A MARKOV MODEL FOR SEDIMENT TRANSPORT IN EPHEMERAL … · 2012. 6. 18. · A MARKOV MODEL FOR SEDIMENT TRANSPORT IN EPHEMERAL CHANNELS Kelly, P. and Thornes, J.B. Geography Department,

A MARKOV MODEL FOR SEDIMENT TRANSPORTIN EPHEMERAL CHANNELS

Kelly, P. and Thornes, J.B.Geography Department, King’s college, London

11, Kingsland Bridge Mansions, Shrewsbury SY1 1 J F [email protected]

Resumen: Este trabajo exploratorio examina la aplicación del método deMarkov para estimar el ransporte de sedimento a través de las redes de canales.El método es revisado y se postula un modelo empleando la teoría de transportede Bagnold y las relaciones de la geometría hidráulica. La ecuación se parame-triza para una aplicación concreta y se simulan los impactos de tres escenariosde cambio con el modelo. Los resultados aún no se han confirmado, pero seidentifican y comentan las limitaciones del método

Abstract: This exploratory paper examines the application of the MarkovApproach to channel networks in estimating the transport of coarse sedimentthrough a network. The methodology is reviewed, and the model formulatedusing the Bagnold transport theory and hydraulic geometry relationships. For aspecific application the equations are parameterised and the impacts of threechange scenarios are simulated using the model. The results have not yet beenverified, but the limitations of the approach are recognised and discussed.

1. Introduction

The transport of water and sediment through ephemeral stream systems is animportant topic in geomorphology, hydrology and channel engineering. A recentcollection of paper on the subject by Bull and Kirkby (2002) provides a welcome andcomprehensive background. The contribution by Shannon et al (2002) suggests threeapproaches to sediment yield modelling using finite-element, random-walk andMarkovian methodologies. It is the purpose of this paper to present an exploration ofthe third of these, the discrete time and space Markov-Chain model and its applica-tion in a practical example. The first part of the paper examines the principles, metho-dology and implementation of the procedure. In the second part we describe a spe-

GEOGRAPHICALIA (2007), 52, 5-30

5

Geographicalia, 52, 2007: 5-30 Kelly, P. y Thornes, J.B.

6

cific application of the approach and the results obtained for two scenarios of envi-ronmental change. The paper concludes with an indication of the research needed toturn it into an operational method.

2. Principles

The principles of Markov modelling, state and transition can best be described bya metaphor (Thornes and Brunsden, 1977) in which a frog living in a lily-pond movesby jumping between the leaves in the pond (see fig.1) The state of the system is des-cribed by which leaf (A, B, C, D, or E) the frog occupies. After a jump the leaf occu-pied depends only on the previous leaf occupied. There has been a state transition.By watching the frog for a few hours we could empirically determine the probabilityof a jump from one leaf to any other leaf. For example the frog might jump from leafA. to the various other leaves as shown in Table 1 in 1000 jumps.

Table 1.The empirical results from the observation of 1000 state transitions.

Correspondingly if the frog is on A we could assert that he has probabilities ofjumping on to the other leaves of 0.2,0.4,0.3.and 0.1. An alternative approach wouldbe to assume he jumps onto the nearest leaf. In other words the transition probabil-ities can be determined empirically or by some law.

If we obtain a row of probabilities for jumps from each leaf we have a matrix,which is a Markov transition matrix. In the sediment transport model that follows thetransition matrix is obtained from the laws of sediment transport.

Furthermore the frog-jumping can be simulated. Assume the frog starts on leaf A,and then we can use a random-number table to decide which leaf he jumps to. Byrepeating this process we can simulate the frog’s behaviour. Each sequence of jumpsis called a realisation of the process.

In fact we don’t have to simulate. Markov showed that each transition’s outcomecan be obtained by post- multiplying the transition matrix by the state vector. Theresult of this multiplication is a new state vector, the next condition. Eventually this

To B To C To D To EFrom A 200 400 300 100

leads to a steady-state vector which gives the probability of the number of times eachstate is occupied. If the transitions are at regular intervals (e.g. one minute), the row(state vector) indicates the average amount of time spent in each state. This specialcase of the Markov process is called a Markov Chain. Thornes and Morley (1972) usedthis idea to investigate how many cars, on average occupied the road links in a trans-port network in Dartmoor National Park in S.W. England. A full treatment of themethod and its mathematics can be found in Herman, Bonini and Hausman(1969) andmany other texts.

Thornes (in Shannon et al. 2000) indicated application of this concept to a simplestream network (Fig. 2). This indicates another aspect of Markov’s theory. If a state(leaf, channel-link) cannot be left once entered, the probability of staying in that stateis 1.0, so the probability of leaving is zero. Such a state is called an absorbing state.The check-dam in fig.2 is such an absorbing state. Once in it a sediment particle can-not leave.

In Figure 2 Kelly (2004) illustrates how in a simple stream network the initial sedi-ment in link g starts with a few particles then, after 6 transitions a wave of sedimenthas reached g from the upstream links. So a sediment wave passes through g but can-not escape from the check-dam represented as link h.

The Markov strategy has not been widely adopted in geomorphology but it pro-vides a useful model for semi-arid channel systems where the lack of data to run more

A Markov model for sediment transport in ephemeral channelsse Geographicalia, 52, 2007: 5-30

7

Figure 1. The jumping-frog metaphor of a Markov Process.


8

Figure 2. Proposed Markov model of Thornes in Shannon et al(2000).

sophisticated models is a major problem. Rodriguez-Iturbe and Valdes(1979) work,applied by Conesa - Garcia and Alonso- Sarria (1997) is based on using the conti-nuous-time Markov process to route water through perennial river networks and drai-nage basins. They relate the instantaneous unit hydrograph (I.U.H.) to the Hortoniancharacteristics of the network to produce the Geomorphic Unit Hydrograph (G.U.H.).Rodriguez-Iturbe (1994) developed the theory to show how the Hortonian Lawsinfluence the G.U.H. and how the I.U.H. can thereby be predicted. The G.U.H. canthen be coupled to the rainfall amounts and intensities to produce the distributionalcharacteristics of families of G.U.H.s in different regimes. Finally the theory is used toderive the frequency distribution of peak flows.

Schick et al. (1987) used the Markov technique to provide a model of the role ofburial and excavation of coarse particles. They found from experiments in the Negevthat particles had a 50% probability of burial and a 25% probability of re-excavation.They also showed that for the model to reach equilibrium i.e. the point where the(vertical) distribution of particles no longer changes, would take at least 5 events.Field observations suggested that between 6 and 14 events would have been requiredto reach the present vertical distribution of marked particles.

Shannon in Shannon et al. (2000) used a random –walk model for routing waterthrough an ephemeral channel. Shannon allowed the water to move forward, for-ward-left or forward-right and by changing the probabilities of these steps she couldsimulate the different types of flow behaviours. Assigning higher probabilities to thedifferent options simulated a more sinuous flow and rougher channel reaches wouldhave higher probabilities of no movement.

Hoogendorn and Weltje (2001) extended the Markov process model to simulatethe sediment discharge record of the Terek River, on the western shore of the CaspianSea in the Russian Federation. This was successful and the simulated values fell closeto the measured values.

Care must be taken not to generate probabilities from a data record and then usethe same records to validate the model! This has been done in using vertical transi-tions between different lithologies in depositional sequences in well-cores to estimatethe steady state composition of rock types and to predict for a given lithology whatwill follow next in the sequence.

3. Model Development.

The empirical approach to assigning the probabilities in an ephemeral channelsystem would involve the tracking of marker pebbles. We have experimented with


9

this at our field site (described below) using resin ’pebbles’ implanted with strongmagnets as tracers and having the same size and mass characteristics as the actualpebbles in the link into which the tracers were planted. The experiment over severalyears since 2002 has involved searching for the dummy pebbles using a magnetome-ter. Recovery rates were quite high but the discharge events were less frequent andthe magnitudes lower than normal. As a results movement distances were small, cer-tainly not through entire links. The results showed that the movements typically havea Gamma-type distribution with a mode of 2-5 metres. This is a useful result becauseit can be used to supply the probabilities of passage through a link of a given length.Fig.3 shows the characteristics of the Gamma distribution.


10

Figure 3. Re-distribution of tracer particles by number(Y) and distance from source(X) from fieldexperiment.

In light of the difficulties of establishing the movement probabilities empirically,we decided to approach the problem theoretically. We start with Bagnold’s (1966)proposition that the transport capacity of a stream depends on the energy availablefor transport. In the first instance this is given by the potential energy of elevation,and in the second by the kinetic energy of flowing water. Losses in the network come

from frictional effects, and the residual is available for transport. These elements arecombined in Bagnold’s famous formula:

ib=w [(eb/ tan a+0.01(u/v)] —————————- (1)

Where ib is the total transport rate,w is the stream power (λqs) where λ is the spe-cific weight of water, q is the discharge per unit width of the bed, s is the energyslope, eb is the efficiency of bedload transport, tan a is the dynamic friction, u is themean flow velocity and v is the effective fall velocity of suspended sediments (sedi-ment in motion). Because power is the rate of expenditure of energy, this approachis normally called the stream power theory and it has been widely used in sedimenttransport studies (Yalin, 1982). The most common form is the cross-sectional streampower that provides a physically –based index of the sediment transporting capacityof the water flow:

w=gQS ————————— (2)

w is stream power per unit length (Watts/m), g is equal to rho*g, where rho is thedensity of the water-sediment mixture, and S is the dimensionless energy gradient,usually taken to be the channel slope. Note the similarity to the standard equation fothe erosive power of running water in hillslope studies (Kirkby, 1972)

W. Bull (1979) introduced the concept of network power by relating the terms ofequation (2) to the hydraulic geometry of the network system. Hydraulic geometryassumes that the relationship between discharge and network variables (width, depthand velocity) can be described by power functions.

Lawler (1992) substituted a power function describing downstream changes in dis-charge and a negative exponential function describing slope to give:

w=g(kLm)So e-rl) ——————————————————— (3)

Where L is the channel distance from the divide, So is the initial channel slopeupstream (i.e. of the highest tributaries). This equation produces a curve wherebycross-sectional stream power increases downstream. Substituting a power function forchannel width into the mean stream power gives:

W=[g(kLn)(Soe-rL)]/aLb —————————————————-(4)

In these equations, a, b and m, r are the hydraulic geometry coefficients and expo-nents respectively. Equation 4 shows that mean stream power (W) also peaks down-stream but closer to the drainage divide (Lawler, 1992; Lecce,1997). Knighton (1987)suggested that cross-sectional stream power should increase and mean stream powershould remain relatively constant in the downstream direction.


11

Empirical evidence from Lecce (1997) and Graf (1983a) show that maximumstream power occurs in mid-basin links

4. Methodology and procedures

The following is a list of the parameters and variables required to implement themodel for estimating the probability of transport in each reach (a link comprises manysmaller reaches).

1. Area upstream (used to estimate discharge).2. Channel slope.3. Particle size.4. Channel length.5. Channel width.6. Channel Roughness.7 Normalised probability of transport.

Particle size analysis was carried on ten randomly-selected cross sections in eachlink. One particle was measured every metre of channel width. Where the channelswas <10m wide particles were measured every 0.5m and <1m every 10cm. Thesecross-sections also provided 10 channel widths. Particles <1mm were classed as finesand organic matter and bare-rock were noted as such. Identification of particle lithol-ogy was unnecessary as blue mica schist dominates the basin.

The length of each link was surveyed from the bottom, walking upstream, in themiddle of the channel to the boundary point, taking in all the changes in height, usinga laser theodolite, which also then gave channel slope. Detailed ground photographywas also taken and this, in combination with the method of Barnes (1967) andArcement and Schneider (1989) were used to obtain the Manning roughness(n) coef-ficient (Bull, W.1979).

An estimate of discharge was calculated using the relationship of channel lengthand area following Shreve (1975) who obtained the relation:

L=kAb ————————————————————— (5)

Where A is the area above the point L down the main channel. b is the exponentfound by Shreve to be 0.6. Thornes (1976) estimated the value of b for two ramblasnear Ugijar in the Alpujarras and obtained values of b of 0.84 and 0.6 which are inkeeping with those theoretically obtained by Shreve. On the basis of topological argu-


12

ments Shreve found that the exponents change from 0.5 in small catchments to anexpected value of 0.6 for larger catchments. Avalue of 1.0 is used here for the expo-nent and 0.35 for the coefficient k. This yields:

A=L/0.35 ——————————————————— (6)

This was used in the hydraulic geometry relation for discharge and area given by:

Qb=Ad0.75 ———————————————————- (7)

Where Ad is the drainage area above a point. These manipulations are unavoid-able in ephemeral channels where acceptable actual measurements of Q are rarelypossible.

The index (Eq.8), which represents the main controls on stream power and sedi-ment transport, provides the results used to estimate the probabilities for sedimentmovement in each link. (A in m. is a proxy for discharge)

As/ (Lphi n w) ——————————————————- (8)

S is link slope, phi is the average particle size (b-axis) in phi units. The phi scaleis used because particle size is presumed to be log-normally distributed. Reach lengthL is implicated because the probability of leaving a long reach must be less than thatof leaving a shorter reach.

Figure 4 shows the application procedure to a simple case, with a hypotheticaltransition matrix in the middle of the figure. The graph at the foot shows the set oftime iterations (X) against the particle numbers for the final link (g) and the checkdam (h) which is an absorbing state.

The initial particle distribution vector comprised 400 particles. The check dam fillsafter 30 iterations of the model following the passage of a sediment wave throughlink g. Once the check-dam is filled there is no further change. The peak of the wavein g occurs after 6 iterations when most of the particles from upstream links are pass-ing through.

The probabilities are obtained using equation 8 and these must be normalised andwe do this by dividing by the largest link value in the network. This link thereforehas a probability on 1.0 of being evacuated at each iteration. In these small irregularchannels, the greatest value is usually a near- vertical drop (waterfall).

For the matrix multiplication we used the Mathcad 7 software that can only accom-modate a 10x10 matrix. For systems with a large number of links, the computingrequirement increases very rapidly so the temptation to have many sub-links has tobe resisted. This can be achieved by averaging over longer links, which we have donein this case.


13


14

Figure 4. (below) schematic representation of a hypothetical channel system, transition matrixand sediment content(number of particles) through time for link g.

Sedim

ent

Conte

nt

Of

Lin

k

Once the probabilities have been assigned the initial state vector defines howmuch sediment or how many particles will start at the top of each link. The initialstate vector is then post-mutiplied by the transition matrix to give the first step of themodel run. This operation is fully described in Bierman et al (1969).

5. Case Study

To exemplify this methodology we chose a field site in which we have subse-quently tracked particle movements. The field site is a small tributary to the lowerreaches of the Rambla Nogalte (Murcian-Almerian) border (see fig. 5) It is locatedapproximately 2km northeast of Puerto Lumbreras (mapa military 975, 1:50000,03605885). The lithology is dominated by blue flaky mica-schist which produces athin soil and has a sparse matorral cover. That contrasts sharply between north andsouth-facing slopes (as shown in fig.6). The south faces (to the left) are much steeperthan those facing north. The vegetation mainly consists of Stipa tenasissima, Anthylliscytisoides with smaller amounts of Rosemarinus officinalis and graminae species.Slopes in the system are too steep to cultivate but not too steep for grazing and treecrops (almonds and olives).The parameter values for the links are given in Table2.

The network has seven well defined links and terminates in a check-dam (fig7).The first-order channel bed slopes range from 18o-27o. Link 1 is very steep, rocky andstepped, 2, 3 and 4 are heavily vegetated and less steep. The two second-order chan-nels are quite different. Link 5 is similar to link1 with steep banks and little vegeta-tion. Link 6 is longer and split into two sections by a 6m headcut. Below the knick-point the channel narrows again to less than 10m. The main channel, link7, is moresinuous than the other six links and has a lower average slope (6o45’). Link4 has thelargest mean particle size with a phi value of 3.85.

Stream power increases overall downstream, agreeing with the results of W. Bull(1979) and Lawer’s (1992) findings. The 2 vegetated links have the highest roughnessvalues (0.11 and 0.12). The lowest estimated values are 0.5. Table2 shows the char-acteristics of the seven links under current conditions. Table 3 shows the correspon-ding transition matrix and table 4 the initial state vector or influx to the system as itis at present shows the matrix of probabilities assigned to the current system. Eachrow and column of the matrix represents a link, except for the right-most (8th) whichrepresents the check-dam and is an absorbing state. The principal diagonal of thematrix are the probabilities of remaining in the same link and are all close to unityThis result is validated by the particle tracking studies that indicate the marked parti-cles only moved very short distances, certainly less than the typical reach length. In


15


16

Figure 5. Location of experimental field site just west of Puerto Lumbreras.

Figure 6. Looking downstream at the experimental field site near Puerto Lumbreras.

the real world this probably reflects the low rainfall in recent autumns. In the modelalso it reflects the low estimated discharges. The probability of moving to a down-stream link is consistently less than remaining in the same link. This suggests that,under average rainfall events, transport of coarse sediment is a relatively rare phe-nomenon and mainly confined to large and infrequent flash floods.

Note that the greatest probability of leaving a link is from link 1 to link 5. This isbecause link 1 is very steep which offsets the small discharge. Reach 16 in link 1 hasa fall of 75.8o in just 6.3 cm (a step) giving a proxy stream power of 958976.5. In spiteof its overall low gradient, link 7 has the next largest probability of movement,because of a step and the higher discharge. The stream power composite index isaveraged over the reaches in each link. The smallest value occurs in link 3 with largeparticle sizes, greater link length, larger average width and higher Manning’s n value.

Table 4 shows a hypothetical initial state vector giving the particle content of eachlink. This is for demonstration purposes only. Each time this vector is pre-multipliedby the transition matrix, the redistribution of the particles is given for an event ofgiven magnitude (e.g. the annual peak rainfall).


17

Figure 7. Schematic plan of network at experimental site ( not to scale).

The repeated multiplication results in the progressive evacuation of the initial par-ticles and it can be.

The graphs show that it needs 4500 iterations for the almost complete evacuationof all the particles in the initial condition. This is the total number of events at theevent magnitude assumed for this run of the model. If this was the annual event, then


18

Table 2. Summary characteristics of the seven links.

Table 3. Transition matrix for case study network under current conditions.

Table 4. The hypothetical initial state vector or influx of sediment to the system as it isat present.

1000 800 300 400 700 200 300 0


19

Figure 8. (below). A Simulated wave of sediment passing through link 7between iterations 30-150.

Figure 9. Graph of the origins od deposited particles in a series of simulations of the existingsytem. The graph shows the impact of the network characteristics on the sediment patterns.

4405

6

3500

0

1500

0

8500

6500

4500

2500950

750

550

350

1507030

Initia

l Vec

tor

Sed

imen

t Con

tent

4405

6

3500

0

1500

0

8500

6500

4500

2500950

750

550

350

1507030

Initia

l Vec

tor

Sed

imen

t Con

tent

Of L

ink

the time axis (x) would represent the number of years. A very large event couldachieve this in a few iterations. It would be interesting and useful to invert the processto observe what magnitude and frequency combination would be needed to rendera freshly constructed check dam useless. In the last decade the large autumn floodsappear to have diminished in size and ‘reliability’. If so, this will give the large num-ber of check dams constructed after the great flood of 1973 a greater length of use-ful life.

We present a compound colour-coded graph (fig9). The very steep loss in capac-ity for the check-dam (blue) is shown by the steep fall in the curve to the left-handside of the graph. In the next section we use similar compound graphs to show theeffects of various scenarios. Notice how the configuration of the system and its initialstate produce a very rapid filling of the check-dam between 150 and 450 iterations.There is no change in runoff conditions. It is the network geometry that is determin-ing the performance of the transport conditions. The composite diagram is a type ofresponse map to a unit impulse of sediments at the initial condition. In this sense, wemight think of it as a kind of Geomorphic Sedigraph. Evidently the sudden filling ofthe check dam is caused by the sediment wave passing through link 7 between iter-ations 30-150 and shown in fig 8.This wave is especially marked in links 5-7.


20

Figure 10. The outcomes of simulation of the ‘fire in links 3 and 4’ scenarios.

2180

7

2100

0

1000

0

8500

7000

5500

4000

2500

1000

850

700

550

400

250

154

150

805020

Initi

al V

ecto

r

Sed

imen

t Con

tent


21

6. Scenarios

Having defined our system we now consider what it can tell us about the resultsof changes in the network or in its environment. The three scenarios that are consid-ered for exploration are (a) change in the network by growth of an extra link, (b) theinfluence of a fire in the headwaters (links3 and 4) and (c) climate change over thebasin.

(a) The first scenario is the addition of an extra link where a gully is assumed tohave developed in the lower reaches of link 6 (see fig.7). This would provide extrasediments to both links 6 and 7 and an expected increase in discharge, particle sizeand roughness in these links. We used a proto-gully to provide parameters for theadditional link (see table 5). The transition matrix is given in Table 6 and the com-pound graph of simulations outcomes is shown as figure 10 (another compoundcolour graph). The main effect is to increase the probability of movement out of link3,50 that, as shown in fig.10, the evacuation of this link is rapid. The result is a ten-dency to build up sediments in link 6 (the red-coloured bars in the compound graph).The total evacuation of the initial and added particles requires 21808 iterations, abouthalf that for the case without a fire. The interesting result again is the way in whichthe network configuration and characteristics play an important role in the sedimen-tational response following an environmental change.

Table 5. Summary data for the small gully labelled in figure 6 now introduced to the system asa fully integrated link.

(b) The second scenario is for a fire in links 3 and 4 of the network. This washypothesised to remove the vegetation in the channel and hence reduce theManning’s n and to increase the movement of coarser particles. These changes areshown in Table 7 and the resulting revised transition matrix is shown in Table 8. Inaddition we added 4000 extra particles to the system to give the initial state vectorgiven in Table 10. The changes in the probability matrix are given in Table 11. Theresults are summarised in the compound graph (Fig. 11) 46 iterations after the

changes were introduced only 63% of the initial particles in link 4 remain there. Theresult is a tendency to build up sediments in link 6 (the red-coloured bars in the com-posite graph). The total evacuation of the initial and added particles required 21808iterations, about half that for the case without a fire. The interesting result again is theway in which the network configuration and characteristics play an important role inthe sedimentational response following an environmental change.

(c) Work by Woodward and Diamond (1988) in conjunction with the Elwell andStocking curves show that climate change could cause a 40% reduction in vegetationcover and a 76% increase in runoff in the next 50 years in south-East Spain. Table 10shows our hypothesised changes to particle size and Manning’s n for all seven linksin the subsystem. Table 11 shows the re-computed transition matrix. Table 12 showsthe initial state vector for climate change.


22

Table 6. The matrix of probabilities used to run the model for the influence of a new link.

Table 7. Changes made to the Manning n and particle size values for links three and four forfire scenario.


23

Table 8. The matrix of probabilities used to run the model after 153 iterations for the influenceof a fire across links three and four.

Table 9. The state vector used to multiply the matrix for scenario two, the influence of a fire inlinks three and four.

0.107 49.676 3257.42 4136.55 478.62 452.158 342.644 1980

Table 10. Changes to the particle size and Manning n for all seven links under the influence ofclimate change.

The model took 5000 iterations to move 99.9% of the total sediment to the checkdam. The outcome of the simulation is shown in the compound Fig. 12. Again thedifferent sources are shown in different colours. The pattern is very similar to thatobtained without climate change adjustments. That could be taken to imply that therole of the network characteristics might be just as important as the climate changein determining the redistribution of sediments.

7. Discussion and Conclusions

The first part of the paper briefly reviewed the methodology of the Markov mod-elling approach and the previous efforts to implement it. Essentially the transitionmatrix has to be generated empirically or from theoretical approaches. The initial con-dition vector is post multiplied by the transition matrix to give a new distribution ofparticles. A change in the environmental conditions can be simulated by an alteredinitial vector and/or a revised transition matrix and the operation recommenced. Thesoftware can be home-grown or a commercial package such as MatCad that allowsmatrix operations.


24

Table11 Probabilities of movement for the subsystem under the influence of climate change.

Table12. The state vector used to multiply the matrix for scenario two, the influence of a fire inlinks three and four.

0.107 49.676 3257.42 4136.55 478.62 452.158 342.644 1980

The application of the methodology in the second half of the paper is intended toillustrate its potential when coupled with parameters in a field situation and to exploreits use in environmental change scenarios. The three applications considered aremeaningful but the actual parameters used represent our best estimates of the likelyparameter changes. The model simply simulates the outcomes of these premises andthey are therefore to be regarded as outcomes rather than predictions.

These outcomes provide three important conclusions:

First that the network characteristics and geometry play a crucial role in thesequence of sediments passing a fixed location.

Second, the sequence can account for the timing of waves of sediments from dif-ferent sources and their relative magnitudes. In turn this could help to explain thesedimentary succession in depositional environments such as the check dam or inspecific links. It should be interesting and possible to ‘match’ sediment sequences atdepositional sites in river networks with the network upstream of the sites.

Third, caution is needed not to mix the controls of the geomorphic network withexternal conditions when interpreting sediment sequences as exhibiting climatechanges.

Of course there is still much to be done to improve the approach. It might be appro-priate in studying contemporary perennial systems to have a different transition matrixfor events of different magnitudes, so that the impacts of different magnitude and fre-quency of event can be simulated. Furthermore, in this example we have averagedparameters over reaches within links. At least in principal the reaches could be muchshorter, but this would require larger matrices and a considerably greater computingpower than that used here.

The subsequent experiments with particle tracking revealed that movement ratesare very small and that an empirical Gamma Function could be used to estimate theprobabilities of the distance travelled to provide a continuous-space Markov modelrather than the discrete time and space version used here. As always with ephemeralchannels however, inferences based on a few years of observation are subjected toextreme fluctuations. Thus using the Bagnold sediment transport model to estimatethe probabilities may be the only available option, especially for historical studies. Wehave not been able to verify the results for the scenario but the tracking experimentsshow how very slow the rates of movement are under the climatic conditions pre-vailing over the last seven years.


25

8. Acknowledgements

The authors are indebted to N.E.R.C. for the travel grant to Peter Kelly. TheLeverhulme Trust Millennium Fund supported the tracking experiment by Thornes.Prof. K. Rowntree from Rhodes University South Africa contributed to the detailedtopographic surveying of the gully network used and Prof. R. Ferguson provided valu-able advice and practical help with the tracking experiment and Mr and Mrs. J.Fallows assisted in the task of recovering the tracked particles under difficult condi-tions.


26

Figure 11. The outcomes for the ‘climatic change’ scenario.

4127

041

000

3000

015

000

9000

7500

6000

4500

3000

150090

075

060

045

030

015

060525020

Initia

l Vec

tor

Sed

imen

t Con

tent

10. References

Andreu, V., Rubio, J. L., Gimeno-Garcia, E.,Llinares, J. V. (2000). Impact of water ero-sion on burnt Mediterranean areas:Spatial and temporal variability, In:Mediterranean Desertification, ResearchResults and Policy Implications, EuropeanCommis-sion, pp. 265-276.

Arcement, G. J. Jr. and Schneider, V. R. (1989).Guide for selecting Manning’s roughnesscoefficients for natural channels and floodplains, USGS Water-Supply Paper, 2339.

Baker, V. R. and Costa, J. E. (1987). FloodPower, In: Mayer, L., Nash, D. (Eds).,Catastrophic Flooding, Allen and Unwin,London, pp. 1-24.

Bagnold, R. A. (1960). Sediment dischargeand stream power- A preliminaryannouncement, USGS Circular 421.

Bagnold, R. A. (1966). An approach to thesediment transport problem from generalphysics, USGS Professional Paper, 422-I.

Bagnold, R. A. (1977). Bed load transport bynatural rivers, Water Resources Research,13, pp. 303-312.

Barnes, H. H. (1967). Roughness characteris-tics of natural channels, USGS Water-Supply Paper, 1849.

Bierman, H.Jr., Bonini, C.P.. and Hausman,W.H. (1969). Quantitative Analysis forBusiness decisions. R.D. Irwin HomewoodIllinois, 496 pp.

Bull, L. J., Kirkby, M. J., Shannon, J., Hooke,J. M. (2000). The impact of rainstorms onfloods in ephemeral channels in S. E.Spain, Catena, 38, pp. 191-209.

Bull, L. J. and Kirkby, M. J. (2002). Drylandriver characteristics and concepts, InDryland Rivers: Hydrology and Geomor-phology of Semi-Arid Channels, Wiley,Chichester.


27

Bull, W. B. (1979). Threshold of critical powerin streams, Geological Society ofAmerican Bulletin, 90, pp. 453-464.

Butcher, G. C. and Thornes, J. B. (1978).Spatial variability in runoff processes inan ephemeral channel, Zeitschrift furGeomorphologie, Supplement Band 29,pp. 83-92.

Chow, V. T. (1964). Handbook of AppliedHydrology, McGraw-Hill, New York.

Conesa-Garcia, C. and Alonso-Sarria, F.(1997). Stochastic matrices applied to theprobabilistic analysis of runoff events in asemi-arid stream. Hydrological Processes,11, pp. 297-310.

Costa, J. E. (1975). Effects of agriculture onerosion and sedimentation in thePiedmont Province, Maryland, GeologicalSociety of America Bulletin, 86, pp. 1281-1286.

Davis, D. J. (1980). Rare and unusual postfireflood events experienced in Los AngelesCounty during 1978 and 1980, InCalifornia Institute Of Technology.

Elwell, H. A. and Stockin, M. A. (1976).Vegetation cover to estimate soil erosionhazard in Rhodesia, Geoderma, 15, pp.61-70.

Ergenzinger, P. and Conrady, J. (1982). A newtracer technique for measuring bedload innatural channels, Catena, 6, pp. 77-80.

Ferguson, R. I. (1987). Hydraulic and sedi-mentary controls of channel pattern, In:Richards, K. S. (Ed.), River Channels:Environment and Process. Blackwell,Oxford, pp. 129-158.

Ferguson, R. I. and Wathen, S. J. (1998).Tracer-pebble movement along a concaveriver profile: Virtual velocity in relation tograin size and shear stress, WaterResources Research, 34:8, pp. 2031-2038.


28

Frostick, L.E., Reid, I., Layman, J. T. (1983).Changing size-distribution of suspendedsediment in arid zone flash floods,International Association ofSedimentologists Special Publication, 6,pp. 97-106.

Graf, W. L. (1983a). Downstream changes instream power in the Henry Mountains,Utah, Annals of the Association ofAmerican Geographers, 73, pp. 373-387.

Gregory, K. J. and Walling, D. E. (1973).Drainage Basin Form and Process,Edward Arnold, London.

Hassan, M. A. (1990). Scour, fill and burialdepth of coarse material in gravel bedstreams, Earth Surface Processes andLandforms, 15, pp. 351-356.

Hickin, E. J. and Nanson, G. C. (1984). Lateralmigration rates of river bends, Journal ofHydraulic Engineering, 110, pp. 1557-1567.

Howard, R. A. (1960). Dynamic programmingand Markov processes, The M.I.T. Press,Massachusetts Institute of Technology,Cambridge, Massachusetts.

Hoogendoorn, R. M. and Weltje, G. J. (2000).Modelling of sediment discharge as aMarkov process,[email protected]

Kirkby, M. J. (1999). Gully characteristics andModelling, In MediterraneanDesertification and Land Use, EphemeralChannels and Rivers, Final Report, pp. 53-112.

Kelly, P.,(2002) To explore the use of theDiscrete Markov process analogy to routesediment through an ephemeral channelsystem, south-east Spain. MSc Dissetationthesis, Geography Department, KingsCollege, London.

Knighton, D. (1987). River channel adjust-ment-the downstream dimension. InRichards, K. S. (Ed.)., River Channels:Environment and Process, Blackwell,Oxford, pp. 95-128.

Knighton, D. (1998). Fluvial Forms andProcess: A New Perspective, Arnold,London.

Knighton, D. (1999). Downstream variationin stream power, Geomorphology, 29, pp.293-306.

Knighton, D. and Nanson, G. C. (1993).Anastomosis and the continuum of chan-nel pattern, Earth Surface Processes andLandforms, 18, pp. 613-625.

Lane, E. W. and Borland, W. M. (1953). River-bed scour during floods, American Societyof Civil Engineers, Paper No. 2712, pp.1072-1079.

Langbein, W. B. (1947). Topographic charac-teristics of drainage basins, U.S.G.S. WaterSupply Paper, 968-C.

Laursen, E. M. (1958). Total sediment load ofa stream, Journal Hydraulics Division,Proceedings of American Society of CivilEngineers, 84 (HY1), pp. 1531-1536.

Lawler, D. M. (1992). Process dominance inbank erosion systems, In: LowlandFloodplain Rivers: GeomorphologicalPerspectives, Carling, P. A. and Petts, G. E.(Eds) Wiley, Chichester, pp. 117-143.

Lecce, S. A. (1997). Nonlinear downstreamchanges in stream power on Wisconsin’sBlue River, Annals of the Association ofAmerican Geographers, 87, pp. 471-486.

Leopold, L. B., and Miller, J. P. (1956).Ephemeral streams-hydraulic factors andtheir relationship to the drainage net.USGS Professional Paper, 282-A.

Leopold, L. B., Emmett, W. W., Myrick, R. M.(1966). Channel and hillslope processes ina semi-arid area, New Mexico, USGSProfessional Paper, 352-G.

Lopez, R. and Batalla, R. J. (2001). Analisis delcomportamiento hidrologico de la cuencamediterranea de Arbucies antes y despuesde un incendio forestal, III CongresoForestal Espanol, Granada.


29

Mackin, J. H. (1948). Concept of the gradedriver, Geological Society of AmericaBulletin, 59, pp. 463-512.

Magilligan, F. J. (1992). Thresholds and thespatial variability of flood power duringextreme floods, Geomorphology, 5, pp.373-390.

Mansanet Terol, C.M. (1987). IncendiosForestales en Alicante. Estudio de laEvolución de la Vegetación Quemada,Publicaciones de la Caja de AhorrosProvincial, Alicante.

McEwen, L. J. (1994). Channel planformadjustment and stream power variationson the Middle River Coe, WesternGrampian Highlands, Scotland, Catena,21, pp. 357-74.

Meade, R, H. (1982). Sources, sinks, and stor-age of river sediment in the Atlanticdrainage of the United States, Journal ofGeology, 90, pp. 235-252.

Melia, J. (In Press). MediterraneanDesertification: A Mosaic Of Land-UseChange, Eds. Brandt, J., Thornes, J. B.,Geeson, N., Wiley, Chichester.

Midgley, P. (2001). www.wrc.org.za

Moreno, J. M. (1999). Forest fires: trends andimplications in desertification prone areasof southern Europe, MediterraneanDesertification Research Results AndPolicy Implications, Volume 1, ConferenceProceedings, European Commission, pp.115-150.

Morisawa, M. E. (1968). Steams: Their dynam-ics and morphology, New York.

Mosley, M. P. (1981). Semi-determinatehydraulic geometry of river channels,South Island, New Zealand, Earth SurfaceProcesses and Landforms, 6, pp. 127-137.

Nanson, G. C. and Hickin, E. J. (1986). A sta-tistical analysis of bank erosion and chan-nel migration in western Canada,

Geological Society of America Bulletin, 97,pp. 497-504.

Palutikof, J. (1998). Climate change scenariosfor the Mediterranean, Atlas OfMediterranean Environments: TheDesertification Context, Eds. Mairota, P,Thornes, J. B., Geeson, N., Wiley,Chichester, pp. 68-69.

Perry, G. (2002). PVA Toolkit, UnpublishedMicrosoft Excel Software, Department OfGeography, King’s College London.

Phillips, J. D. (1989). Fluvial sediment storagein wetlands, Water Resources Bulletin, 25,pp. 867-873.

Reid, I. and Frostick, L. E. (1984). Flash floodhydraulics and sediments in the arid zoneof northern Kenya, Universita Firenze,Departimento di ingegneria civile, Sezionegelogia applicata 3/84.

Reid, I. and Fostick, L. (1989). Channel form,flows and sediments in deserts, In: AridZone Geomorphology in Drylands,Process and Change, Thomas, L.D.S.G.(ed.), Second Edition, J. Wiley, Chichester,pp. 205-229.

Renard, K. G. and Keppel, R. V. (1962).Transmission losses in ephemeral streambeds, Journal of the Hydraulics Division,Proceeding’s of the American Society ofCivil Engineers, pp. 59-68.

Rhoades, B. L. (1987). Stream power terminol-ogy, The Professional Geographer, 39, pp.189-195.

Richards, K. (1982). Rivers: Form and Processin Alluvial Channels, Methuen, London.

Rodriguez-Iturbe, I. (1994). The geomorpho-logical unit hydrograph, In Bevan K. andKirkby, M. J. (Eds) Channel NetworkHydrology. Wiley, Chichester, pp. 43-69.

Shannon, J. Richardson, R. Thornes, J. B.(2000). Short term dynamics of ephemeralstreams, In Kirkby, M. J. and Bull, L.


30

Ephemeral Channels in the Mediterranean,Wiley, London.

Shannon, J., Richardson, R. Thornes, J. B.(2002). Modelling event-based fluxes inephemeral streams, In Dryland Rivers:Hydrology and Geomorphology of Semi-Arid Channels, Wiley, Chichester.

Shreve, R. L. (1975). Variation of mainstreamlength with basin area in river networks,Water Resources Research, 10:6, pp. 1167-1177.

Schmidt, K. H. and Ergenzinger, P. (1992).Bedload entrainment, travel lengths, steplengths, rest periods-studied with passive(iron, magnetic) and active (radio) tracertechniques, Earth Surface Processes andLandforms, 17, pp. 147-165.

Schick A.P., Lekach, J. and Hassan, M.A.(1987). Bedloasd transport in desertfloods: Observations in the Negev. In C.R.Thorne, J.C. Bathurst and R.D. Hey (Eds.)Sedient Transport in Gravel Bed Rivers,Wiley, Chichester, 617-636

Schumm, S. A. and Khan, H. R. (1972).Experimental study of channel patterns,Geological Society of America Bulletin, 83,991755-1770.

Simons, D. B. and Richardson, E. V. (1966).Resistance to flow in alluvial channels,USGS Professional Paper, 422-J.

Summerfield, M. A. (1991). GlobalGeomorphology, Longman Scientific andTechnical, Harlow, Essex, p. 198.

Thornes, J. B. (1976). Semi-arid erosional sys-tems. London School of EconomicsGeographical Paper 7.

Thornes, J. B. and Brunsden, D. (1977).Geomorphology and Time, Methuen andCo Ltd. London.

Thornes, J.B. and Morley D. (1972). A Markovdecision model for network flows,Geographical Analysis, Apr. 1972, 180-91.

Thornes, J. B. (1980). Structural instabilityand ephemeral channel behaviour,Zeitschrift fur Geomorphologie, N. F.,Supplement band 36, pp. 233-244.

Thornes, J. B. (1994a). Catchment and chan-nel hydrology, In: Abrahams, A.D. andParsons, A.J. (eds.), Geomorphology ofDesert Environments, Chapman and Hall,London, pp. 257-287.

Thornes, J.B. (1994b). Channel processes, evo-lution and history, In: Abrahams A.D. andParsons, A.J. (eds.), Geomorphology ofdesert environments, Chapman and Hall,London, pp. 288-317.

Tooth, S. (2000). Process, form and change indryland rivers: a review of recent research,Earth Science Reviews, 51, pp. 67-107.

Rodriguez-Iturbe, I. and Valdes, J. (1979). Thegeomorphic structure of the hydrologicresponse. Water Resources Research, 156,1409-20.

Woodward, I. and Diamond, S. (1998). Climatechange scenarios for the Mediterranean,Atlas Of Mediterranean Environments: TheDesertification Context, Eds. Mairota, P,Thornes, J. B., Geeson, N., Wiley,Chichester, pp. 68-69.

Yalin, M. S. (1982). Mechanics of sedimenttransport, Pergamon Press.

Documents

A MARKOV MODEL FOR SEDIMENT TRANSPORT IN EPHEMERAL … · 2012. 6. 18. · A MARKOV MODEL FOR SEDIMENT TRANSPORT IN EPHEMERAL CHANNELS Kelly, P. and Thornes, J.B. Geography Department,