles] = pthandler(grid,C,xpts,ypts,n) %% PTHANDLER.m takes a list of x coordinates and y coordinates that record % alternating spring locations and profile endpoints, sorts % these origins and endpoints into separate lists, and then % passes these points to strtprof.m individually for processing. % pthandler then takes the output from each called to strtprof and % appends it to the end of the data matrix PROFILES.

% To easily extract certain types of data from the data matrix use % EXTRACTOR.m.

% ARGUMENT CHECKING if nargin < 5 n=O; end

% SORT SPRING LOCATIONS FROM ENDPOINTS IN XPTS,YPTS pairs=length(xpts)/2; origins=zeros(pairs,2); ends=zeros(pairs,2); for i=1:pairs origins (i, :)=[xpts (2*i-1)+2000 ypts(2*i-1)]; ends(i,:)=[xpts(2*i)+2000 ypts(2*i)]; end

% MEMORY ALLOCATION profiles=zeros(10*pairs,200);

% MAIN LOOP for i=1:pairs profile-i=strtprof(grid,C,origins(i,1),origins(i,2),ends(i,1),ends(i,2),n+i); profiles(10*(i-1)+1:10*(i-1)+10,:)=profile-i; end

% SAVE DATA fOuti = sprintf('profiles.mat'); save(fOutl, 'profiles'); end

34 7.2 strtprof.m

function [profile]=strtprof(grid,C,xO,yO,xl,yi,n)

%% STRTPROF.m % [profile]=strtprof(grid,C,xO,yO,xi,yl,n) draws a linear % profile along the path from (xO,yO) to (xl,yi) in the elevation GRID. % The variable n is used as a reference number when saving and storing % output figures.

% INPUTS Y grid = elevation grid % C = grid of total curvature (laplacian), as calculated by gradcurv.m % (xO,yO) = estimated spring location in elevation grid % (xi,yi) = manually chosen upland endpoint % n = reference integer used to name saved output

% OUTPUTS % [profile] is a 10x100 vector that stores the following information relating % to the sampled profile: % row 1: x coordinates, referenced to entire LIDAR grid % row 2: y coordinates, referenced to entire LIDAR grid % row 3: cumulative distances from origin to upland end-point % row 4: interpolated absolute elevations % row 5: relative elevations % row 6: -(log(max(row5)-row5)-log(row5)) transformed elevations % row 7: unconstrained fitline values % row 8: origin-forced fitline values % row 9: other info. see below. % row 10: calculated gradients along profile

% row 9 is used to store the calculated fit parameters from various % methods. profile(9,4) is the lambda value relied on in final % interpretations, computed using the revised method described in % Thesis Section 2.6. The associated standard error is in % profile(9,6).

% profile(9,:)=[unconstrained-fit.intercept 7 unconstrained-fit-slope % thru-origin-fit-slope % lambda % zinf-star % stats.se(1) % stats.se(2) % stats.s % zeros(1,91)];

%% ARGUMENT CHECKING if nargin < 7 % 11 length(bounds)~=4 help strtprof; return; end

% DEFINE LOCAL VARIABLES lengtho=floor(sqrt((xO-xl)^2+(y0-y)^2)); xvals=linspace(x0,xl,lengthO); yvals=linspace(y0,y1,length0);

%% ADJUST VECTORS FOR CURVATURE curvature-profile=interp2(C,xvals-2000,yvals); negcurv=find(curvature-profile<0);

% Check curvature profile and aborts if no inflection point found if numel(negcurv)==0 profile=zeros(10,200); % Creates empty data matrix 'no curvature inflection found' % Displayed message to user return; % Returns to calling function else inflection=negcurv(1); % Stores the index of the curvature % inflection point end

% Truncate x and y to begin profile at curvature inflection xvals=xvals(inflection:end); yvals=yvals(inflection:end); lengthO=length(xvals); % new length of coordinate vector

%% COMPUTE PROFILE DATA

% Cumulative horizontal distances dist=sqrt((xvals-xvals(1)).^2+(yvals-yvals(1)).^2);

36 % Elevations elev=interp2(grid,xvals,yvals);

% Truncate uphill portions that slope away from spring cuthere=0; i=1; check=1;

while check==1 i=i+1;

if elev(i-1)>elev(i) 0/ elevation decreases away from spring cuthere=i-1; /0 record index of elevation decrease check=0; exists while loop elseif i==lengthO check=0; % no elevation decrease found; exit loop end end

if cuthere~=0 % If elevation decrease found, truncate profile % to omit that section xvals=xvals(1:cuthere); yvals=yvals(1:cuthere);

lengthO=length(xvals); % new length of coordinate vector

dist=sqrt((xvals-xvals(1)).^2+(yvals-yvals(1)).^2);

elev=interp2(grid,xvals,yvals); end

% Reject extremely short profiles as inadequate; not enough points for % robust regression. if lengthO<10 profile=zeros(10,200); % Returns empty data matrix to calling 'short profile rejected' % function return end

% Relative elevations relev=elev-min(elev);

%% Log transform data zinf=max(relev); % Elevation of far-field % upland plain above A spring

37 trans=-(log(zinf-relev(1:end-1))-log(zinf)); % Skips final elevation value trans=[trans NaN]; % to avoid inf; later robustfitting % on trans will result in % error if inf value is % included

% Caculate UNCONSTRAINED FIT line to log-transformed data % (for details see thesis Section 2.4) % Robustfit forced to ignores last 30% of points that are likely % to show significant upward skew. fitdata=robustfit(dist(1:floor(O.70*lengthO)),trans(1:floor(0.70*lengthO))); fitline=fitdata(1)+fitdata(2).*dist;

% Find slope of fit if forced through origin originfit-slope=dist(1:floor(O.70*lengthO))'\trans(1:floor(0.70*lengthO))'; origin-fitline=dist*origin-fit-slope;

% Taylor's method for getting v/k without zinf (see thesis Section 2.6) z-prime=elev2grad(relev); [fit2,stats] =robustf it (dist, -log(z-prime)); lambda=fit2(2); I=fit2(1); zinfstar=exp(-I)/lambda; % zinf estimate backcalculated from lambda

% Store data to output data matrix profile(1,:)=xvals; % X coordinates profile(2,:)=yvals; % Y coordinates profile(3,:)=dist; % Cumulative horizontal distance from spring profile(4,:)=elev; % elev(i) records elevation at point [x(i) y(i)] profile(5,:)=relev; % relative elevations above spring profile(6,:)=trans; % -(log(zinf-relev)-log(zinf)) transformation profile(7,:)=fitline; % linear fit line to trans profile(8,:)=originfitline; % linear fit line to % -(log(zinf-relev)-log(zinf) forced through % origin profile(9,1)=fitdata(1); profile(9,2)=fitdata(2); profile(9,3)=origin-fit-slope; profile(9,4)=lambda; profile(9,5)=zinf-star; profile(9,6)=stats.se(1); profile(9,7)=stats.se(2); profile(9,8)=stats.s; profile(10,:)=z-prime;

38 % Pad data to fit in data matrix if lengthO<9 profile=[profile zeros(10,191)]; else profile=[profile zeros(10,200-lengthO)]; end

%% SAVE DATA % Create a filenames using sprintf fOuti = sprintf('p%03d.mat',n);

% save data as matlab binary save(fOutl, 'p'); end

39 7.3 extractor.n

function [subset] = extractor(DataMatrix,r,profsize)

%EXTRACTOR is used to select all of a particular row from each nine-row % block in the DataMatrix output by pthandler.m.

% As currently written, strtprof.m will output profiles with ten rows, % so users of extractor.m should specify profsize=10. profsize is a % parameter so that extractor can be used without modification if % strtprof is modified to generate larger output matrices with additional % methods.

% Each individual profile is represented by nine rows of the data % matrix. The ten rows are as follows:

%row 1: x coordinates, referenced to entire LIDAR grid % row 2: y coordinates, referenced to entire LIDAR grid % row 3: cumulative distances from origin to upland end-point % row 4: interpolated absolute elevations % row 5: relative elevations % row 6: -(log(max(row5)-row5)-log(row5)) transformed elevations % row 7: unconstrained fitline values % row 8: origin-forced fitline values % row 9: other info. see below. % row 10: calculated gradients along profile

% prof ile(9, :)=[unconstrained_fit-intercept % unconstrainedfit.slope % thruorigin.fit-slope % lambda % zinfstar % stats.se(1) % stats.se(2) % stats.s % zeros (1,91)]

40 % USAGE EXAMPLES:

% To obtain the unconstrained v/k fit slopes for all profiles: % >> row9s=extractor(DataMatrix,9); % >> slopes=row9s(:,2);

% To obtain the v/k slopes that result from a fit forced through the % origin: % >> row9s=extractor(DataMatrix,9); % >> slopes=row9s(:,3);

% To get the total length of each profile: % >> distances=extractor(DataMatrix,3); % >> totlength=distances(:,end);

% Relief measured on each profile: % >> relev=extractor(DataMatrix,5); % relative elevations % >> relief=relev(:,end); % maximum elevation relative to zero % % by definition comes at the end of the % % profile

[n m]=size(DataMatrix);

% ARGUMENT CHECKING if mod(n,profsize)~=O error('Data matrix must have profsize*n rows, where n is a positive integer.') elseif r>profsize error('row number, r, must be less than or equal to 9') end

numprofs=n/profsize; % Number of distinct profiles present in matrix

for i=1:numprofs subset(i, :)=DataMatrix(profsize*(i-1)+r,:); end end

41 7.4 qualitycontrol.m

function [goodlist] = qualitycontrol(distances,profiles,gradients) =X QUALITYCONTROL.m % qualitycontrol takes distance, profile, and gradient matrices % generated by extractor.m and displays them for one by one for % visual inspection. Each row in the input matrix should correspond % to a different profile.

% Top plot: distance versus elevation above spring % Middle plot: distance vs -ln(z') transform of gradients, with fit % line superimposed % Bottom plot: stem plot of residuals

% A prompt asks the user whether the displayed profile visually % meets our criteria for analysis. When all profiles have been % viewed, returns boolean goodlist. This list is used to quickly % select useable profiles from the data matrix. numprofs=size(profiles,1); figure; for i=1:numprofs

%find end last=find(distances(i,:)==0); last=last(2);

subplot 311 plot(distances(i,1:last),profiles(i,1:last),'*') fOut=sprintf('%03d',i); text(10,1,fOut)

subplot 312 grad=-log(gradients(i,1:last-1)); plot(distances(i,1:last-1),grad,'*') hold on if last>10 fit=robustfit(distances(i,1:last-1),grad); resid=-(distances(i,1:last-1)*fit(2)+fit(1))+grad; plot(distances(i,1:last-1),(distances(i,1:last-1)*fit(2)+fit(1))) fOut=sprintf("Xd',fit(2)); text(10,1,fOut) end

hold off

42 subplot 313 if last>10 stem(distances (i,1:last-1) ,resid, 'fill', '--'); end

goodlist(i)=input('Is this a good profile? 1=yes, O=no ); end

end

8 Appendix 2: Data Tables

43 9 Appendix 3: Elevation Plots

44 index easting (UTM) northing (UTM) length 0 index |X (UTM) Y (UTM) length 0 "1. ' ine'atn UM otig(T ) lnt ne X'(UM) ' (UM legt 1 695072.6460 3374622.1864 62.47 72.65 27 695283.8514 3374766.7079 30.53 283.85 2 695077.9052 3375880.0178 58.72 77.91 28 695076.8425 3374681.1058 49.73 76.84 3 695323.1301 3373363.6738 50.66 323.13 29 695073.9092 3372729.0329 27.71 73.91 4 695032.7352 3372626.9112 63.88 32.74 30 695336.9149 3374994.8158 18.23 336.91 5 695045.5563 3372681.2595 57.60 45.56 31 695314.3097 3375322.0240 24.47 314.31 6 695303.1470 3375150.1224 48.10 303.15 32 695355.0608 3373887.3182 33.40 355.06 7 695068.0394 3374861.1802 25.38 68.04 33 695292.1219 3374420.7545 48.48 292.12 8 695352.1847 3373880.6357 47.94 352.18 34 695049.2364 3372514.4926 30.44 49.24 9 695274.2364 3374696.6299 40.83 274.24 35 695033.8870 3372753.7307 41.40 33.89 10 695088.8542 3375125.5794 60.84 88.85 36 695009.4623 3375961.8862 23.30 9.46 11 695282.4259 3374684.1358 46.20 282.43 37 695041.0548 3374714.4259 24.80 41.05 12 695344.8271 3374959.0369 47.12 344.83 38 695021.0375 3373815.7510 18.17 21.04 13 695282.4500 3374673.1994 50.61 282.45 39 695075.9638 3375816.3688 31.32 75.96 14 695080.3948 3374019.5122 29.40 80.39 40 695270.0000 3374274.7527 26.44 270.00 15 695327.3808 3375014.5628 42.97 327.38 41 695081.0760 3374348.1101 38.36 81.08 16 695349.1145 3374386.5043 43.48 349.11 42 695317.2546 3372563.6738 29.66 317.25 17 695321.3402 3374871.4806 28.57 321.34 43 695033.9965 3375436.7386 21.35 34.00 18 695275.7106 3374740.1155 29.58 275.71 44 695311.0548 3374675.0884 25.41 311.05 19 695280.5735 3374695.2743 59.68 280.57 45 695274.0856 3375495.1562 41.58 274.09 20 695287.8189 3373916.5354 69.98 287.82 46 695289.6538 3375284.1662 28.42 289.65 21 695332.9044 3374869.2405 32.72 332.90 47 695273.3665 3373956.2770 14.15 273.37 22 695023.8602 3375380.6088 16.23 23.86 48 695340.1679 3372486.8698 21.59 340.17 23 695003.0128 3373894.4355 46.56 3.01 49 695057.7835 3374371.1008 26.46 57.78 24 695274.1596 3373609.5063 12.12 274.16 50 695325.7510 3375392.5767 25.35 325.75 25 695010.7389 3374680.1200 42.73 10.74 51 695322.2243 3373640.6326 9.12 322.22 26 695276.3990 3374196.1959 60.42 276.40

Table 1: Table shows the UTM location of each spring in the Florida sapping network as well as the total measured length and the direction toward the upland plain (given as a polar angle in degrees). Index numbers correspond to the profile numbers shown on data plots. index lambda RSE Drainage area (km2) index lambda RSE |Drainage area (m2 1 0.00770 13.2872% 0.676280008 0.00770 27 0.04465 6.4307% 0.826488358 0.04465 2 0.01126 18.5683% 0.910624262 0.01126 28 0.05506 8.6350% 0.570042067 0.05506 3 0.02313 14.9470% 0.865022033 0.02313 29 0.05661 11.2353% 0.417664394 0.05661 4 0.02325 7.8153% 0.80709928 0.02325 30 0.05681 14.2389% 0.758873088 0.05681 5 0.02522 7.6518% 0.832224513 0.02522 31 0.05887 16.2366% 0.649763179 0.05887 6 0.02728 8.6798% 0.884092001 0.02728 32 0.05962 10.0234% 0.513629575 0.05962 7 0.02771 14.1913% 0.445353895 0.02771 33 0.06380 3.6461% 0.610791334 0.06380 8 0.02835 8.8300% 0.76526 0.02835 34 0.06442 4.2076% 0.774417937 0.06442 9 0.02891 12.4568% 0.337356087 0.02891 35 0.06557 5.4741% 0.656974264 0.06557 10 0.02898 9.5425% 0.55394317 0.02898 36 0.06698 9.4962% 0.896272485 11 0.02955 7.1061% 0.694303354 0.02955 37 0.07265 5.7125% 0.771025236 0.07265 12 0.03105 5.7914% 0.937220307 0.03105 38 0.07387 9.7354% 0.695731584 0.07387 13 0.03172 9.2775% 0.590802616 0.03172 39 0.07495 5.2801% 0.532480727 0.07495 14 0.03371 13.4981% 0.758848908 0.03371 40 0.07764 10.2871% 0.774154482 0.07764 15 0.03422 3.0931% 0.443644269 0.03422 41 0.07939 5.6449% 0.931074483 0.07939 16 0.03737 10.9782% 0.558349482 0.03737 42 0.08768 7.6708% 0.696176868 0.08768 17 0.03844 7.8856% 0.559313774 0.03844 43 0.08787 14.0275% 0.765594241 0.08787 18 0.03882 17.5240% 0.904735801 0.03882 44 0.09007 7.1487% 0.93025342 0.09007 19 0.03891 6.3884% 0.673735704 0.03891 45 0.09304 7.3398% 0.829085454 0.09304 20 0.03924 5.5992% 0.874098368 0.03924 46 0.10039 4.9921% 0.450260004 0.10039 21 0.03945 9.6617% 0.778950255 0.03945 47 0.10328 13.8828% 0.87770954 0.10328 22 0.04055 9.7116% 0.836256879 0.04055 48 0.10418 12.3851% 0.830602447 0.10418 23 0.04146 7.2192% 0.601323357 0.04146 49 0.13081 8.1575% 0.34373778 0.13081 24 0.04197 28.6550% 0.721007235 0.04197 50 0.13821 5.6967% 0.924030972 0.13821 25 0.04280 5.7498% 0.946837792 0.04280 51 0.14611 9.1136% 0.831285062 0.14611 26 0.04283 6.4351% 0.563255693 0.04283

Table 2: Table shows a summary of computed A values, as well as associated relative standard errors (RSE), geometric drainage area, and R2 values. Relative standard error is the ratio of standard error to A, where standard error is equal to stats. se (2) returned by robustf it when fitting to values of - In z vs x. 2 Profile: 1, R2 - 0.6762M0, Xz 0.021 1 21 Profile: 2, R = 0.910624,0. = 0.034216

3 . * 00

4 N .+.0 IL-.4 * * 0 1'0 2W 3 40 50 i0 70 1 5 10 15 20 25 30 35 40 45 Distance (meters) Distance (meters)

2 2 Profile: 3, R . 0.866022, I 0.031063 Profile: 4, R a 0.80709, x = 0.039460 . - -- -- Ir - - I --- 1 - - 1 15 25, --- -- r - r------10

- * *.4 1,

.5 .* -55 1- 4

-2

0+- -- - 2 ------1 ------O 0 0 5 10 1'5 2O 2 -530 -35 40 -45 50 5 10 15 20 25 30 35 Distance (meters) Distance (meters)

2 2 Profile: 5, R = 0.832225, X a 0.038446 Profile: 6, R = 0.884092, X a 0.042804

184- 2

12- -4

3. . .0

DA - - + 10

0.8% 0 5 10 15 20 25 3.-0 5 10 15 20 25 30 3 40 45 Distance (meters) Distance (meters)

2 2 Profile: 7, R = 0.44364,0. = 0.007636 Profit.: 8, R = 0.766260,0. = 0.027263 15 3 --r- -r-,------10

2.5

- 10 - t e ( t

E S"E

Distance (meters

0.5 S 10 20 30 4M SO 60 7 Distance (motors)

47 2 2 Profile: 9, R = 0.337366, X = 0.066906 Profile: 10, R 0.53943, X a 0.028911 2. . 6 4. . . 16

2 3

2 U.5 2 2

.4. +

2 4 6 8 10 12 14 16 18 20 5 10 15 20 25 30 35 40 45 Distance (meters) Distance (meters)

2 2 Profile: 11, R * 0.694303, a . 0.066060 Profile: 12, R - 0.937220, a - 0.072647 2 5 c - -- - , ------10

' ..

4 I 4 -2 I I * 2

0 5 ** -2

C 5 s0 (m20 .5). 0 0 10 15 20 25 Distance (meters) Distance (meters)

2 2 Profile: 13, R 0.69803, a = 0.029646 Profile 14, ft 0.76484, a.. 0.038911

, +.-5' . *.. 2 .

5 10 15 20 25 3* 35 40 4'5 50 10 20 30 40 0 60 Distance (meters) Distance (meters)

2 2 Profile: 16, R . 0.443644, a * 0.031720 Profile: 16, ft = 0.666348, a.. 0.036617 3. 4r,

--- r 1------0 - 1

to0

0- 1 20 30 40 - -- -0--o 0 5 10 D a 20 25 30 Distance (meters) Distance (meters)

48 Profile: 17, R2 - 0.659314, A = 0.044663 Profile: 16, R2 = 0.904736, A 0.090067 4,--- 10 - 10

0- 0 0 0 S 10 15 2 2 - 5 0 V5 2 5 4 Distance (meters) Distance (meters)

2 Profile: 19, R2 * 0.673736, A 0.068873 Profile: 20, R - 0.874098, A 0.130807

0 0 ------5 0 10 15 2i -- 25-- - 300 Distance (meters) Distance (meters)

Profile: 21, ft 2. 0.778980, AX 0.077636 Profile: 22, R2 - 0.836257, A * 0.079389 3..- I ------,6 10

2 4

01 1- 0 50- -- - -g- 5 10 15m 2es 25 : D 0 5 10 ~ 15 20 25 30 35 40 Distance (meters) Distance (meters)

Profile: 24, R2 * 0.721007, A 0.063799 0 Profile: 22, ft " 0.836257, Ac 0.079389 10 r ------

4 '10 I I I I U .. S S -S 2-5 I

0 5 10 15 20 00 30 30 40 45 50 JO Distance (meters) 0 5101 0 1 20 25 00 00 40 Distance (meters)

49 2 Profile: 24, R = 0.721007, x = 0.063799P Profile: 26, R2 - 0.946M3, X a 0.146106

4 . * 10 S

2- 5 ID

0 S 5 10 1i 2o 25 30 35 40 5 2 3 4 0 t 7 0 9 10 Distance (meters) Distance (meters)

2 Profile: 26, R2 = 0.663266,). = 0.041971 Profile: 27, R = 0.826488, X = 0.103282 2 5 ------15

2 - 2 3 +

1.5 .02 , 4)

10+

IL 2 4a io - - 4 0 Distance (meters) Distance (meters)

2 Profile: 28, R2 2 0.670042,). = 0.033710 Profile: 29, R = 0.417664, X = 0.023127

2~ N I .0

0 a 5 0 t5 250 1 10 Distance (meters) Distance (meters)

2 Profile: 30, R2 . 0.768873, X = 0.028360 Proflie: 31, R '. 0.648763,).- 0.06617 4, 10, -- - I ------r------4

51 N

(

0 -- 0 -- - s --ls -- 5m a 5- 1.0 Is i0 20 00 4 Distance (meters) Distance (meters)

50 2 Profile: 32, f R 0.513630, A 0.073870 Profile: 33, R2 u 0.910791, 1 a 0.041457

E0 0

2 4 0 0 12 14 16 18 20 0 5 16 15 20 2i5 30 M _ _40 45 -0 Distance (meters) Distance (meters)

2 2 Profile: 34, R 0.774418, A - 0.039242 Profile: 36, R = 0.666974, x = 0.023248 Q

J10 E + .+ 4-

2 4 2.. .. I 2-+ *, e * 5 1- W+_ " *

S 10 20 NO 40 50 60 70 0 10 2- 3o 40 5s 60 - 70 Distance (meters) Distance (meters) 2 Profile: 36, R2 - 0.811,1272, X 0.065M67 Profile: 37, R 0.771026, A = 0.026224

E .

-g S

S 4 0 0 4 5 10 15 20 25 30 35 40 J1 0 10 20 20 40 50 -0 Distance (meters) Distance (meters)

2 2 5 Profle: 38, R - 0.696732, x. 0.087681 10r Profile: 39, R - 0.632481, X a 0.104183

4 ,0* I.

-- 0 Ds tanc 25 30 0 2 _ 2 Distance (motors) 20 Distance (meters)

51 Profile: 40, R2 0.774164 X 0.06M68 Profile. 41, R2 - 0.931074, X z 0.064421 35 5

E25 3 * + E2

- - - - -0 0. - 0 0 5 tO I5 20 2 5 30 5 10 is 20 25 30 35 Distance (meters) Distance (meters) 2 Profile: 42, R = 0.666177, a 0.027707 2. Profile: 43, R2 - 0.766694, A a 0.042838

4' 41 . -, . !..- .

I . JO 0- 5 10 15 20 25 30 Distance (meters) - 70 Distance (meters)

Profile: 44, R2 a 0.930263, X = 0.100396 Profile: 46, R2 a 0.829066, A 0.093038 5 5

4 2 I

0 - ----0 5 10 15 20 25 3) 0 5 00 15 20 20 30 35 40 Distance (meters) Distance (meters)

Profile: 46, R2 A0460260, A 0.087868 Profile: 47, R2 a 0.877710, x a 0.040648 10g 10 1.4,r -- -- ,------I r-

1.2 in'

-. ------

0.6-

10 15 20 2 4 t a 10 12 14 1 Distance (meters) Distance (meters)

52 2 Profile: 48, R2 a 0.830602, X z 0.138206 Profile: 49, R a 0.343738, X = 0.011263 3. , 20

S 10 N -. +., I

01- -' -Jo0 0 0 5 10 15 20 -is 30 10 20 30 4Q w0 Distance (meters) Distance (meters)

2 2 Profile: 60, R = 0.924031, X = 0.074980 Profile: 61, R 0.831285,, = 0.066979 10 4 10

6 N

a Distance (------oi L a O -,1 5 20 2 30 35 0 5 20 25 Distance (meters) Distance (meters)

53 4000 0.2

3500 0.15

2500 - 0.05

IWO -OM 20002500 *~A --- b0 0 %0

1000 -0.1

SW -0.13

-0.2 1000 2000 3000 4000 5000 6000

Figure 12: Total curvature map, Florida seepage network. Values computed from raw elevation data using gradcurv.m. North corresponds to the positive y-direction. Axes marked in meters. 0.9

I 0.0 3500 .

0.7

-0.6

25000 -0.5

-0.4

1000 0.2

0.1

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500

Figure 13: Gradient map, Florida seepage network. Values computed from raw elevation data using gradcurv.m. North corresponds to the positive y-direction. Axes marked in meters.