Seismology Linear Velocity Gradient Travel Time Calculator

Calculator Inputs

Scenario name

Velocity at zero depth, v0

Use km/s.

Linear velocity gradient, g

Use km/s per km, equal to s^-1.

Starting depth, z1

Use km.

Ending depth, z2

Use km.

Horizontal offset

Use km. Leave zero for vertical time.

Ray parameter, p

Use s/km. Leave blank to solve from offset.

Source static correction

Use seconds.

Receiver static correction

Use seconds.

Path legs multiplier

Use 2 for a down-and-up reflection check.

Formula Used

The velocity model is v(z) = v0 + g z. The vertical travel time between two depths is:

T = |ln(v2 / v1) / g|, when the gradient is not zero.

For constant velocity, the calculator uses T = |z2 - z1| / v.

For a curved ray with ray parameter p, it uses:

T(p) = |[acosh(1 / p v1) - acosh(1 / p v2)] / g|

X(p) = |[sqrt(1 - p²v1²) - sqrt(1 - p²v2²)] / (p g)|

The final time is (ray time + source static + receiver static) × legs.

How To Use This Calculator

Enter the velocity at zero depth in kilometers per second.
Enter the linear gradient using kilometers per second per kilometer.
Set starting and ending depths in kilometers.
Enter offset if you want a ray path estimate.
Enter ray parameter if you already know horizontal slowness.
Add source and receiver statics when needed.
Use the legs multiplier for repeated paths.
Submit the form and export the result as CSV or PDF.

Example Data Table

v0 km/s	g s^-1	z1 km	z2 km	Vertical time s	Average velocity km/s
3.00	0.08	0	5	1.5645	3.1958
4.00	0.05	2	12	2.3014	4.3452
2.50	0.00	0	8	3.2000	2.5000
5.00	-0.03	1	6	1.0215	4.8946

Understanding Gradient Travel Time

Seismology often starts with a simple speed model. A linear velocity gradient says wave speed changes steadily with depth. This is useful for quick checks. It also helps students connect geometry, calculus, and Earth structure. The model is not a full Earth model. Still, it gives practical insight before using larger inversion software.

Why The Gradient Matters

A positive gradient means deeper material is faster. A ray entering faster material bends away from the vertical. That bending changes travel time and horizontal reach. The vertical formula uses a logarithm. It comes from integrating slowness through depth. When the gradient is zero, the equation returns the constant velocity case. That makes the method easy to compare.

Useful Calculator Inputs

The main inputs are starting velocity, gradient, starting depth, ending depth, and offset. Depth and velocity must use matching units. This page uses kilometers and seconds. A ray parameter may be supplied. It describes the horizontal slowness of the ray. If it is left blank, the calculator estimates it from the requested offset when possible.

Interpreting The Output

The result includes start velocity, end velocity, vertical time, ray time, and average velocity. It also reports takeoff angles when a valid ray parameter exists. A warning appears when the requested offset is beyond the valid range for the selected endpoint. This usually means the ray would turn earlier, or the model needs a different endpoint.

Practical Use In Study

This calculation is valuable in class exercises. It shows why straight line distance is not always enough. The gradient controls curvature. The ray parameter controls angle. Together, they explain many first arrival patterns. Field teams can also use the estimate as a screening tool. It can reveal unreasonable picks, unit mistakes, or unrealistic gradient choices.

Limits And Good Practice

Use this calculator for learning and early planning. Real surveys may need layered, anisotropic, or laterally varying models. Near surface corrections can also matter. Always compare results with local geology and measured picks. Keep units consistent. Record assumptions with each exported result. Clear assumptions make later interpretation safer, faster, and easier for teams. Save each scenario name so future reviews match the exact calculation context. This reduces confusion during later reporting.

FAQs

What is a linear velocity gradient?

It is a model where seismic wave speed changes at a constant rate with depth. The equation is v(z) = v0 + gz. It is simple, but useful for teaching and early travel time checks.

Which units should I use?

Use kilometers for depth and offset. Use kilometers per second for velocity. Use kilometers per second per kilometer for the gradient. This gradient unit is also written as per second.

What happens when the gradient is zero?

The calculator switches to a constant velocity calculation. Vertical time becomes depth interval divided by velocity. If offset is supplied, it uses straight slant distance divided by velocity.

What is a ray parameter?

Ray parameter is horizontal slowness. It links velocity and ray angle. In this model, it helps calculate curved ray time, predicted offset, and takeoff angles at the selected depths.

Can the calculator solve ray parameter from offset?

Yes. Leave the ray parameter blank and enter a valid offset. The calculator uses a bisection search to estimate the ray parameter that matches the selected endpoint offset.

Why do I get an offset warning?

The chosen offset may be too large for the selected depth endpoint and gradient. The ray may turn before reaching that endpoint. Try a smaller offset, deeper endpoint, or different gradient.

When should I use the legs multiplier?

Use it for repeated paths. For example, a basic reflection estimate may use two legs, one downward and one upward. Keep assumptions clear when interpreting multiplied travel times.

Is this suitable for real survey design?

Use it for study, screening, and early planning. Real survey design may require layered models, anisotropy, lateral variation, statics, and calibration with measured first arrivals.